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User`s Guide to PARI / GP

1. 118 NFSLCMO door ls eed ee A 118 nfeltmul o Arosa dee Gea ek ae 118 nfeltmulmodpr 118 NPSL POW ua o Re AO BA Se 118 nfeltpowmodpr 118 nfeltreduce 119 nfeltreducemodpr 119 nfeltval hdr OS Bite tee 119 nffactor 80 108 119 122 nffactormod 119 nfgaloisapply 119 nfgaloisconj 109 120 NThermite naa ee OS BS 121 nfhermitemod 121 HTBILDEFT sek hie p ee Ee RI a 121 MENNL is Gos Gages Hak cee ee bee 121 nfhnfmod 440 se ak eke fe ae OA ee 121 FINIE fwd a wee 96 109 121 126 e oo eR eS Ae ee ad 122 nfisideal raid 122 o A hee PaO ai i 122 123 NETSUSOM PA ee ey Es 123 ni kermodprs s ry 6 ah ld 123 NEMO Ca Aes HR ee A EAA 118 nfmodprinit 118 123 nfnewprec 4 122 123 nfreducemodpr 119 NETOOUS yoo fe ace Mk eRe Bd ee ee be 123 NETOOCSOPL hei Dk ape Stee FE 123 Ni Smith ora ng aw he ee Sa ae A 124 MISO Ve hha ee Oe Are ee Ad A 124 nfsolvemodpr 124 nfsubfield cia dr p e p 109 nfsubfields 123 163 TO sci eet Sige te BOAR ae ok ENS 98 NOEM e Alok AN Ok ty gh Bo 65 normalize song AA 218 normalizepol 218 HOTMiLZ oh ls a o 65 TOC A e AA ih ie E 57 DUCOMP ars did ds 86 MUCUPIL cimas Bro an El aes 86 DUMDATV d y aeee e i aet A na a A 84 number field 26 NUMDPAT i ca a a o ad ae 84
2. K automorphisms of L K X R local polabs N H R Mod 1 nfK pol convert coeffs to polmod elts of K polabs rnfequation nfK R N nfgaloisconj polabs R Q automorphisms of L H for i 1 N select the ones that fix K if subst R variable R Mod N i R 0 120 H concat H N i K nfinit y 2 7 polL x 4 y x 3 3 x72 y x 1 rnfgaloisconj K polL K automorphisms of L The library syntax is galoisconjO nf flag d prec Also available are galoisconj nf for flag 0 galoisconj2 nf n prec for flag 2 where n is a bound on the number of conjugates and galoisconj4 nf d corresponding to flag 4 3 6 90 nfhilbert nf a b pr if pr is omitted compute the global Hilbert symbol a b in nf that is 1 if z ay bz has a non trivial solution x y z in nf and 1 otherwise Otherwise compute the local symbol modulo the prime ideal pr as output by idealprimedec The library syntax is nfhilbert nf a b pr where an omitted pr is coded as NULL 3 6 91 nfhnf nf x given a pseudo matrix A I finds a pseudo basis in Hermite normal form of the module it generates The library syntax is nfhermite nf x 3 6 92 nfhnfmod nf x detz given a pseudo matrix A J and an ideal detx which is contained in read integral multiple of the determinant of A J finds a pseudo basis in Hermite normal form of the module generated by A J This avoids coefficient
3. 137 intersect na ts eo a pie wy bod eee 146 intf rma l cda te A 136 At ptos ad ate betes 154 inverseimage 146 o O ee te ea 208 ANVUMO ds ii aa e 208 TScompleR waa ata hE ok 217 isdiagonal 146 PSEXACEZETO a ia cee i 57 216 isfundamental 82 side inmi ee Sa 122 TSMONOME ots Foe he ds a a 217 iepr ino seo ante ado Senay cha idad 82 83 isprincipalall 102 isprincipalrayall 107 ispseudoprime 82 83 84 241 TESQU TE ori Mae ge Se ea as 83 issquarefree 76 83 SUNET id A ert ache oy dete Need Coes 103 DEORE ss eye ee ie Eee te ya Se LE Geen te Be 0 210 ICOS 5 28 Ara il ae 179 202 210 J T teak fae el lng ates gle ot iether a 89 JACODE port Se A tale acy Mand odes 149 jbesselh o ooo ooo 71 JELNE ers Selah cg Gee AM ase Rae 93 K kbesselas ove 6 a eye hd da Ae 71 Kbessel2 ec koe iia ee eS 71 OT cae etre aie Nee ee Bog o dew te E 146 kerbras ete Sava Bw hie So BS ees 146 KOVINE gaip ian as a 146 keyword ai dph a 8 y a Gl a maa a bu 40 Fea Dry ire rs E lt ia 166 Kodalta nia Rie a E a aA 93 Kronecker symbol 83 kronecker iaa ote aoe ad 83 84 L Laplalte sien ek oe a 140 Telon ati a e kt eve BS 178 Lem 3 34 Sing Uae Be Vere 84 LCOpy eri if ita na ae ee ae iS 178 leadingcoeff 138 leave 2 2h aoe bk Bae a a a 8 leaves imita Saale ie ee ee as E 7 Legendre polynomi
4. 2 0 matsolvemod m 3 1 m X 1 1 over F_3 143 1 1 If flag 1 all solutions are returned in the form of a two component row vector x u where x is a small integer solution to the system of congruences and u is a matrix whose columns give a basis of the homogeneous system so that all solutions can be obtained by adding x to any linear combination of columns of u If no solution exists returns zero The library syntax is matsolvemod0O m d y flag Also available are gaussmodulo m d y flag 0 and gaussmodulo2 m d y flag 1 3 8 40 matsupplement assuming that the columns of the matrix x are linearly independent if they are not an error message is issued finds a square invertible matrix whose first columns are the columns of x i e supplement the columns of x to a basis of the whole space The library syntax is suppl z 3 8 41 mattranspose x or x transpose of x This has an effect only on vectors and matrices The library syntax is gtrans z1 148 3 8 42 qfgaussred q decomposition into squares of the quadratic form represented by the sym metric matrix g The result is a matrix whose diagonal entries are the coefficients of the squares and the non diagonal entries represent the bilinear forms More precisely if a denotes the output one has g x Y ale aiz j gt t The library syntax is sqred z 3 8 43 qfjacobi z x being a real symmetric matrix this gives a vec
5. 3 10 8 plotfile s set the output file for plotting output The special filename redirects to the same place as PARI output This is only taken into account by the gnuplot interface 3 10 9 ploth X a b expr flag 0 n 0 high precision plot of the function y f x represented by the expression expr x going from a to b This opens a specific window which is killed whenever you click on it and returns a four component vector giving the coordinates of the bounding box in the form amin emaz ymin ymaz Important note Since this may involve a lot of function calls it is advised to keep the current precision to a minimum e g 9 before calling this function n specifies the number of reference point on the graph 0 means use the hardwired default values that is 1000 for general plot 1500 for parametric plot and 15 for recursive plot If no flag is given expr is either a scalar expression f X in which case the plane curve y f X will be drawn or a vector f X f X and then all the curves y f X will be drawn in the same window The binary digits of flag mean e 1 Parametric parametric plot Here expr must be a vector with an even number of components Successive pairs are then understood as the parametric coordinates of a plane curve Each of these are then drawn For instance ploth X 0 2 Pi sin X cos X 1 will draw a circle ploth X 0 2 Pi sin X cos X will draw two entwined si
6. After the copyright the computer works for a few seconds it is in fact computing and storing a table of primes writes the top level help information some initial defaults and then waits after printing its prompt by default 7 Note that at any point the user can type Ctrl C that is press simultaneously the Control and C keys the current computation will be interrupted and control given back to the user at the GP prompt The top level help information tells you that as in many systems to get help you should type a When you do this and hit return a menu appears describing the eleven main categories of available functions and what to do to get more detailed help If you now type n with 1 lt n lt 11 you will get the list of commands corresponding to category n and simultaneously to Section 3 n of this manual If you type functionname where functionname is the name of a PARI function you will get a short explanation of this function If extended help see Section 2 2 1 is available on your system you can double or triple the sign to get much more respectively the complete description of the function e g sqrt or a list of GP functions relevant to your query e g 7 elliptic curve or quadratic field If GP was compiled with the right options see Appendix A a line editor will be available to correct the command line get automatic completions and so on See Section 2 11 1 for a short summary of av
7. If flag is non zero reduce the result using idealred The library syntax is idealmul nf x y flag 0 or idealmulred nf x y prec flag 4 0 where as usual prec is a C long integer representing the precision 3 6 56 idealnorm nf x computes the norm of the ideal x in the number field nf The library syntax is idealnorm nf x 3 6 57 idealpow nf x k flag 0 computes the k th power of the ideal x in the number field nf k can be positive negative or zero The result is NOT reduced it is really the k th ideal power and is given in HNF If flag is non zero reduce the result using idealred Note however that this is NOT the same as as idealpow nf x k followed by reduction since the reduction is performed throughout the powering process The library syntax corresponding to flag 0 is idealpow nf x k If k is a long you can use idealpows nf zx k Corresponding to flag 1 is idealpowred nf vp k prec where prec is a long 114 3 6 58 idealprimedec nf p computes the prime ideal decomposition of the prime number p in the number field nf p must be a positive prime number Note that the fact that p is prime is not checked so if a non prime number p is given it may lead to unpredictable results The result is a vector of 5 component vectors each representing one of the prime ideals above p in the number field nf The representation vp p a e f b of a prime ideal means the following The prime ideal is e
8. The library syntax is diagonal z 3 8 15 mateigen x gives the eigenvectors of x as columns of a matrix The library syntax is eigen z 3 8 16 mathess x Hessenberg form of the square matrix z The library syntax is hess z 3 8 17 mathilbert x x being a long creates the Hilbert matrix of order zx i e the matrix whose coefficient i j is 1 i j 1 The library syntax is mathilbert x 144 3 8 18 mathnf x flag 0 if x is a not necessarily square matrix finds the upper triangular Hermite normal form of x If the rank of x is equal to its number of rows the result is a square matrix In general the columns of the result form a basis of the lattice spanned by the columns of z If flag 0 uses the naive algorithm This should never be used if the dimension is at all large larger than 10 say It is recommanded to use either mathnfmod x matdetint x when x has maximal rank or mathnf x 1 Note that the latter is in general faster than mathnfmod and also provides a base change matrix If flag 1 uses Batut s algorithm which is much faster than the default Outputs a two component row vector H U where H is the upper triangular Hermite normal form of x defined as above and U is the unimodular transformation matrix such that zU 0 H U has in general huge coefficients in particular when the kernel is large If flag 3 uses Batut s algorithm but outputs H U P such that H and U are as
9. for1 lt i lt g we have 0 lt e lt oi Gigs Gn If present den must be a suitable value for gal 5 The library syntax is galoisinit gal den 3 6 36 galoisisabelian gal fl 0 gal being as output by galoisinit return 0 if gal is not an abelian group and the HNF matrix of gal over gal gen if fl 0 1 if fl 1 The library syntax is galoisisabelian gal fl where fl is a C long integer 110 3 6 37 galoispermtopol gal perm gal being a Galois field as output by galoisinit and perm a element of gal group return the polynomial defining the Galois automorphism as output by nfgaloisconj associated with the permutation perm of the roots gal roots perm can also be a vector or matrix in this case galoispermtopol is applied to all components recursively Note that G galoisinit pol galoispermtopol G G 6 is equivalent to nfgaloisconj pol if degree of pol is greater or equal to 2 The library syntax is galoispermtopol gal perm 3 6 38 galoissubcyclo N H fl 0 v computes the subextension of Q fixed by the subgroup H C Z nZ By the Kronecker Weber theorem all abelian number fields can be generated in this way uniquely if n is taken to be minimal The pair n H is deduced from the parameters N H as follows e N an integer then n N H is a generator i e an integer or an integer modulo n or a vector of generators e N the output of znstar n H as in the first case abov
10. 147 Path svi ee are ets Ya Sy sk Yad 19 Pailin tS saat a ae A id 117 POLE ea BAe toy ee OS ee et 150 Perl pots st eat 46 permtonum o 65 66 DDS ara nda e 79 Paba an dd 31 69 PEL tits nt o dass 157 PlobDOX suma o a a o ds 157 pPLOTCILIP pia ma e e ees 157 PLOLCOLOE 000 a is ee ee aS 157 PLOTCOPY wha saidi ee ee es 157 158 PLOEUESODO rroen Ste Deana ine 158 PLOtdFAaW ci a A RA 158 Pplotfile cocos ir Oe EO 158 DIOEDS 018 ak Lao a aR ates 54 158 pilotira Fennia a A 159 PLOGHSIZES e ve eee e po ee 159 pLotinit sow a arete MY ae eee 159 A kei aetd in Be Re aaa 160 plotlines 2 be ee a 1 160 plotlinetype 160 PLOCMOVE 2252 cits Re aa an eh ere as 160 plotpoint s 2 648 ea wie gn es 160 plotpointsize 22 La St e 160 plotpointtype 160 Plotrbox sorar ey ee ew e 160 Plotrecth a sb Salah etn Aas 159 161 plotrecthraw 161 PLlotrline cierres 161 PLOCEMOVE Cel pis wee Ds Ge aed Bd 161 PlOtrpoint moi vasa Rees 161 plotscale s worde pee ee 159 161 plotstring 42 161 plotterm cosa ee oe PS eed as 42 161 PO Ma hoe eo A a ue ee Bee 2 78 POTN CH ech a ele dd Bes 91 pointell aia 96 POMETE ti Oe ee 54 POINGET 42h wok a a A ee IN 54 Bol fo stein Goin Sy Att oak ae os ld as amp 59 POLCOCET ic eo a A 63 136 POLCOS LO os oct Ges Seb ir cs 137 polcompositum 124 polcompositumO 125 POUCY CIO Auber de tt dos
11. 17 LOGS a teeta Gane ie Goes SLE Hy Relies 53 IL ISOXPT eo soc 192 193 A his Seek ee 192 193 ELUSSOXPE macs ale 191 FLOOR cia a E A ee 64 POO a A ale a ete ae ee OA 53 POW oe te had sty ae a a e a ow A gies Geet 162 forcecopy 178 179 215 Ford a be pnd Sama van Sele eh Ged 117 LOLA V Seay hw cea Ge 2 eh SE EG ye 162 formal integration 136 FORM At o yet te Se ee a A 193 195 format 6x0 ad ach pbs ees Boge a es 17 TOrpripe ii ep en es ae 162 LOSE dics woe Mea ay A wed 162 forsubgroup 134 163 TOTVEC leelo dia ete ad Oe BS 163 FPM KT ite bs i es ae aoe a oe e 146 EPprIO STO one a BN kes 196 203 PLAC a ra ida ia ae 64 FreeBSD orto iaa a a eta 13 EU ok ee hae a Pe ee aa 98 fundamental units 88 98 99 PUNGUN TES A ias 88 TUBE AAA 98 G Ca tina o a Ai Ne A 70 Pac ica a A ent 70 PACOS co i cr Ad toa MO 70 gadd ibas ata a 54 gaddgs o ee 173 PAddGSZ is at a a as 173 paddgs z marcara 218 PADIS Omnis cS ahhh A psa E 173 PAdIS OZ few ad bal Baya ae See ad 173 gaddsg z 218 PaddZ nn tie na re Sine EAS 173 178 185 gadd z erase gay a ee eyes 208 218 Patfect haus bet AAs 177 178 217 Paths Mh ft ee ee eee ee 178 217 Galois 102 119 120 123 125 132 163 BAVOTS pida oh Pea a oe Rowe oS 126 galoisapply 120 galoiscony ve veg LY ht aS Peace 121 galoisconjO 0 121 galos con
12. See also the discrete summation methods below sharing the prefix sum 3 9 1 intnum X a b expr flag 0 numerical integration of expr smooth in Ja b with respect to X Set flag 0 or omit it altogether when a and b are not too large the function is smooth and can be evaluated exactly everywhere on the interval a b If flag 1 uses a general driver routine for doing numerical integration making no particular assumption slow flag 2 is tailored for being used when a or b are infinite One must have ab gt 0 and in fact if for example b 00 then it is preferable to have a as large as possible at least a gt 1 If flag 3 the function is allowed to be undefined but continuous at a or b for example the function sin x x at z 0 The library syntax is intnum void E GEN eval GEN void GEN a GEN b long flag long prec Where eval z E returns the value of the function at x You may store any additional information required by eval in F or set it to NULL 3 9 2 prod X a b expr x 1 product of expression expr initialized at x the formal parameter X going from a to b As for sum the main purpose of the initialization parameter x is to force the type of the operations being performed For example if it is set equal to the integer 1 operations will start being done exactly If it is set equal to the real 1 they will be done using real numbers having the default precision If it is set e
13. The arguments of the following functions are processed as keywords alias default first argument install all arguments but the last trap first argument type second argument whatnow 2 7 Errors and error recovery 2 7 1 Errors There are two kind of errors syntax errors and errors produced by functions in the PARI library Both kinds will be fatal to your computation GP will report the error perform some cleanup restore variables modified while evaluating the erroneous command close open files re claim unused memory etc and will output its usual prompt When reporting a syntax error GP tries to give meaningful context by copying the sentence it was trying to read whitespace and comments stripped out indicating an error with a little caret like in factor x 2 1 eK expected character instead of factor x 2 1 a 42 possibly enlarged to a full arrow given enough trailing context if siN x lt eps do_something xxx expected character instead of if siN x lt eps do_something GP error messages will often be mysterious because GP cannot guess what you were trying to do and the error usually occurs once GP has been sidetracked Let s have a look at the two messages above The first error is a missing parenthesis but from GP s point of view you might as well have intended to give further arguments to factor this is possible and often useful see the description of the
14. y The library syntax is nfmod nf x y 3 6 79 nfeltmul nf x y given two elements x and y in nf computes their product x x y in the number field nf The library syntax is element_mul nf x y 3 6 80 nfeltmulmodpr nf x y pr given two elements x and y in nf and pr a prime ideal in modpr format see nfmodprinit computes their product x x y modulo the prime ideal pr The library syntax is element_mulmodpr nf x y pr 118 3 6 81 nfeltpow nf x k given an element x in nf and a positive or negative integer k computes x in the number field nf The library syntax is element_pow nf x k 3 6 82 nfeltpowmodpr nf x k pr given an element x in nf an integer k and a prime ideal pr in modpr format see nfmodprinit computes x modulo the prime ideal pr The library syntax is element_powmodpr nf x k pr 3 6 83 nfeltreduce nf x ideal given an ideal in Hermite normal form and an element x of the number field nf finds an element r in nf such that x r belongs to the ideal and r is small The library syntax is element_reduce nf x ideal 3 6 84 nfeltreducemodpr nf x pr given an element x of the number field nf and a prime ideal pr in modpr format compute a canonical representative for the class of modulo pr The library syntax is nfreducemodpr nf x pr 3 6 85 nfeltval nf x pr given an element x in nf and a prime ideal pr in the format output by idealprimedec computes their the valu
15. 0 quotient x y of the two ideals x and y in the number field nf The result is given in HNF If flag is non zero the quotient x y is assumed to be an integral ideal This can be much faster when the norm of the quotient is small even though the norms of x and y are large The library syntax is idealdivO nf x y flag Also available are idealdiv nf x y flag 0 and idealdivexact nf x y flag 1 3 6 47 idealfactor nf x factors into prime ideal powers the ideal x in the number field nf The output format is similar to the factor function and the prime ideals are represented in the form output by the idealprimedec function i e as 5 element vectors The library syntax is idealfactor nf x 3 6 48 idealhnf nf a b gives the Hermite normal form matrix of the ideal a The ideal can be given in any form whatsoever typically by an algebraic number if it is principal by a Zx system of generators as a prime ideal as given by idealprimedec or by a Z basis If b is not omitted assume the ideal given was aZk bZ kx where a and b are elements of K given either as vectors on the integral basis nf 7 or as algebraic numbers The library syntax is idealhnf0 nf a b where an omitted b is coded as NULL Also available is idealhermite nf a b omitted 3 6 49 idealintersect nf x y intersection of the two ideals x and y in the number field nf When z and y are given by Z bases this does not depend on nf and can b
16. 2 instead The library syntax is matkerintO z flag Also available is kerint x flag 0 3 8 30 matmuldiagonal z d product of the matrix x by the diagonal matrix whose diagonal entries are those of the vector d Equivalent to but much faster than x matdiagonal d The library syntax is matmuldiagonal z d 146 3 8 31 matmultodiagonal x y product of the matrices x and y assuming that the result is a diagonal matrix Much faster than xx y in that case The result is undefined if x y is not diagonal The library syntax is matmultodiagonal z y 3 8 32 matpascal x q creates as a matrix the lower triangular Pascal triangle of order x 1 i e with binomial coefficients up to x If q is given compute the q Pascal triangle i e using g binomial coefficients The library syntax is matqpascal z q where x is a long and q NULL is used to omit q Also available is matpascal x 3 8 33 matrank x rank of the matrix z The library syntax is rank x and the result is a long 3 8 34 matrix m n X Y expr 0 creation of the m x n matrix whose coefficients are given by the expression expr There are two formal parameters in expr the first one X corresponding to the rows the second Y to the columns and X goes from 1 to m Y goes from 1 to n If one of the last 3 parameters is omitted fill the matrix with zeroes The library syntax is matrice GEN nlig GEN ncol entree el entree e2 char expr 3 8 35 ma
17. 3 1 10 shift x n flag 0 or z lt lt n x gt gt n shifts componentwise left by n bits if n gt 0 and right by n bits if n lt 0 A left shift by n corresponds to multiplication by 2 A right shift of an integer x by n corresponds to a Euclidean division of x by 2 with a remainder of the same sign as x hence is not the same when x lt 0 as x 2 If flag is non zero this behaviour is modified and right shift of a negative x is the same as x 2 which is consistent with 2 complement semantic of negative numbers The library syntax is gshift3 x n flag where n is a long Also available is gshift x n for the case flag 0 3 1 11 shiftmul x n multiplies x by 2 The difference with shift is that when n lt 0 ordinary division takes place hence for example if x is an integer the result may be a fraction while for shift Euclidean division takes place when n lt 0 hence if x is an integer the result is still an integer The library syntax is gmul2n x n where n is a long 3 1 12 Comparison and boolean operators The six standard comparison operators lt lt gt gt are available in GP and in library mode under the names gle glt gge ggt geq gne respectively The library syntax is co x y where co is the comparison operator The result is 1 as a GEN if the comparison is true O as a GEN if it is false For the purpose of comparison t_STR objects are strictly larger than any ot
18. As you can see in simple conditions the use of gerepile is not really difficult However make sure you understand exactly what has happened before you go on use the figure from the preceding section Important remark as we will see presently it is often necessary to do several gerepiles during a computation However the fewer the better The only condition for gerepile to work is that the garbage be connected If the computation can be arranged so that there is a minimal number of connected pieces of garbage then it should be done that way For example suppose we want to write a function of two GEN variables x and y which creates the vector x y y x Without garbage collecting one would write p1 gsqr x p2 gadd pl y p3 gsqr y p4 gadd p3 x z cgetg 3 t_VEC z 1 long p2 z 2 long p4 This leaves a dirty stack containing in this order z p4 p3 p2 p1 The garbage here consists of pi and p3 which are separated by p2 But if we compute p3 before p2 then the garbage becomes connected and we get the following program with garbage collecting ltop avma pl gsqr x p3 gsqr y lbot avma z cgetg 3 t_VEC z 1 ladd p1 y z 2 ladd p3 x z gerepile ltop lbot z Finishing by z gerepileupto ltop z would be ok as well But when you have the choice it is usually clearer to brace the garbage between 1top 1bot pairs Beware that ltop avma pl gadd gsqr x y p3
19. Finally an optional argument between braces followed by a star like z means that any number of such arguments possibly none can be given This is in particular used by the various print routines Flags A flag is an argument which rather than conveying actual information to the routine intructs it to change its default behaviour e g return more or less information All such flags are optional and will be called flag in the function descriptions to follow There are two different kind of flags e generic all valid values for the flag are individually described If flag is equal to 1 then e binary use customary binary notation as a compact way to represent many toggles with just one integer Let po Pn be a list of switches i e of properties which take either the value 0 or 1 the number 2 2 40 means that p3 and ps are set that is set to 1 and none of the others are that is they are set to 0 This is announced as The binary digits of flag mean 1 po 2 pi 4 p2 and so on using the available consecutive powers of 2 53 Mnemonics for flags Numeric flags as mentionned above are obscure error prone and quite rigid should the authors want to adopt a new flag numbering scheme for instance when noticing flags with the same meaning but different numeric values it would break backward compatibility The only advantage of explicit numeric values is that they are fast to type so their use is
20. The library syntax is ggcd z y For general polynomial inputs srgcd x y is also available For univariate rational polynomials one also has modularged z y 3 4 28 hilbert x y p Hilbert symbol of x and y modulo p If x and y are of type integer or fraction an explicit third parameter p must be supplied p 0 meaning the place at infinity Otherwise p needs not be given and x and y can be of compatible types integer fraction real integermod a prime result is undefined if the modulus is not prime or p adic The library syntax is hil z y p 3 4 29 isfundamental x true 1 if x is equal to 1 or to the discriminant of a quadratic field false 0 otherwise The library syntax is gisfundamental zx but the simpler function isfundamental x which returns a long should be used if x is known to be of type integer 82 3 4 30 isprime z flag 0 true 1 if x is a proven prime number false 0 otherwise This can be very slow when x is indeed prime and has more than 1000 digits say Use ispseudoprime to quickly check for pseudo primality If flag 0 use a combination of Baillie PSW pseudo primality test see ispseudoprime Selfridge p 1 test if 1 is smooth enough and Adleman Pomerance Rumely Cohen Lenstra APRCL for general x If flag 1 use Selfridge Pocklington Lehmer p 1 test and output a primality certificate as follows return 0 if x is composite 1 if x is small enough that pa
21. compatible 0 no backward compatibility In this mode a very handy function to be described in Section 3 11 2 27 is whatnow which tells you what has become of your favourite functions which GP suddenly can t seem to remember compatible 1 warn when using obsolete functions but otherwise accept them The output uses the new conventions though and there may be subtle incompatibilities between the behaviour of former and current functions even when they share the same name the current function is used in such cases of course We thought of this one as a transitory help for GP old timers Thus to encourage switching to compatible 0 it is not possible to disable the warning compatible 2 use only the old function naming scheme as used up to version 1 39 15 but taking case into account Thus I y 1 is not the same as i user variable unbound by default and you won t get an error message using i as a loop index as used to be the case compatible 3 try to mimic exactly the former behaviour This is not always possible when functions have changed in a fundamental way But these differences are usually for the better they were meant to anyway and will probably not be discovered by the casual user One adverse side effect is that any user functions and aliases that have been defined before changing compatible will get erased if this change modifies the function list i e if you move between groups 0 1 and 2 3
22. 114 idealprimedec 114 idealprincipal 115 A A ghd oP ie OOS tine 4 115 dealstar oca kee ek Ae a 115 idealstar0 nos otn 5 is Be Ss 115 ide altwo8e lt i 252 8 wee ra 116 idedlval wis ise hp a Os 116 ideal_two_eltO 116 delete rta a dl a ates INS Lo A ETETEA 96 ideleprincipal 116 TAM a ee aoe a A 145 TES A Lg ones Siege Gok ldots 163 imag i ih a Bhat a we ed os the 64 image sean a le ese A 145 imagecompl 146 imag i rd a o Patria dia amp 64 imprecise object 8 inegal aie at Geet ed lends goede Sets 72 73 AMC PAMO ss oe 4 Heal es Bi Bad ate Ss 73 incaminata i Be a ed tee e A 73 AN CPAMD Sd Avi thee Ae te A Ma Moh TE 73 AMOPAM GC go boo a Se we te Bath te Pet eet On ae 73 inclusive or 57 indefinite binary quadratic form 189 indexrank sie is She Sok ted 146 INAEXSOT Lic A 152 INP PVC a o Pa ee ees 195 infinite product 154 infinite sum 155 WAM eat e aoe Eee a 153 INES wo ore sheesh a Nites a 93 initz ta cyte da a A A 135 PULL Pee NE Se hs Sow ae Gees 192 input ii Bae oe awe S 166 install 42 46 166 193 195 204 intep an eee Bee ae Rn eS 136 integer swe Ya e EA Bd ee Se 7 25 186 integermod 7 25 188 integral basis 117 internal longword format 24 internal representation 24 interpolating polynomial
23. 2 2 forsubgroup H G 2 print H 1 1 1 2 2 1 1 O 1 1 The last one for instance is generated by 91 91 g2 This routine is intended to treat huge groups when subgrouplist is not an option due to the sheer size of the output For maximal speed the subgroups have been left as produced by the algorithm To print them in canonical form as left divisors of G in HNF form one can for instance use G matdiagonal 2 2 forsubgroup H G 2 print mathnf concat G H 2 1 0 1 1 0 0 2 2 0 0 1 1 0 0 1 Note that in this last representation the index G H is given by the determinant See galois subcyclo and galoisfixedfield for nfsubfields applications to Galois theory Warning the present implementation cannot treat a group G if one of its p Sylow subgroups has a cyclic factor has more than 2 resp 2 elements on a 32 bit resp 64 bit architecture 3 11 1 7 forvec X v seq flag 0 v being an n component vector where n is arbitrary of two component vectors a b for 1 lt i lt n the seq is evaluated with the formal variable X 1 going from a to by X n going from a to bn The formal variable with the highest index moves the fastest If flag 1 generate only nondecreasing vectors X and if flag 2 generate only strictly increasing vectors X 163 3 11 1 8 if a seq1 seq2 if a is non zero the expression sequence seq1 is evaluated otherwise the expressi
24. 3 7 15 pollead z v leading coefficient of the polynomial or power series x This is computed with respect to the main variable of x if v is omitted with respect to the variable v otherwise The library syntax is pollead z v where v is a long and an omitted v is coded as 1 Also available is leadingcoeff x 3 7 16 pollegendre n v x creates the nt Legendre polynomial in variable v The library syntax is legendre n where x is a long 3 7 17 polrecip pol reciprocal polynomial of pol i e the coefficients are in reverse order pol must be a polynomial The library syntax is polrecip z 3 7 18 polresultant z y vu flag 0 resultant of the two polynomials x and y with exact entries with respect to the main variables of x and y if v is omitted with respect to the variable v otherwise The algorithm used is the subresultant algorithm by default If flag 1 uses the determinant of Sylvester s matrix instead here x and y may have non exact coefficients If flag 2 uses Ducos s modified subresultant algorithm It should be much faster than the default if the coefficient ring is complicated e g multivariate polynomials or huge coefficients and slightly slower otherwise The library syntax is polresultant0 z y v flag where v is a long and an omitted v is coded as 1 Also available are subres z y flag 0 and resultant2 x y flag 1 3 7 19 polroots pol flag 0 complex roots o
25. 3 8 1 algdep z k flag 0 x being real complex or p adic finds a polynomial of degree at most k with integer coefficients having x as approximate root Note that the polynomial which is obtained is not necessarily the correct one it s not even guaranteed to be irreducible One can check the closeness either by a polynomial evaluation or substitution or by computing the roots of the polynomial given by algdep If x is p adic flag is meaningless and the algorithm LLL reduces the dual lattice correspond ing to the powers of x Otherwise if flag is zero the algorithm used is a variant of the LLL algorithm due to Hastad Lagarias and Schnorr STACS 1986 If the precision is too low the routine may enter an infinite loop If flag is non zero use a standard LLL flag then indicates a precision which should be between 0 5 and 1 0 times the number of decimal digits to which x was computed The library syntax is algdepO z k flag prec where k and flag are longs Also available is algdep z k prec flag 0 141 3 8 2 charpoly A v x flag 0 characteristic polynomial of A with respect to the variable v i e determinant of v x J A if A is a square matrix determinant of the map multiplication by A if A is a scalar in particular a polmod e g charpoly 1 x x 2 1 Note that in the latter case the minimal polynomial can be obtained as minpoly A local y y charpoly A y gcdly
26. 3 8 6 listinsert list x n inserts the object x at position n in list which must be of type t_LIST All the remaining elements of list from position n 1 onwards are shifted to the right This and listput are the only commands which enable you to increase a list s effective length as long as it remains under the maximal length specified at the time of the listcreate This function is useless in library mode 3 8 7 listkill list kill list This deletes all elements from list and sets its effective length to 0 The maximal length is not affected This function is useless in library mode 3 8 8 listput list x n sets the n th element of the list list which must be of type t_LIST equal to x If n is omitted or greater than the list current effective length just appends x This and listinsert are the only commands which enable you to increase a list s effective length as long as it remains under the maximal length specified at the time of the listcreate If you want to put an element into an occupied cell i e if you don t want to change the effective length you can consider the list as a vector and use the usual list n x construct This function is useless in library mode 143 3 8 9 listsort list flag 0 sorts list which must be of type t_LIST in place If flag is non zero suppresses all repeated coefficients This is much faster than the vecsort command since no copy has to be made This function is
27. C _ will undo your last changes incrementally M r undoes all changes made to the current line C t and M t will transpose the character word preceding the cursor and the one under the cursor Keeping the M key down while you enter an integer a minus sign meaning reverse behaviour gives an argument to your next readline command for instance M C k will kill text back to the start of line If you prefer Vi style editing M C j will toggle you to Vi mode Of course you can change all these default bindings For that you need to create a file named inputrc in your home directory For instance notice the embedding conditional in case you would want specific bindings for GP if Pari GP set show all if ambiguous 50 C h backward delete char e C h backward kill word C xd dump functions Ce N 7 C b can be annoying when copy pasting Le C v C b endif C x C r will re read this init file incorporating any changes made to it during the current session Note By default and are bound to the function pari matched insert which if electric parentheses are enabled default off will automatically insert the matching closure respectively and This behaviour can be toggled on and off by giving the numeric argument 2 to M 2 which is useful if you want e g to copy paste some text into the calculator If you don t want a toggle you can use M 0 M 1 to specifically switch it on or o
28. S 24 1 1 Aa x Co 24 1 2 S 24 1 1 G3 36 1 1 G 36 1 1 S4 x C2 48 1 1 As PSLa 5 60 1 1 Gro 72 1 1 Ss PGL2 5 120 1 1 Ag 360 1 1 Ss 720 1 1 In degree T C7 7 1 1 Dz 14 1 1 Mo 21 1 1 Maz 42 1 1 PSLa 7 PSL3 2 168 1 1 Ay 2520 1 1 S7 5040 1 1 This is deprecated and obsolete but for reasons of backward compatibility we cannot change this behaviour yet So you can use the default new_galois_format to switch to a consistent naming scheme namely k is always the standard numbering of the group among all transitive subgroups of Sn If this default is in effect the above groups will be coded as In degree 1 S 1 1 1 In degree 2 S2 2 1 1 125 In degree 3 43 C3 3 1 1 S3 6 1 2 In degree 4 Cy 4 1 1 Va 4 1 2 Da 8 1 3 Ag 12 1 4 S4 24 1 5 In degree 5 Cs 5 1 1 Ds 10 1 2 Moo 20 1 3 As 60 1 4 S5 120 1 5 In degree 6 Ce 6 1 1 S3 6 1 2 Dg 12 1 3 Aa 12 1 4 Gig 18 1 5 Ay x C2 24 1 6 S 24 1 7 SI 24 1 8 G3 36 1 9 Gi 36 1 10 S4 x Cy 48 1 11 As PSL2 5 60 1 12 G72 72 1 13 Ss PGLa 5 120 1 14 Ag 360 1 15 S 720 1 16 In degree 7 C7 7 1 1 Dz 14 1 2 Mo 21 1 3 Mas 42 1 4 PS
29. Subgroups are given as HNF left divisors of the SNF matrix corresponding to G Warning the present implementation cannot treat a group G where any cyclic factor has more than 231 resp 2 elements on a 32 bit resp 64 bit architecture forsubgroup is a bit more general and can handle G if all p Sylow subgroups of G satisfy the condition above If flag 0 default and bnr is as output by bnrinit 1 gives only the subgroups whose modulus is the conductor Otherwise the modulus is not taken into account If bound is present and is a positive integer restrict the output to subgroups of index less than bound If bound is a vector containing a single positive integer B then only subgroups of index exactly equal to B are computed For instance subgrouplist 6 2 1 6 0 0 2 2 0 0 2 6 3 0 1 2 1 0 1 3 0 0 2 1 0 0 2 6 O O 1 2 0 O 1 3 0 O 1 1 O O 1 subgrouplist 6 2 3 index less than 3 12 2 1 0 1 1 0 O 2 2 0 O 1 3 0 O 1 1 0 O 1 subgrouplist 6 2 3 Ml index 3 43 3 0 O 1 bnr bnrinit bnfinit x 120 1 1 L subgrouplist bnr 8 In the last example L corresponds to the 24 subfields of Q C120 of degree 8 and conductor 12000 by setting flag we see there are a total of 43 subgroups of degree 8 134 vector L i galoissubcyclo bnr L i will produce their equations For a general base fiel
30. Warning this routine only works for subgroups of prime index It uses Kummer theory adjoining necessary roots of unity it needs to compute a tough bnfinit here and finds a generator via Hecke s characterization of ramification in Kummer extensions of prime degree If your extension does not have prime degree for the time being you have to split it by hand as a tower compositum of such extensions The library syntax is rnfkummer bnr subgroup deg prec where deg is a long and an omit ted subgroup is coded as NULL 3 6 139 rnflllgram nf pol order given a polynomial pol with coefficients in nf defining a relative extension L and a suborder order of L of maximal rank as output by rnfpseudobasis nf pol or similar gives neworder U where neworder is a reduced order and U is the unimodular transformation matrix The library syntax is rnflllgram nf pol order prec 3 6 140 rnfnormgroup bnr pol bnr being a big ray class field as output by bnrinit and pol a relative polynomial defining an Abelian extension computes the norm group alias Artin or Takagi group corresponding to the Abelian extension of bnf bnr 1 defined by pol where the module corresponding to bnr is assumed to be a multiple of the conductor i e pol defines a subextension of bnr The result is the HNF defining the norm group on the given generators of bnr 5 3 Note that neither the fact that pol defines an Abelian extension nor the fact that the modul
31. a3 flag 0 al a2 a3 defining a big ray number field L over a ground field K see bnr at the beginning of this section for the meaning of al a2 a3 outputs a 3 component row vector N R D where N is the absolute degree of L R the number of real places of L and D the discriminant of L Q including sign if flag 0 If flag 1 as above but outputs relative data N is now the degree of L K R is the number of real places of K unramified in L so that the number of real places of L is equal to Ri times the relative degree N and D is the relative discriminant ideal of L K If flag 2 does as in case 0 except that if the modulus is not the exact conductor corre sponding to the L no data is computed and the result is 0 gzero If flag 3 as case 2 outputting relative data The library syntax is bnrdiscO a1 a2 a3 flag 3 6 24 bnrdisclist bnf bound arch flag 0 bnf being a big number field as output by bnfinit the units are mandatory computes a list of discriminants of Abelian extensions of the number field by increasing modulus norm up to bound bound where the ramified Archimedean places are given by arch unramified at infinity if arch is void or omitted If flag is non zero give arch all the possible values See bnr at the beginning of this section for the meaning of al a2 a3 The alternative syntax bnrdisclist bnf list is supported where list is as output by ideal list or ideallistarch
32. called boldfg is provided if you find the former too pale In the present version this default is incompatible with Emacs Changing it will just fail silently the alternative would be to display escape sequences as is since Emacs will refuse to interpret them On the other hand you can customize highlighting in your emacs so as to mimic exactly this behaviour See emacs pariemacs txt If you use an old readline library version number less than 2 0 you should do as in the example above and leave a3 and a4 prompt and input line strictly alone Since old versions of readline did not handle escape characters correctly or more accurately treated them in the only sensible way since they did not care to check all your terminal capabilities it just ignored them changing them would result in many annoying display bugs The hacker s way to check if this is the case would be to look in the readline h include file wherever your readline include files are for the string RL_PROMPT_START_IGNORE If it s there you are safe A more sensible way is to make some experiments and get a more recent readline if yours doesn t work the way you would like it to See the file misc gprc dft for some examples 16 2 1 3 compatible default 0 The GP function names and syntax have changed tremendously between versions 1 xx and 2 00 To help you cope with this we provide some kind of backward compatibility depending on the value of this default
33. default realprecision P series precision default seriesprecision global variable precd1 for the library 202 e Return type GEN by default otherwise the following can appear at the start of the code string i return int 1 return long Vv return void No more than 8 arguments can be given syntax requirements and return types are not con sidered as arguments This is currently hardcoded but can trivially be changed by modifying the definition of argvec in anal c identifier This limitation should disappear in future versions When the function is called under GP the prototype is scanned and each time an atom corresponding to a mandatory argument is met a user given argument is read GP outputs an error message it the argument was missing Each time an optional atom is met a default value is inserted if the user omits the argument The automatic atoms fill in the argument list transparently supplying the current value of the corresponding variable or a dummy pointer For instance here is how you would code the following prototypes which do not involve default values GEN name GEN x GEN y long prec gt GGp void name GEN x GEN y long prec gt vGGp void name GEN x long y long prec gt vGLp long name GEN x gt 1G int name long x gt GL If you want more examples GP gives you easy access to the parser codes associated to all GP functions just type h function You can then c
34. flag 1 and llgramkerim x flag 4 3 8 46 qfminim z b m flag 0 x being a square and symmetric matrix representing a posi tive definite quadratic form this function deals with the minimal vectors of x depending on flag If flag 0 default seeks vectors of square norm less than or equal to b for the norm defined by x and at most 2m of these vectors The result is a three component vector the first component being the number of vectors the second being the maximum norm found and the last vector is a matrix whose columns are the vectors found only one being given for each pair v at most m such pairs If flag 1 ignores m and returns the first vector whose norm is less than b In both these cases x is assumed to have integral entries and the function searches for the minimal non zero vectors whenever b 0 If flag 2 x can have non integral real entries but b 0 is now meaningless uses Fincke Pohst algorithm The library syntax is qfminim0 z b m flag prec also available are minim x b m flag 0 minim2 x b m flag 1 and finally fincke_pohst b m prec flag 2 3 8 47 qfperfection x x being a square and symmetric matrix with integer entries representing a positive definite quadratic form outputs the perfection rank of the form That is gives the rank of the family of the s symmetric matrices v vt where s is half the number of minimal vectors and the v 1 lt i lt s are the mini
35. gadd gsqr y x z cgetg 3 t_VEC z 1 long p1 z 2 long p3 183 z gerepileupto ltop z WRONG would be a disaster since p1 and p3 would be created before z so the call to gerepileupto would overwrite them leaving z 1 and z 2 pointing at random data We next want to write a program to compute the product of two complex numbers x and y using the 3M method which takes only 3 multiplications instead of 4 Let z x y and set x ixx and similarly for y and z We compute p r Yr p2 Li Yj P3 Lp X Yr ty and then we have zr pi pa zi p3 pi p2 The program is essentially as follows ltop avma pi gmul x 1 y 1 p2 gmul x 21 y 21 p3 gmul gadd x 1 x 2 gadd y 1 y 2 p4 gadd p1 p2 lbot avma z cgetg 3 t_COMPLEX z 1 1sub p1 p2 z 2 1sub p3 p4 z gerepile ltop lbot z Essentially because for instance x 1 is a long and not a GEN so we need to insert many annoying typecasts p1 gmul GEN x 1 GEN y 1 and so on Let us now look at a less trivial example where more than one gerepile is needed in practice At the expense of efficiency one can always use only one using gerepilecopy Suppose that we want to write a function which multiplies a line vector by a matrix Such a function is of course already part of gmul but let us ignore this for a moment Then the most natural way is to do a cgetg of the result imme
36. ideal flag 0 bnf being a big number field as output by bnfinit the units are mandatory unless the ideal is trivial and ideal being either an ideal in any form or a two component row vector containing an ideal and an r component row vector of flags indicating which real Archimedean embeddings to take in the module computes the ray class group of the number field for the module ideal as a 3 component vector as all other finite Abelian groups cardinality vector of cyclic components corresponding generators If flag 2 the output is different It is a 6 component vector w w 1 is bnf w 2 is the result of applying idealstar bnf J 2 w 3 w 4 and w 6 are technical components used only by the function bnrisprincipal wl5 is the structure of the ray class group as would have been output with flag 0 If flag 1 as above except that the generators of the ray class group are not computed which saves time The library syntax is bnrclass0 bnf ideal flag 3 6 19 bnrclassno bnf 1 bnf being a big number field as output by bnfinit units are manda tory unless the ideal is trivial and J being either an ideal in any form or a two component row vec tor containing an ideal and an r component row vector of flags indicating which real Archimedean embeddings to take in the modulus computes the ray class number of the number field for the modulus This is faster than bnrclass and should be used if only the ray class number is
37. mpent zi wale howe die eee wie a 211 Mpeuler eones eet 69 231 MpLACt rpa eke aa e 4 81 DPT AOE ai ke ee o aed odes ES 81 pnr ora ii tweed Woe Ss 211 MPAIOVI Tio e A 212 MPIOIVS E ii a e a 211 MPIADVZ aia oy a a ge es 211 1 Los A e te Mee ck a Be NS 187 MPPE e 2 eek So ee io Set eth e 69 231 MPE QS olas BA Gost a Bos 76 81 mpshift 2 Loira eee A has 211 mptrunie z uc Ros dehy Bias poh Pe ee A 211 MSQ TIMER coco io o Wein a ce sede Begs 196 MSgtimer a wake ey A es 196 ME ne ge ten Heed Gite Bo 84 MV oi ie pea Oe Redes a A 207 MUSA s dak whe cated oe ce ke OE Ged 215 multivariate polynomial 38 muluumod 00 208 N mame_var 004 192 Nbessel uti ee Foe eR ees 71 newtonpoly 40 116 news chunk cid a 175 243 new_galois_format 17 18 125 126 NOX foe wh Pee Ba ees 44 164 NOXEPTAIMS sos be EE ems she ap de ed Bek 84 NEXT_PRIME_VIADIFF 232 NEXT_PRIME_VIADIFF_CHECK p ptr 232 MU da E 96 Di dr a o tio rt EE 39 98 nfalgtobasiS 117 PEDAS S come GM erie Deak ES 117 122 NibasisO vas a kody eee a 117 nfbasistoalg 117 Nidetint s ohare HA ie ie Ge a eoa 117 NEAISC ito Se Se ee e AA 117 NFATSCLO es dg hols ee a ee ae aS 117 NEAIVEUC 6k le ee ee el eee ee 118 NEGATIVES sanl ee ee se Ys 118 telti fo ce boc n Bee PRO A N a 118 nfeltdiveuc 118 nfeltdivmodpr ey oa we eee a ee eek 118 nfeltdivrem
38. of x x must be a power series whose valuation is exactly equal to one The library syntax is recip z 3 7 31 subst z y z replace the simple variable y by the argument z in the polynomial ex pression x Every type is allowed for x but if it is not a genuine polynomial or power series or rational function the substitution will be done as if the scalar components were polynomials of degree zero In particular beware that subst 1 x 1 2 3 4 1 1 0 0 1 subst 1 x Mat 0 1 eK forbidden substitution by a non square matrix If x is a power series z must be either a polynomial a power series or a rational function The variable y is in fact allowed to be any polynomial in which case the substitution is done as per the following script subst_poly pol from to local t subst_poly_t M from t subst lift Mod pol M variable M t to For instance subst x 4 x72 1 x72 y 41 y 2 ytil subst x 4 x72 1 x73 y 2 x 2 y x 1 subst x 4 x72 1 x 1 72 y 13 4 y 6 x y 2 3 y 3 The library syntax is gsubst v z where v is the number of the variable y for regular usage Also available is gsubst0 x y z where y is a GEN polynomial 140 3 7 32 taylor x y Taylor expansion around 0 of x with respect to the simple variable y x can be of any reasonable type for example a rational function The number of terms of the expa
39. precision loss in truncation e g when trying to convert 1E1000 known to 28 digits of accuracy to an integer or division by 0 e g inverting 0E1000 when all accuracy has been lost and no significant digit remains It would be enough to restart part of the computation at a slightly higher precision We now describe error trapping a useful mechanism which alleviates much of the pain in the first situation and provides a satisfactory way out of the second one Everything is handled via the trap function whose different modes we now describe 2 7 3 Break loop A break loop is a special debugging mode that you enter whenever an error occurs freezing the GP state and preventing cleanup until you get out of the loop Any error syntax error library error user error from error even user interrupts like C c Control C When a break loop starts a prompt is issued break gt You can type in a GP command which is evaluated when you hit the lt Return gt key and the result is printed as during the main GP loop except that no history of results is kept Then the break loop prompt reappears and you can type further commands as long as you do not exit the loop If you are using readline the history of commands is kept and line editing is available as usual If you type in a command that results in an error you are sent back to the break loop prompt errors does not terminate the loop To get out of a break loop you can use next bre
40. t This is a forbidden assignment in PARI so an error message is issued 210 void affrr GEN x GEN z assigns the real x into the real z GEN itor GEN x long prec assigns the t_INT x into a t_REAL of length prec and return the latter long itos GEN x converts the PARI integer x to a C long if possible otherwise an error message is issued GEN stoi long s creates the PARI integer corresponding to the long s GEN stor long s long prec assigns the long s into a t_REAL of length prec and return the latter GEN mptrunc z GEN x GEN z truncates the integer or real x not the same as the integer part if x is non integer and negative GEN mpent z GEN x GEN z true integer part of the integer or real x i e the floor function 5 2 4 Valuation and shift long vals long s 2 adic valuation of the long s Returns 1 if s is equal to 0 with no error long vali GEN x 2 adic valuation of the integer x Returns 1 if s is equal to 0 with no error GEN mpshift z GEN x long n GEN z shifts the real or integer x by n If n is positive this is a left shift i e multiplication by 2 If n is negative it is a right shift by n which amounts to the truncation of the quotient of x by 27 GEN shifts long s long n converts the long s into a PARI integer and shifts the value by n GEN shifti GEN x long n shifts the integer x by n GEN shiftr GEN x long n shifts the real x by n 5 2 5 Unary operations L
41. void pari_init long size ulong maxprime is called this table is initialized with the successive differences of primes up to just a little beyond maxprime see Section 4 1 The prime table will occupy roughly maxprime log maxprime bytes in memory so be sensible when choosing maxprime it is 500000 by default under gp In any case the implementation requires that maxprime lt 4294965248 resp 18446744073709549568 on 32 bit resp 64 bit machines whatever memory is available The largest prime computable using this table is available as the output of ulong maxprime 231 After the following initializations the names p and ptr are arbitrary of course byteptr ptr diffptr ulong p 0 calling the macro NEXT_PRIME_VIADIFF_CHECK p ptr repeatedly will assign the successive prime numbers to p Overrunning the prime table boundary will raise the error primer1 which will just print the error message not enough precomputed primes and then abort the computations The alternative macro NEXT_PRIME_VIADIFF operates in the same way but will omit that check and is slightly faster It should be used in the following way byteptr ptr diffptr ulong p 0 if maxprime lt goal err primer1 not enough primes while p lt goal run through all primes up to goal NEXT_PRIME_VIADIFF p ptr Here we use the general error handling function err see Section 4 7 3 with the codeword primer1 raisi
42. we would have liked to define a type GEN to be a pointer to itself This unfortu nately is not possible in C except by using structures but then the names become unwieldy The result of this is that when we use a component of a PARI object it will be a long hence will need to be typecast to a GEN again if we want to avoid warnings from the compiler This will sometimes be quite tedious but of course is trivially done See the discussion on typecasts in the next section After declaring the use of the file pari h the first executable statement of a main program should be to initialize the PARI system and in particular the PARI stack which will be both a scratchboard and a repository for computed objects This is done with a call to the function void pari_init long size ulong maxprime The first argument is the number of bytes given to PARI to work with it should not reasonably be taken below 500000 and the second is the upper limit on a precomputed prime number table If you do not want prime numbers just put maxprime 2 Be careful because lots a PARI functions need this table certainly all the ones of interest to number theorists If you wind up with the error message not enough precomputed primes try to increase this value We have now at our disposal e a large PARI stack containing nothing It is a big connected chunk of memory whose size you chose when invoking pari_init All your computations are going to take place h
43. while i lt a i 1 time 4 478 ms i 0 while i lt a i time 3 639 ms For the same reason the shift operators should be preferred to multiplication a 1 lt lt 20000 i 1 while i lt a i i 2 time 5 255 ms i 1 while i lt a i lt lt 1 time 988 ms 2 5 The general GP input line 2 5 1 Generalities User interaction with a GP session proceeds as follows a sequence of charac ters is typed by the user at the GP prompt This can be either a command a function definition an expression or a sequence of expressions i e a program In the latter two cases after the last expression has been computed its result is put into an internal history array and printed The successive elements of this array are called 1 2 As a shortcut the latest computed expression can also be called the previous one the one before that and so on If you want to suppress the printing of the result for example because it is a long unimportant intermediate result end the expression with a sign This same sign is used as an instruction separator when several instructions are written on the same line note that for the pleasure of BASIC addicts the sign can also be used but we will try to stick to C style conventions in this manual The final expression computed even if not printed will still be assigned to the history array so you may have to pay close attention when you intend to refer
44. with units The output format is as follows The output v is a row vector of row vectors allowing the bound to be greater than 21 for 32 bit machines and v i j is understood to be in fact V 2 i 1 j of a unique big vector V note that 21 is hardwired and can be increased in the source code only on 64 bit machines and higher Such a component Vk is itself a vector W maybe of length 0 whose components correspond to each possible ideal of norm k Each component W i corresponds to an Abelian extension L of bnf whose modulus is an ideal of norm k and no Archimedean components hence the extension is 106 unramified at infinity The extension Wi is represented by a 4 component row vector m d r D with the following meaning m is the prime ideal factorization of the modulus d L Q is the absolute degree of L r is the number of real places of L and D is the factorization of the absolute discriminant Each prime ideal pr p a e f 3 in the prime factorization m is coded as p n f 1 n j 1 where n is the degree of the base field and j is such that pr idealprimedec nf p j m can be decoded using bnfdecodemodule The library syntax is bnrdisclistO al a2 a3 bound arch flag 3 6 25 bnrinit bnf ideal flag 0 bnf is as output by bnfinit ideal is a valid ideal or a module initializes data linked to the ray class group structure corresponding to this module This is the same as bnrclass
45. y 3 8 26 matinverseimage z y gives a column vector belonging to the inverse image of the column vector y by the matrix x if one exists the empty vector otherwise To get the complete inverse image it suffices to add to the result any element of the kernel of x obtained for example by matker The library syntax is inverseimage z y 3 8 27 matisdiagonal x returns true 1 if x is a diagonal matrix false 0 if not The library syntax is isdiagonal x and this returns a long integer 3 8 28 matker z flag 0 gives a basis for the kernel of the matrix x as columns of a matrix A priori the matrix can have entries of any type If x is known to have integral entries set flag 1 Note The library function FpM_ker z p where x has integer entries reduced mod p and p is prime is equivalent to but orders of magnitude faster than matker x Mod 1 p and needs much less stack space To use it under GP type install FpM_ker GG first The library syntax is matker0 x flag Also available are ker x flag 0 keri x flag 1 and FpM_ker z p 3 8 29 matkerint z flag 0 gives an LLL reduced Z basis for the lattice equal to the kernel of the matrix x as columns of the matrix x with integer entries rational entries are not permitted If flag 0 uses a modified integer LLL algorithm If flag 1 uses matrixqz x 2 If LLL reduction of the final result is not desired you can save time using matrixqz matker x
46. you should use the function void switchout char name where name is a character string giving the name of the file you are going to use The output will be appended at the end of the file In order to close the file simply call switchout NULL Similarly errors are sent to the stream errfile stderr by default and input is done on the stream infile which you can change using the function switchin which is analogous to switchout Advanced Remark All output is done according to the values of the pariOut pariErr global variables which are pointers to structs of pointer to functions If you really intend to use these this probably means you are rewriting GP In that case have a look at the code in language es c init80 or GENtostr for instance 4 7 3 Errors If you want your functions to issue error messages you can use the general error handling routine err The basic syntax is err talker error message This will print the corresponding error message and exit the program in library mode go back to the GP prompt otherwise You can also use it in the more versatile guise err talker format where format describes the format to use to write the remaining operands as in the printf function however see the next section The simple syntax above is just a special case with a constant format and no remaining arguments The general syntax is void err numerr where numerr is a codeword which indicates wha
47. 0 0 0000 88 246 quadregulator 88 Quadunit fa ea Gass ea et oe AS 88 Quit sk Hak als fee a ee 24 167 QUODE gt fee te e he eat 4 167 d otient si ek ay od Sees as Ee Bee 55 R TaGine mima aa a E a AG 88 tandom te mia a a e the heh ane e 66 Fank oh iat ese be a a eet 147 rational function 7 27 189 rational number 7 25 188 TOW JORNAL pao We peck eee Sa E A 18 rayclassno 00 105 rayclassnolist 105 POAC st he the at te ta ee a h 24 Heads soe Seek wha ks amp Adee 21 42 168 169 readline s scr 6 8 SM wet eae ae gues 50 readline s oe eso e aeck a a ed a a ena 20 real number 7 25 187 Teal o gg se O 66 realprecision 20 24 TOA ZO eur ene cared siren eee eee tie E ea 199 real i iid ten Ra PE ea seas 2 a aa 66 FOCI anA ee vee Goby yd ad Be be 140 recursion depth 38 FECUESION hikes sod te e a 38 recursive plot 0 eee ee 158 recursiveness 200 6 redimag 2 eee a ee 86 Leda ito an ca o e Sacer e 86 redrealnod occ o Sions Toana a 86 reduceddiscsmith 137 TECUCCI N mara ha E A k 85 86 reference Card o o 22 TOG ia Ske yh Ae eS o bn 98 A eS en ee 88 HESULACOK e eat sh a pcan 104 removeprimes 88 reorder s awia sae Aan wee 34 63 168 resultant o h N ol esha ees 138 TON te ehh He OMe Ee Fes 44 164 ThOKEAIY 3 Ay ent dee in Se ea e od 86 TOCAN oi e
48. 0 use Doud s algorithm bound torsion by computing F for small primes of good reduction then look for torsion points using Weierstrass parametrization and Mazur s classification If flag 1 use Lutz Nagell much slower E is allowed to be a medium vector The library syntax is elltors0 E flag 3 5 28 ellwp E z x flag 0 Computes the value at z of the Weierstrass p function attached to the elliptic curve E as given by ellinit alternatively E can be given as a lattice w1 w2 If z is omitted or is a simple variable computes the power series expansion in z starting z7 O 2 The number of terms to an even power in the expansion is the default serieslength in GP and the second argument C long integer in library mode Optional flag is for now only taken into account when z is numeric and means 0 compute only p z 1 compute p 2 p z The library syntax is ellwp0 E z flag prec precdl Also available is weipell E precdl for the power series in x po1x 0 3 5 29 ellzeta E z value of the Weierstrass function of the lattice associated to E as given by ellinit alternatively E can be given as a lattice w1 w2 The library syntax is ellzeta E z 95 3 5 30 ellztopoint z E being a long vector computes the coordinates x y on the curve E corresponding to the complex number z Hence this is the inverse function of ellpointtoz In other words if the curve is put in
49. 1 29 strictmatch default 1 this is a toggle which can be either 1 on or 0 off If on unused characters after a sequence has been processed will produce an error Otherwise just a warning is printed This can be useful when you re not sure how many parentheses you have to close after complicated nested loops 2 1 30 TeXstyle default 0 the bits of this default allow GP to use less rigid TeX formatting commands in the logfile This default is only taken into account when log 3 The bits of TeXstyle have the following meaning 1 use frac instead of over for fractions 2 insert right left pairs where appropriate 4 insert discretionary breaks in polynomials to enhance the probability of a good line break 2 1 31 timer default 0 this is a toggle which can be either 1 on or 0 off If on every instruction sequence anything ended by a newline in your input is timed to some accuracy depending on the hardware and operating system The time measured is the user CPU time not including the time for printing the results see and 2 1 32 Note on output formats A zero real number is printed in e format as 0 Exx where xx is the usually negative decimal exponent of the number cf Section 1 2 6 3 This allows the user to check the accuracy of the zero in question this could also be done using x but that would be more technical When the integer part of a real number x is not known exactly because the exponent of
50. 2 6 3 void setvalp GEN z long e sets the p adic valuation of z to e In addition to this codeword z 2 points to the prime p z 3 points to pP P W3 and z 4 points to an integer representing the p adic unit associated to z modulo z 3 and points to zero if z is zero To summarize if z 4 0 we have the equality z pi y 2 4 O 2 3 pr y z 4 O pPre 4 5 7 Type t_QUAD quadratic number z 1 points to the polynomial defining the quadratic field z 2 to the real part and z 3 to the imaginary part which are to be taken as the coefficients of z with respect to the canonical basis 1 w see Section 1 2 3 Complex numbers are a particular case of quadratics but deserve a separate type 188 4 5 8 Type t_POLMOD polmod exactly as for integermods z 1 points to the modulus and z 2 to a polynomial representing the class of z Both must be of type t_POL in the same variable However z 2 is allowed to be a simplification of such a polynomial e g a scalar This is quite tricky considering the hierarchical structure of the variables in particular a polynomial in variable of lesser priority see Section 2 6 2 than the modulus variable is valid since it can be considered as the constant term of a polynomial of degree 0 in the correct variable On the other hand a variable of greater priority would not be acceptable see Section 2 6 2 for the problems which may arise 4 5 9 Type t_POL polynomial th
51. 3 10 Plotting functions Although plotting is not even a side purpose of PARI a number of plotting functions are provided Moreover a lot of people felt like suggesting ideas or submitting huge patches for this section of the code Among these special thanks go to Klaus Peter Nischke who suggested the recursive plotting and the forking resizing stuff under X11 and Ilya Zakharevich who undertook a complete rewrite of the graphic code so that most of it is now platform independent and should be relatively easy to port or expand These graphic functions are either e high level plotting functions all the functions starting with ploth in which the user has little to do but explain what type of plot he wants and whose syntax is similar to the one used in the preceding section with somewhat more complicated flags e low level plotting functions where every drawing primitive point line box etc must be specified by the user These low level functions called rectplot functions sharing the prefix plot work as follows You have at your disposal 16 virtual windows which are filled independently and can then be physically ORed on a single window at user defined positions These windows are numbered from 0 to 15 and must be initialized before being used by the function plotinit which specifies the height and width of the virtual window called a rectwindow in the sequel At all times a virtual cursor initialized at 0 0 is associated to
52. 4 is Shanks s distance function and should be of type real 4 5 13 Type t_QFI definite binary quadratic form z 1 z 2 z 3 point to the three coefficients of the form All three should be of type integer 4 5 14 Type t_VEC and t_COL vector z 1 z 2 z 1g z 1 point to the components of the vector 4 5 15 Type t_MAT matrix z 1 z 2 z 1g z 1 point to the column vectors of z i e they must be of type t_COL and of the same length The next two types were introduced for specific GP use and you will be much better off using the standard malloc ed C constructs when programming in library mode We quote them just for completeness advising you not to use them 4 5 16 Type t_LIST list This one has a second codeword which contains an effective length handled through lgef setlgef z 2 z 1gef z 1 contain the components of the list 4 5 17 Type t_STR character string char GSTR z z 1 points to the first char acter of the NULL terminated string Implementation note for the types including an exponent or a valuation we actually store a biased non negative exponent bit ORing the biased exponent to the codeword obtained by adding a constant to the true exponent either HIGHEXPOBIT for t_REAL or HIGHVALPBIT for t_PADIC and t_SER Of course this is encapsulated by the exponent valuation handling macros and need not concern the library user 4 6 PARI variables 4 6 1 Multivariate obje
53. 4 0 a Warning is issued so that you can set flag 1 to check whether L K is known to be Galois according to T Example bnf bnfinit y 3 y72 2 y 1 p x 2 Mod y 2 2 y 1 bnf pol T rnfisnorminit bnf p rnfisnorm T 17 checks whether 17 is a norm in the Galois extension Q 8 Q a where a a 2a 1 0 and B a 2a 1 0 it is The library syntax is rnfisnorm T x flag 3 6 137 rnfisnorminit pol polrel flag 2 let K be defined by a root of pol and L K the extension defined by the polynomial polrel As usual pol can in fact be an nf or bnf etc if pol has degree 1 the base field is Q polrel is also allowed to be an nf etc Computes technical data needed by rnfisnorm to solve norm equations Na a for x in L andain K If flag 0 do not care whether L K is Galois or not If flag 1 L K is assumed to be Galois unchecked which speeds up rnfisnorm If flag 2 let the routine determine whether L K is Galois The library syntax is rnfisnorminit pol polrel flag 132 3 6 138 rnfkummer bnr subgroup deg 0 bnr being as output by bnrinit finds a relative equation for the class field corresponding to the module in bnr and the given congruence subgroup the full ray class field if subgroup is omitted If deg is positive outputs the list of all relative equations of degree deg contained in the ray class field defined by bnr with the same conductor as bnr subgroup
54. 4 GP operators 2 5 The general GP input line 2 6 The GP PARI programming language 2 7 Errors and error recovery 2 8 Interfacing GP with other languages 2 9 The preferences file 2 10 Using GP under GNU Emacs 2 11 Using GP with readline Chapter 3 Functions and Operations Ayailsbles in PARI and GP 3 1 Standard monadic or dyadic operators 3 2 Conversions and similar elementary functions or commands 3 3 Transcendental functions 3 4 Arithmetic functions 3 5 Functions related to elliptic curves 3 6 Functions related to general number fields 3 7 Polynomials and power series 3 8 Vectors matrices linear algebra and sets 3 9 Sums products integrals and similar functions 3 10 Plotting functions 3 11 Programming under GP er Chapter 4 Programming PARI in Tibras Mode 4 1 Introduction initializations universal objects 4 2 Important technical notes 4 3 Creation of PARI objects assignments conversions 4 4 Garbage collection 4 5 Implementation of the PARI types 4 6 PARI variables 4 7 Input and output 4 8 A complete program 4 9 Adding functions to PARI Chapter 5 Technical Reference Guide for Loe Level Futo 5 1 Level 0 kernel operations on unsigned longs 5 2 Level 1 kernel operations on longs integers and reals 5 3 Level 2 kernel operations on general PARI objects Appendix A Installation Guide for the UNIX Versions Appendix B A Sample program and Makefile Appendix C Summary of Available Cons
55. EMX RSX runtime package install excluded under DOS since DLLs are not supported by the OS For Windows 95 and higher you can also use the Cygwin compatibility library to run GP almost as if running a genuine Unix system Note that a native Linux binary will be faster than one using any of these compatibility packages see the MACHINES benchmark file included in the distribution If you have GNU Emacs you can work in a special Emacs shell see Section 2 10 which is started by typing M x gp where as usual M is the Meta key if you accept the default stack prime and buffer sizes or C u M x gp which will ask you for the name of the gp executable the stack size the prime limit and the buffer size Specific features of this Emacs shell will be indicated by an EMACS sign If a preferences file or gprc to be discussed in Section 2 9 can be found GP will then read it and execute the commands it contains This provides an easy way to customize GP without having to delve into the code to hardwire it to your likings The files argument is processed right after the gprc On the Macintosh even after clicking on the gp icon once in the MPW Shell you still need to type explicitly a command of the above form 13 UNIX A copyright message then appears which includes the version number and a lot of useful technical information The present manual is written for version 2 2 7 and has undergone major changes since the 1 39 xx versions
56. For example to build a package for a Linux distribution you may want to use Configure prefix usr This phase extracts some files and creates a directory Oxxx where the object files and exe cutables will be built The xxx part depends on your architecture and operating system thus you can build GP for several different machines from the same source tree the builds are completely independent so can be done simultaneously Technical note The precise default destinations are as follows the gp binary the scripts gphelp and tex2mail go to prefix bin The pari library goes to prefix lib and include files to prefix include pari Other system dependant data go to prefix lib pari As for architecture independent files they go to various subdirectories of share prefix which defaults to prefix share and can be specified via the share prefix argument Man pages go into share_prefix man Emacs files into share_prefix emacs site lisp pari and other system independant data to various subdirectories of share_prefix pari documentation sample GP scripts and C code extra packages like galdata You can also set directly bindir executables libdir library includedir include files mandir manual pages datadir other architecture independent data and finally sysdatadir other architecture dependent data Technical note 2 Configure lets the following environment variable override the defaults if set AS Assembl
57. In other words x is isomorphic to ZK d ZK dy and di divides d _ for i gt 2 The link between x and A I J is as follows if e is the canonical basis of K I bi bn and J a1 an then x is isomorphic to bre O Odnen a1 41 0 DAanAn where the A are the columns of the matrix A Note that every finitely generated torsion module can be given in this way and even with b Zg for all i The library syntax is nfsmith nf x 3 6 104 nfsolvemodpr nf a b pr solution of a x b in Zx pr where a is a matrix and ba column vector and where pr is in modpr format see nfmodprinit The library syntax is nfsolvemodpr nf a b pr 3 6 105 polcompositum P Q flag 0 P and Q being squarefree polynomials in Z X in the same variable outputs the simple factors of the tale Q algebra A Q X Y P X Q Y The factors are given by a list of polynomials R in Z X associated to the number field Q X R and sorted by increasing degree with respect to lexicographic ordering for factors of equal degrees Returns an error if one of the polynomials is not squarefree Note that it is more efficient to reduce to the case where P and Q are irreducible first The routine will not perform this for you since it may be expensive and the inputs are irreducible in most applications anyway Assuming P is irreducible of smaller degree than Q for efficiency it is in general much faster to proceed as follow
58. MUMNGTV eo a a bul a 84 MUMS sone Gok oe E ade cae Sek 65 Numerator ers ra Bye eed as 35 65 numerical derivation 29 numerical integration 153 numtoperm 04 65 66 DUPON 4 0 Silk A a en ke A 86 N ron Tate height 91 O Da it Son Be tan 31 135 OMEGA 2 tek a Bee ae ees aoe A 87 OMEGI bic el a we ae ae eee ee A 84 89 ONCUFVE Se ce keel id bi a 93 Operator trish feo See ep a ald de a 28 OF a ach a a Seed Bee os 57 OWS estate ehh ate Oa a 62 OTOL eit nce sad aie a OR 94 onder 2 2 9 3 8 4 2 fs eee ek 89 Order dy ioi e 404 hee de eee ae A a 94 a si aoe a ae Beale lhe Alb a 127 outbeaut sse a ek dee bes 194 QUEDTUtS fad heat Bot eed Wee des 194 OUGL ALCS mia e pide las ee eS a 194 OUtMa ts et tie aks te er a A Eee 194 output formats 0 14 OUUPUC 5 6 4 A a ee ee ee 193 OUTPUT Gn ka ee aks 18 23 194 195 P p adic number 7 25 188 padicappr o oo 136 244 padicpre 2 2 2 2 2 Ya gn ee Sea ds 66 parametric plot 2 ee ee ee 158 Parise shod on de p aa wnt coke ep ee Sink 171 PARTE A E i EAS 195 pariksl iia A AAN 198 parido a we A 4 195 PariPerl ity ia e Soak a 46 PaPIputs ea a a aA 203 PArTP YOMO aes 46 ParisiZe ia a ee BAL ee 19 pari init rocio 171 172 231 pari Sp way ehh pian ce e di ets RO Sig te aa 179 pari tipar os see hws a A ae oS 196 parser code 2 201 204 Pascal triangle
59. Mec Ml wate tee 137 poldegree 2 we i oe Re eee A 137 AA le te Ba ee ena ae a 137 POLATSCO lo enol ee bal wee Se a a 137 poldiscreduced 137 POldIVEeS ui eS A ised 219 pol val i458 ey EF ae ets 136 porini La Ss GA ah ee ar 108 polgalois 0 18 125 polhensellift 137 polinto 1 ea an add poa 137 polinterpolate 137 polisirreducible 137 Pollard Rh0 76 81 polla GOR oe a ee how A 138 pollegendre 138 polmod 0 7 26 188 polmodrecip 116 POLTOCIP inician Eos BAH a aes 138 POLEO g ra did 126 POLLO ea ee de a ei ae 126 polredabs o o o o 126 polredabs0 sos airada 127 polredord ooo oo ooo 127 polresultant 138 polresultantO 138 Rolrev gt me ol e a ad 59 POLTOOCS i mocos e bog we en de 138 polrootsmod eiii Se ee 8 138 polrootspadic 139 POLStUIM ocn a ets 139 polsubcyClo see a Be eS 139 polsylvestermatrix 139 POLSA esmas Ala PR ee es 139 POLtGHEDI kira ees Sak eis ded Begs 139 poltschirnhaus 127 POLUM asta soa Bek eek Eg 172 191 polvan id ee eos Wee 191 245 POLK ss 8 ke a a ce Bs 172 191 POLOP circ as Geers am nice we thie 73 POLY TOGO ove spre he GSE a Ghat addin eee ied 74 polynomial 7 8 27 189 POlZag i ark Behe Kok he A 139 polzagier 139 polzagre
60. Note also that det T is equal to the discriminant of the field K e The columns of MD nf diff express a Z basis of the different of K on the integral basis e TI is equal to d K T which has integral coefficients Note that understood as as ideal the matrix T7 generates the codifferent ideal e Finally MDI is a two element representation for faster ideal product of d K times the codifferent ideal nf discx nf codiff which is an integral ideal MDI is only used in idealinv nf 6 is the vector containing the r1 r2 roots nf roots of nf 1 corresponding to the rl r2 embeddings of the number field into C the first r1 components are real the next r2 have positive imaginary part nf 7 is an integral basis for Zg nf zk expressed on the powers of 0 Its first element is guaranteed to be 1 This basis is LLL reduced with respect to To strictly speaking it is a permutation of such a basis due to the condition that the first element be 1 nf 8 is the n x n integral matrix expressing the power basis in terms of the integral basis and finally nf 9 is the n x n matrix giving the multiplication table of the integral basis If a non monic polynomial is input nfinit will transform it into a monic one then reduce it see flag 3 It is allowed though not very useful given the existence of nfnewprec to input a nf or a bnf instead of a polynomial The special input format x B is also accepted where x is a polynom
61. Q van Hoeij s method is used which is able to cope with hundreds of modular factors Note that PARI tries to guess in a sensible way over which ring you want to factor Note also that factorization of polynomials is done up to multiplication by a constant In particular the factors of rational polynomials will have integer coefficients and the content of a polynomial or rational function is discarded and not included in the factorization If needed you can always ask for the content explicitly factor t 2 5 2 t 1 1 79 2xt 1 1 t 2 1 content t 2 5 2 t 1 12 1 2 See also factornf and nffactor The library syntax is factor0 x lim where lim is a C integer Also available are factor x factor0 x 1 smallfact x factor0 z 0 3 4 19 factorback f e nf gives back the factored object corresponding to a factorization If the last argument is of number field type e g created by nfinit or bnfinit assume we are dealing with an ideal factorization in the number field The resulting ideal product is given in HNF form If e is present e and f must be vectors of the same length e being integral and the corre sponding factorization is the product of the ffi If not and f is vector it is understood as in the preceding case with e a vector of 1 the product of the f i is returned Finally f can be a regular factorization as produced with any factor command A few examples f
62. SHARC oy eed ba oe Aya et te SOE YB YY 57 SHUG dust ada id 211 SHITtl asih e ema a h Bo a A 207 Shif tli aaia Goel a as 207 SHVPtMUL daros el tee ee 57 SOLEDAD ine a 211 Shifts 24 Asp is o 211 SIMA se god ate E bie iP a 88 155 SIG tebe ae Ke a E 58 STON 2 PM ce as Soe ee He es 58 98 SIGMA ek ee Awe BY ahs ek 151 A Ae a 187 189 209 SIGNUNITS sos cca Kido a Ss tia 103 simplefactmod 82 Simplify Sis hs else 20 22 67 A Bak he E ee 74 Sindexlexsort in whe case ee A ets 152 Sindexsort io lare ee Ab ek Se kt 152 Sinh 03h bo ko AW ck es HN Geran 74 sizebyte o ooo ooo oo 67 SIZEdICIC ns ao rio 67 216 smallfact nc ir naa 80 smallinitelle 0d a ee Bees 93 98 101 103 115 124 134 147 163 Smith normal form SMA A A a 148 SOLVE ita oe eS eed ke ein os ee 154 SOMME LA a 155 SOL a ts Ea det ae a ad 194 SOL A dime Ba ed E 152 SUE pt eds 74 sared A kas Reale a OR 149 SUE E 2 Peake wl ate he eS 74 SQFUING i ana shies Beek ees 88 SQN sense daca Un Barts dak ek ee ee 74 ST BCA se 8s aed Bet By eae ey GER Ee 82 Satkit el aay Se fod ei PS ae 171 SACK ac hy dese A a in saad 24 231 stackmalloc cui g bens a e 176 Stacksize aa A a 38 Stack Vim oia a ra ed 185 Stark wits deca a oia amp 87 107 SAUD 2 rs tal OS 47 stderr e aaraa ee ha aa 195 stdoute e in a tot Hate kt 194 Steinitz class 134 SEA ek ra o en 179 211 SUOT oa kti a act uea ha Pode gO He be te 211 Sty
63. See the discussion on typecasts in Chapter 4 3 2 26 conj x conjugate of x The meaning of this is clear except that for real quadratic numbers it means conjugation in the real quadratic field This function has no effect on integers reals integermods fractions or p adics The only forbidden type is polmod see conjvec for this The library syntax is gconj z 3 2 27 conjvec x conjugate vector representation of x If x is a polmod equal to Mod a q this gives a vector of length degree q containing the complex embeddings of the polmod if q has integral or rational coefficients and the conjugates of the polmod if q has some integermod coefficients The order is the same as that of the polroots functions If x is an integer or a rational number the result is x If x is a row or column vector the result is a matrix whose columns are the conjugate vectors of the individual elements of x The library syntax is conjvec z prec 3 2 28 denominator x lowest denominator of x The meaning of this is clear when x is a rational number or function When x is an integer or a polynomial the result is equal to 1 When x is a vector or a matrix the lowest common denominator of the components of x is computed All other types are forbidden Warning multivariate objects are created according to variable priorities with possibly surprising side effects x y is a polynomial but y x is a rational function See Section 2 6 2 The library
64. The local statements can be omitted as usual seq is any expression sequence name is the name given to the function and is subject to the same restrictions as variable names In addition variable names are not valid function names you have to kill the variable first the converse is true function names can t be used as variables see Section 3 11 2 14 Previously used function names can be recycled you are just redefining the function The previous definition is lost of course list of formal variables is the list of variables corresponding to those which you will actually use when calling your function The number of actual parameters supplied when calling the function has to be less than the number of formal variables Uninitialized formal variables will be given a default value An equal sign following a variable name in the function definition followed by any expression gives the variable a default value The said expression gets evaluated the moment the function is called hence may involve the function parameters A variable for which you supply no default value will be initialized to zero list of local variables is the list of the additional local variables which are used in the function body Note that if you omit some or all of these local variable declarations the non declared variables will become global hence known outside of the function and this may have undesirable side effects On the other hand in some cases it ma
65. The result is the sign of x y int cmpsi long s GEN x compares the long s to the integer x int cmpsr long s GEN x compares the long s to the real x int cmpis GEN x long s compares the integer x to the long s int cmpii GEN x GEN y compares the integer x to the integer y int cmpir GEN x GEN y compares the integer x to the real y int cmprs GEN x long s compares the real x to the long s int cmpri GEN x GEN y compares the real x to the integer y int cmprr GEN x GEN y compares the real x to the real y int egalii GEN x GEN y compares the integers x and y The result is 1 if x y 0 otherwise int absi_cmp GEN x GEN y compares the integers x and y The result is the sign of x y int absi_equal GEN x GEN y compares the integers x and y The result is 1 if x ly 0 otherwise int absr_cmp GEN x GEN y compares the reals x and y The result is the sign of x yl 212 5 2 7 Binary operations Let op be some operation of type GEN GEN GEN The names and prototypes of the low level functions corresponding to op will be as follows In this section the z argument in the z functions must be of type t_INT or t_REAL GEN mpop z GEN x GEN y GEN z applies op to the integer or reals x and y GEN opss z long s long t GEN z applies op to the longs s and t GEN opsi z long s GEN x GEN z applies op to the long s and the integer x GEN opsr z long s GEN x GEN z applies
66. You can create more temporary variables using long fetch_var This returns a variable number which is guaranteed to be unused by the library at the time you get it and as long as you do not delete it we will see how to do that shortly This has lower number i e higher priority than any temporary variable produced so far MAXVARN is assumed to be the first such This call updates all the aforementioned internal arrays In particular after the statement v fetch var you can use polun v and polx v The variables created in this way have no identifier assigned to them though and they will be printed as lt number gt except for MAXVARN which will be printed as You can assign a name to a temporary variable after creating it by calling the function void name_var long n char s after which the output machinery will use the name s to represent the variable number n The GP parser will not recognize it by that name however and calling this on a variable known to GP will raise an error Temporary variables are meant to be used as free variables and you should never assign values or functions to them as you would do with variables under GP For that you need a user variable All objects created by fetch_var are on the heap and not on the stack thus they are not subject to standard garbage collecting they will not be destroyed by a gerepile or avma ltop statement When you do not need a variable number anymore you can delete
67. as an integer bitmap That is if x 1 2 with the x in 0 1 this routine returns the integer y Lny 2 0 lt i lt e Bits at negative offsets are 0 A negative value of c means that negative values of x are treated in the spirit of 2 complement arithmetic i e modulo a big power of 2 To extract several bits or groups of bits if c gt 1 separately at once as a vector pass a vector for n The library syntax is gbittest3 z n c Also available are gbittest x n default case c 1 and for simple cases bittest x n where n and the result are longs 3 2 21 bitxor z y bitwise exclusive or of two integers x and y that is the integer ae xor y 2 Negative numbers behave as if modulo a huge power of 2 The library syntax is gbitxor z y 62 3 2 22 ceil x ceiling of x When z is in R the result is the smallest integer greater than or equal to x Applied to a rational function ceil x returns the euclidian quotient of the numerator by the denominator The library syntax is gceil x 3 2 23 centerlift x v lifts an element z a mod n of Z nZ to a in Z and similarly lifts a polmod to a polynomial This is the same as lift except that in the particular case of elements of Z nZ the lift y is such that n 2 lt y lt n 2 If x is of type fraction complex quadratic polynomial power series rational function vector or matrix the lift is done for each coefficient Reals are forbidden The library syntax
68. at least 2 2 is preferred but older versions should be usable e GNU gzip gunzip gzcat package enables GP to read compressed data e GNU emacs GP can be run in an Emacs buffer with all the obvious advantages if you are familiar with this editor Note that readline is still useful in this case since it provides a much better automatic completion than is provided by Emacs GP mode e perl provides extended online help full text from this manual about functions and concepts which can be used under GP or independently http www perl com will direct you to the nearest CPAN archive site e A colour capable xterm which enables GP to use different user configurable colours for its output All xterm programs which come with current X11 distributions satisfy this requirement One notable exception is the native AIX C compiler on IBM RS 6000 workstations which generates fast code even without any special help from the PARI kernel sources 221 2 Compiling the library and the GP calculator 2 1 Basic configuration First have a look at the MACHINES file to see if anything funny applies to your architecture or operating system Then type Configure in the toplevel directory This attempts to configure GP PARI without outside help Note that if you want to install the end product in some nonstandard place you can use the prefix option as in Configure prefix an exotic directory the default prefix is usr local
69. back to it by number since this number does not appear explicitly Of course if you just want to use it on the next line use as usual Any legal expression can be typed in and is evaluated using the conventions about operator priorities and left to right associativity see the previous section using the available operator symbols function names including user defined functions and member functions see Section 2 6 4 and special variables Please note that from version 1 900 on there is a distinction between lowercase and uppercase Also note that outside of constant strings blanks are completely ignored in the input to GP The special variable names known to GP are Euler Euler s constant y 0 577 I the square root of 1 Pi 3 14 which could be thought of as functions with no arguments and which may therefore be invoked without parentheses and O which obeys the following syntax OCexpr k When expr is an integer or a rational number this creates an expr adic number zero in fact of precision k When ezpr is a polynomial a power series or a rational function whose main variable is X say this creates a power series also zero of precision v k where v is the X adic valuation of expr see 2 3 6 and 2 3 9 31 2 5 2 Special editing characters A GP program can of course have more than one line Since GP executes your commands as soon as you have finished typing them there must be a way to tell it t
70. before and P is a permutation of the rows such that P applied to xU gives H The matrix U is smaller than with flag 1 but may still be large If flag 4 as in case 1 above but uses a heuristic variant of LLL reduction along the way The matrix U is in general close to optimal in terms of smallest L norm but the reduction is slower than in case 1 The library syntax is mathnfO z flag Also available are hnf x flag 0 and hnfall x flag 1 To reduce huge say 400 x 400 and more relation matrices sparse with small entries you can use the pair hnfspec hnfadd Since this is rather technical and the calling interface may change they are not documented yet Look at the code in basemath alglin1 c 3 8 19 mathnfmod z d if x is a not necessarily square matrix of maximal rank with integer entries and d is a multiple of the non zero determinant of the lattice spanned by the columns of x finds the upper triangular Hermite normal form of z If the rank of x is equal to its number of rows the result is a square matrix In general the columns of the result form a basis of the lattice spanned by the columns of x This is much faster than mathnf when d is known The library syntax is hnfmod z d 3 8 20 mathnfmodid z d outputs the upper triangular Hermite normal form of x concate nated with d times the identity matrix The library syntax is hnfmodid z d 3 8 21 matid n creates the n x n identity matrix The lib
71. by ideallist flag for instance flag is optional its binary digits are toggles meaning 1 give generators as well 2 list format is L U see ideallist The library syntax is ideallistarchO nf list arch flag where an omitted arch is coded as NULL 3 6 53 ideallog nf x bid nf being a number field bid being a big ideal as output by ideal star and x being a non necessarily integral element of nf which must have valuation equal to 0 at all prime ideals dividing J bid 1 computes the discrete logarithm of x on the generators given in bid 2 In other words if g are these generators of orders d respectively the result is a column vector of integers x such that 0 lt a lt d and x o mod I Note that when J is a module this implies also sign conditions on the embeddings The library syntax is zideallog nf x bid 3 6 54 idealmin nf x vdir computes a minimum of the ideal x in the direction vdir in the number field nf The library syntax is minideal nf x vdir prec where an omitted vdir is coded as NULL 3 6 55 idealmul nf x y flag 0 ideal multiplication of the ideals x and y in the number field nf The result is a generating set for the ideal product with at most n elements and is in Hermite normal form if either x or y is in HNF or is a prime ideal as output by idealprimedec and this is given together with the sum of the Archimedean information in x and y if both are given
72. can translate GP code to C and load them into the GP interpreter A typical script compiled by GP2C will typically run 3 to 10 times faster The C code can also be edited and optimized by hand The use of GP is explained in chapters 2 and 3 and the programming in library mode is explained in chapters 3 4 and 5 In the present Chapter 1 we give an overview of the system Important note A tutorial for GP is provided in the standard distribution tutorial dvi and you should read this first at least the beginning of it you can skip the specialized topics you re not interested in You can then start over and read the more boring stuff which lies ahead But you should do that eventually at the very least the various Chapter headings You can have a quick idea of what is available by looking at the GP reference card refcard dvi or refcard ps In case of need you can then refer to the complete function description in Chapter 3 How to get the latest version This package can be obtained by anonymous ftp from quite a number of sites ask your favourite Web search engine for the site nearest to you But if you want the very latest version including development versions you should use the anonymous ftp address ftp pari math u bordeaux fr pub pari where you will find all the different ports and possibly some binaries A lot of version information mailing list archives and various tips can be found on PARI s home page http pari m
73. creating it if needed and returns its variable number long v fetch_user_var y GEN gy polx v This function raises an error if s is already known as a function name to the interpreter Caveat it is possible to use flissexpr see Section 4 7 1 to execute a GP command and create GP variables on the fly as needed GEN gy flissexpr y supposedly returns polx v for some v long v gvar gy This is dangerous especially when programming functions that will be used under GP The code above reads the value of y as it is currently known by the GP interpreter possibly creating it in the process All is well and good if y has not been tampered with in previous GP commands But if y has been modified e g y 1 then the value of gy is not what you expected it to be and corresponds instead to the current value of the GP variable e g gun More precisely the first time a given identifier is read by the GP parser and is not immediately interpreted as a function a new variable is created and it is assigned a strictly lower priority than any variable in use at this point On startup before any user input has taken place x is defined in this way and will thus always have maximal priority and variable number 0 191 4 6 2 2 Temporary variables MAXVARN is available but is better left to pari internal functions some of which do not check that MAXVARN is free for them to use which can be considered a bug
74. defined only when x and y are longs or integers The result is the true Euclidean remainder i e non negative and less than the absolute value of y 213 5 2 8 Division with remainder the following functions return two objects unless specifically asked for only one of them a quotient and a remainder The remainder will be created on the stack and a GEN pointer to this object will be returned through the variable whose address is passed as the r argument GEN dvmdss long s long t GEN r creates the Euclidean quotient and remainder of the longs s and t If r is not NULL or ONLY_REM this puts the remainder into r and returns the quotient If r is equal to NULL only the quotient is returned If r is equal to ONLY_REM the remainder is returned instead of the quotient In the generic case the remainder is created after the quotient and can be disposed of individually with a cgiv r The remainder is always of the sign of the dividend s GEN dvmdsi long s GEN x GEN r creates the Euclidean quotient and remainder of the long s by the integer x Obeys the same conventions with respect to r GEN dvmdis GEN x long s GEN r create the Euclidean quotient and remainder of the integer x by the long s GEN dvmdii GEN x GEN y GEN r returns the Euclidean quotient of the integer x by the inte ger y and puts the remainder into r If r is equal to NULL the remainder is not created and if r is equal to ONLY_REM only the remainder
75. desired The library syntax is rayclassno bnf I 3 6 20 bnrclassnolist bnf list bnf being a big number field as output by bnfinit units are mandatory unless the ideal is trivial and list being a list of modules as output by ideallist of ideallistarch outputs the list of the class numbers of the corresponding ray class groups The library syntax is rayclassnolist bnf list 105 3 6 21 bnrconductor ay a2 a3 flag 0 conductor f of the subfield of a ray class field as defined by a1 az az see bnr at the beginning of this section If flag 0 returns f If flag 1 returns f Cly H where Clp is the ray class group modulo f as output by bnrclass f 1 and H is the subgroup of Cl defining the extension If a is a bnr the algorithm requires that it has been computed by bnrinit 1 i e with generators If flag 2 returns f bnr f H as above except Cl is replaced by a bnr structure as output by bnrinit f 1 The library syntax is conductor bnr subgroup flag where an omitted subgroup trivial sub group i e ray class field is input as NULL and flag is a C long 3 6 22 bnrconductorofchar bnr chi bnr being a big ray number field as output by bnrclass and chi being a row vector representing a character as expressed on the generators of the ray class group gives the conductor of this character as a modulus The library syntax is bnrconductorofchar bnr chi 3 6 23 bnrdisc al a2
76. divides the discriminant 2 use the round 2 algorithm instead of the default round 4 This should be slower except maybe for polynomials of small degree and coefficients The library syntax is nfdiscfO z flag fa where to omit fa you should input NULL You can also use discf x flag 0 3 6 74 nfeltdiv nf x y given two elements x and y in nf computes their quotient x y in the number field nf The library syntax is element_div nf x y 3 6 75 nfeltdiveuc nf x y given two elements x and y in nf computes an algebraic integer q in the number field nf such that the components of x qy are reasonably small In fact this is functionally identical to round nfeltdiv nf x y The library syntax is nfdiveuc nf x y 3 6 76 nfeltdivmodpr nf x y pr given two elements x and y in nf and pr a prime ideal in modpr format see nfmodprinit computes their quotient y modulo the prime ideal pr The library syntax is element_divmodpr nf x y pr 3 6 77 nfeltdivrem nf x y given two elements x and y in nf gives a two element row vector q r such that x qy r q is an algebraic integer in nf and the components of r are reasonably small The library syntax is nfdivres nf x y 3 6 78 nfeltmod nf x y given two elements x and y in nf computes an element r of nf of the form r x qy with q and algebraic integer and such that r is small This is functionally identical to x nfeltmul nf round nfeltdiv nf x y
77. ee 33 36 39 localreduction 93 TOG Aik eat ed Att 3 18 22 23 54 73 167 logfile o 2a eae eee Ba ee a aA 167 VOGELS tic os Ahh ae oy See od 18 LONG_IS_64BIT 174 POLA ata il ii Ale be AS 173 lsertesell ciao ees Wo ida ci 93 M MACHINES e S hi aLa a ce es ee a 13 MAC eke a Soe ea 63 173 paali s by oh ao ee Rak ae wn be a 215 makebigbnf 103 Matias e ora hl Md is do 27 58 142 Matad jOInt i rs do ad as 143 matalgtobasiS 116 matbasistoal8 116 Matbrute o 194 matcompanion 144 242 Matdet 2204 nya ee Mia a o 144 matdetinto als Share leh e Ae 144 matdiagonal 144 Mateigen xr ao SAS 144 Matvextract e ii bw Saw A 152 mathell 3 aco ey o it Ws 92 MATHESS o oa doe Awad awa ae dad 144 mathilbert 144 Mathnt Gather aa Mt Geer A esd 141 144 matni 3 ay dart a a ee AW Ses 145 mathnfmod 145 mathnfmodid 145 matid esa heh erase hoe a 145 Matimage s e cal we Gea ee ee ea 145 MatimageO o 145 matimagecompl 145 matindexrank 146 Matintersect maraa Gee aot eye ans 146 matinverseimage 146 matisdiagonal 146 MALES as sn a be Be eS 146 matkerO ana dd de ea 146 MatkeTidt cion a a 146 Matkerinto css bs A ee be 146 matmuldiagonal 146 matmultodiagonal 146
78. efficient The library syntax is factorpadic4 pol p r where r is a long integer 3 7 5 intformal z v formal integration of x with respect to the main variable if v is omitted with respect to the variable v otherwise Since PARI does not know about abstract logarithms they are immediately evaluated if only to a power series logarithmic terms in the result will yield an error x can be of any type When x is a rational function it is assumed that the base ring is an integral domain of characteristic zero The library syntax is integ x v where v is a long and an omitted v is coded as 1 3 7 6 padicappr pol a vector of p adic roots of the polynomial pol congruent to the p adic number a modulo p or modulo 4 if p 2 and with the same p adic precision as a The number a can be an ordinary p adic number type t_PADIC i e an element of Qp or can be an element of a finite extension of Qp in which case it is of type t_POLMOD where at least one of the coefficients of the polmod is a p adic number In this case the result is the vector of roots belonging to the same extension of Q as a The library syntax is apprgen9 pol a but if a is known to be simply a p adic number type t_PADIC the syntax apprgen pol a can be used 136 3 7 7 polcoeff x 5 v coefficient of degree s of the polynomial x with respect to the main variable if v is omitted with respect to v otherwise Also applies to power series scalars po
79. empty The library syntax is vecmin z 58 3 2 Conversions and similar elementary functions or commands Many of the conversion functions are rounding or truncating operations In this case if the argu ment is a rational function the result is the Euclidean quotient of the numerator by the denomi nator and if the argument is a vector or a matrix the operation is done componentwise This will not be restated for every function 3 2 1 List x transforms a row or column vector x into a list The only other way to create a t_LIST is to use the function listcreate This is useless in library mode 3 2 2 Mat x transforms the object x into a matrix If x is not a vector or a matrix this creates a 1 x 1 matrix If x is a row resp column vector this creates a 1 row resp 1 column matrix If x is already a matrix a copy of x is created This function can be useful in connection with the function concat see there The library syntax is gtomat z 3 2 3 Mod z y flag 0 creates the PARI object x mod y i e an integermod or a polmod y must be an integer or a polynomial If y is an integer x must be an integer a rational number or a p adic number compatible with the modulus y If y is a polynomial x must be a scalar which is not a polmod a polynomial a rational function or a power series This function is not the same as x y the result of which is an integer or a polynomial If flag is equal
80. equal to the product of the determinant of A by all the ideals of J The determinant of a pseudo matrix is the determinant of any pseudo basis of the module it generates Now a last set of definitions concerning the way big ray number fields or bnr are input using class field theory These are defined by a triple al a2 a3 where the defining set a1 a2 a3 can have any of the following forms bnr bnr subgroup bnf module bnf module subgroup where e bnf is as output by bnfclassunit or bnfinit where units are mandatory unless the ideal is trivial bnr by bnrclass with flag gt 0 or bnrinit This is the ground field e module is either an ideal in any form see above or a two component row vector containing an ideal and an r component row vector of flags indicating which real Archimedean embeddings to take in the module e subgroup is the HNF matrix of a subgroup of the ray class group of the ground field for the modulus module This is input as a square matrix expressing generators of a subgroup of the ray class group bnr clgp on the given generators The corresponding bnr is then the subfield of the ray class field of the ground field for the given modulus associated to the given subgroup 97 All the functions which are specific to relative extensions number fields big number fields big number rays share the prefix rnf nf bnf bnr respectively They are meant to take as first argument a number field of th
81. extension L K as output by rnfinit and x being an ideal of K gives the ideal 1Z 1 as an absolute ideal of L Q in the form of a Z basis given by a vector of polynomials modulo rnf pol The following routine might be useful return y rnfidealup rnf as an ideal in HNF form associated to nf nfinit rnf pol idealgentoHNF nf y local z z nfalgtobasis nf y z 1 Mat z 1 mathnf concat z The library syntax is rnfidealup rnf x 3 6 134 rnfinit nf pol nf being a number field in nfinit format considered as base field and pol a polynomial defining a relative extension over nf this computes all the necessary data to work in the relative extension The main variable of pol must be of higher priority see Section 2 6 2 than that of nf and the coefficients of pol must be in nf The result is a row vector whose components are technical In the following description we let K be the base field defined by nf m the degree of the base field n the relative degree L the large field of relative degree n or absolute degree nm r and ra the number of real and complex places of K rnf 1 contains the relative polynomial pol rnf 2 is currently unused rnf 3 is a two component row vector d L K s where 0 L K is the relative ideal discrimi nant of L K and s is the discriminant of L K viewed as an element of K K in other words it is the output of rnfdisc rnf 4 is the ideal index f ie suc
82. field of discriminant D which can also be a bnf using analytic methods For D lt 0 uses the o function flag has the following meaning if it s an odd integer outputs instead the vector of ideal corresponding root It can also be a two component vector A flag where flag is as above and A is the technical element of bnf necessary for Schertz s method In that case returns 0 if A is not suitable For D gt 0 uses Stark s conjecture If flag is non zero try hard to get the best modulus The function may fail with the following message Cannot find a suitable modulus in FindModulus See bnrstark for more details about the real case The library syntax is quadray D f flag 3 4 59 quadregulator x regulator of the quadratic field of positive discriminant x Returns an error if x is not a discriminant fundamental or not or if x is a square See also quadclassunit if x is large The library syntax is regula z prec 3 4 60 quadunit D fundamental unit of the real quadratic field Q VD where D is the positive discriminant of the field If D is not a fundamental discriminant this probably gives the funda mental unit of the corresponding order D must be an integer congruent to 0 or 1 modulo 4 which is not a square the result is a quadratic number see Section 3 4 56 The library syntax is fundunit z 3 4 61 removeprimes z removes the primes listed in x from the prime number table In particular removepr
83. for i or r s for an ordinary signed long whereas z as a suffix means that the result is not created on the PARI stack but assigned to a preexisting GEN object passed as an extra argument For completeness Chapter 5 gives a description of all these low level functions Please note that in the present version 2 2 7 the names of the functions are not always consis tent This will be changed Hence anyone programming in PARI must be aware that the names of almost all functions that he uses might be subject to change If the need arises i e if there really are people out there who delve into the innards of PARI updated versions with no name changes will be released 4 2 3 Portability 32 bit 64 bit architectures PARI supports both 32 bit and 64 bit based machines but not simultaneously The library will have been compiled assuming a given architecture a priori following a guess by the Configure program see Appendix A and some of the header files you include through pari h will have been modified to match the library Portable macros are defined to bypass most machine dependencies If you want your programs to run identically on 32 bit and 64 bit machines you will have to use these and not the corre sponding numeric values whenever the precise size of your long integers might matter Here are the most important ones 64 bit 32 bit BITS_IN_LONG 64 32 LONG_IS_64BIT defined undefined DEFAULTPREC 3 4 19 decimal digits see
84. format without ending with a newline note that you can still embed newlines within your strings using the n notation 3 11 2 17 printp str outputs its string arguments in prettyprint beautified format ending with a newline 3 11 2 18 printp1 str outputs its string arguments in prettyprint beautified format without ending with a newline 3 11 2 19 printtex str outputs its string arguments in TEX format This output can then be used in a T X manuscript The printing is done on the standard output If you want to print it to a file you should use writetex see there Another possibility is to enable the log default see Section 2 1 You could for instance do default logfile new tex default log 1 printtex result You can use the automatic string expansion concatenation process to have dynamic file names if you wish 167 UNIX 3 11 2 20 quit exits GP 3 11 2 21 read str reads in the file whose name results from the expansion of the string str If str is omitted re reads the last file that was fed into GP The return value is the result of the last expression evaluated Ifa GP binary file is read using this command see Section 3 11 2 30 the file is loaded and the last object in the file is returned 3 11 2 22 reorder z x must be a vector If x is the empty vector this gives the vector whose components are the existing variables in increasing order i e in decreasin
85. formula below MEDDEFAULTPREC 4 6 38 decimal digits BIGDEFAULTPREC 5 8 57 decimal digits For instance suppose you call a transcendental function such as GEN gexp GEN x long prec The last argument prec is only used if x is an exact object otherwise the relative precision is determined by the precision of x But since prec sets the size of the inexact result counted in long words including codewords the same value of prec will yield different results on 32 bit and 64 bit machines Real numbers have two codewords see Section 4 5 so the formula for computing the bit accuracy is bit_accuracy prec prec 2 x BITS_IN_LONG this is actually the definition of a macro The corresponding accuracy expressed in decimal digits would be bit_accuracy prec x log 2 log 10 For example if the value of prec is 5 the corresponding accuracy for 32 bit machines is 5 2 x log 232 log 10 28 decimal digits while for 64 bit machines it is 5 2 log 2 log 10 57 decimal digits 174 Thus you must take care to change the prec parameter you are supplying according to the bit size either using the default precisions given by the various DEFAULTPRECs or by using conditional constructs of the form ifndef LONG_IS_64BIT prec 4 ttelse prec 6 endif which is in this case equivalent to the statement prec MEDDEFAULTPREC Note that for parity reasons half the accuracies available on 32 bit ar
86. function Since GP did not see the closing parenthesis it tried to read a second argument first looking for the comma that would separate it form the first The error occurred at this point So GP tells you that it was expecting a comma and saw a blank The second error is even weirder It is a simple typo siN instead of sin and GP tells us that is was expecting an equal sign a few characters later What happens is this siN is not a recognized identifier but from the context it looks like a function it is followed by an open parenthesis then we have an argument then a closing parenthesis Then if siN were a known function we would evaluate it but it is not so GP assumes that you were trying to define it as in if siN x sin x This is actually allowed and defines the function siN as an alias for sin As any expression a function definition has a value which is 0 hence the test is meaningful and false so nothing happens Admittedly this doesn t look like a useful syntax but it can be interesting in other contexts to let functions define other functions Anyway it is allowed by the language definition So GP tells you in good faith that to correctly define a function you need an equal sign between its name and its body Error messages from the library will usually be much clearer since by definition they an swer a correctly worded query otherwise GP would have protested first Also they have more mathematical co
87. gcevtoi GEN x long e same as grndtoi except that rounding is replaced by truncation GEN gred z GEN x GEN z reduces x to lowest terms if x is a fraction or rational function types t_FRAC t_FRACN t_RFRAC and t_RFRACN otherwise creates a copy of x GEN content GEN x creates the GCD of all the components of x GEN normalize GEN x applied to an unnormalized power series x i e type t_SER with all coef ficients correctly set except that x 2 might be zero normalizes x correctly in place Returns x For internal use GEN normalizepol GEN x applied to an unnormalized polynomial x i e type t_POL with all coefficients correctly set except that x 2 might be zero normalizes x correctly in place and returns x For internal use 5 3 5 Binary operators GEN gmax z GEN x GEN y GEN z yields the maximum of the objects x and y if they can be compared GEN gmaxsglz long s GEN x GEN z yields the maximum of the long s and the object x GEN gmaxgs z GEN x long s GEN z yields the maximum of the object x and the long s GEN gmin z GEN x GEN y GEN z yields the minimum of the objects x and y if they can be compared GEN gminsgl z long s GEN x GEN z yields the minimum of the long s and the object x GEN gmings z GEN x long s GEN z yields the minimum of the object x and the long s GEN gadd z GEN x GEN y GEN z yields the sum of the objects x and y GEN gaddsg z long s GEN x GEN z yiel
88. have typed any GP command not only the name of a variable of course There is no special set of commands becoming available during a break loop as they would in most debuggers Important Note upon startup this mechanism is off Type trap or include it in a script to start trapping errors in this way By default you will be sent back to the prompt Technical Note When you enter a break loop due to a PARI stack overflow the PARI stack is reset so that you can run commands otherwise the stack would immediately overflow again Still as explained above you do not lose the value of any GP variable in the process 2 7 4 Error handlers The break loop described above is a sophisticated example of an error handler a function that is executed whenever an error occurs supposedly to try and recover The break loop is quite a satisfactory error handler but it may not be adequate for some purposes for instance when GP runs in non interactive mode detached from a terminal So you can define a different error handler to be used in place of the break loop This is the purpose of the second argument of trap to specify an error handler We will discuss the first argument at the very end For instance trap note the comma argl is omitted print reorder writebin crash After that whenever an error occurs the list of all user variables is printed and they are all saved in binary format in file crash ready for insp
89. header file if you expect somebody will access them from C For example if dynamic loading is not available you may need to modify PARI to access these functions so put them in paridecl h The other functions should be declared static in your file 203 Your function is now ready to be used in library mode after compilation and creation of the library If possible compile it as a shared library see the Makefile coming with the matexp example in the distribution It is however still inaccessible from GP 4 9 4 Integration with GP as a shared module To tell GP about your function you must do the following First find a name for it It does not have to match the one used in library mode but consistency is nice It has to be a valid GP identifier i e use only alphabetic characters digits and the underscore character _ the first character being alphabetic Then you have to figure out the correct parser code corresponding to the function prototype This has been explained above Section 4 9 2 Now assuming your Operating System is supported by install simply write a GP script like the following install libname code gpname library addhelp gpname some help text see Section 3 11 2 1 and 3 11 2 13 The addhelp part is not mandatory but very useful if you want others to use your module libname is how the function is named in the library usually the same name as one visible from C Read that file from your GP sessio
90. in particular the new function can be called from GP 201 4 9 2 The calling interface from GP parser codes A parser code is a character string describing all the GP parser needs to know about the function prototype It contains a sequence of the following atoms e Syntax requirements used by functions like for sum etc separator required at this point between two arguments e Mandatory arguments appearing in the same order as the input arguments they describe G GEN amp GEN L long we implicitly identify int with long S symbol i e GP identifier name Function expects a entree V variable as S but rejects symbols associated to functions n variable expects a variable number a long not an entree I string containing a sequence of GP statements a seq to be processed by lisseq useful for control statements E string containing a single GP statement an expr to be processed by lisexpr r raw input treated as a string without quotes Quoted args are copied as strings Stops at first unquoted or Special chars can be quoted using Example aa b n c yields the string aab n c s expanded string Example Pi x 2 yields 3 142x2 Unquoted components can be of any PARI type converted following current output format e Optional arguments sx any number of strings possibly 0 see s Dexx argument has a default value The s code is technical and you probably do not need it but we gi
91. is clear When lt z is an integer or a polynomial the result is x itself When x is a vector or a matrix then numerator x is defined to be denominator x x x All other types are forbidden 65 Warning multivariate objects are created according to variable priorities with possibly surprising side effects x y is a polynomial but y x is a rational function See Section 2 6 2 The library syntax is numer z 3 2 37 numtoperm n k generates the k th permutation as a row vector of length n of the numbers 1 to n The number k is taken modulo n i e inverse function of permtonun The library syntax is numtoperm n k where n is a long 3 2 38 padicprec zx p absolute p adic precision of the object x This is the minimum precision of the components of x The result is VERYBIGINT 2 1 for 32 bit machines or 263 1 for 64 bit machines if x is an exact object The library syntax is padicprec z p and the result is a long integer 3 2 39 permtonum given a permutation x on n elements gives the number k such that x numtoperm n k i e inverse function of numtoperm The library syntax is permtonum z 3 2 40 precision z n gives the precision in decimal digits of the PARI object x If x is an exact object the largest single precision integer is returned If n is not omitted creates a new object equal to x with a new precision n This is to be understood as follows For exact types no change For x a vector or a matri
92. is a function usually associated to basic arithmetic operations whose name contains only non alphanumeric characters In practice most of these are simple functions which take arguments and return a value assignment operators also have side effects Each of these has some fixed and unchangeable priority which means that in a given expression the operations with the highest priority will be performed first Operations at the same priority level will always be performed in the order they were written i e from left to right Anything enclosed between parenthesis is considered a complete subexpression and will be resolved independently of the surrounding context For instance assuming that op op2 0p3 are standard binary operators with increasing priorities think of for instance T Op Y 0P2 Z 0P2 T 0P3 Y is equivalent to op y Opa 2 opa x ops y GP knows quite a lot of different operators some of them unary having only one argument some binary plus special selection operators Unary operators are defined for either prefix pre ceding their single argument op x or postfix following the argument x op position never both some are syntactically correct in both positions but with different meanings Binary operators all use the syntax x op y Most of them are well known some are borrowed from C syntax and a few are specific to GP Beware that some GP operators may differ slightly from their C counterparts 2
93. is an exact zero so it is almost always faster to test for pointer equality first and call isexactzero or gemp0 only when the first test fails int gcmpl GEN x returns 1 true if x is equal to 1 0 false otherwise int gcmp _1 GEN x returns 1 true if x is equal to 1 0 false otherwise long gcmp GEN x GEN y comparison of x with y returns the sign of x y long gempsg long s GEN x comparison of the long s with x long gcmpgs GEN x long s comparison of x with the long s long lexcmp GEN x GEN y comparison of x with y for the lexicographic ordering long gegal GEN x GEN y returns 1 true if x is equal to y 0 otherwise long gegalsg long s GEN x returns 1 true if the long s is equal to x 0 otherwise long gegalgs GEN x long s returns 1 true if x is equal to the long s 0 otherwise long iscomplex GEN x returns 1 true if x is a complex number of component types embed dable into the reals but is not itself real 0 if x is a real not necessarily of type t_REAL or raises an error if x is not embeddable into the complex numbers long ismonome GEN x returns 1 true if x is a non zero monomial in its main variable 0 oth erwise long ggval GEN x GEN p returns the greatest exponent e such that p divides x when this makes sense long gval GEN x long v returns the highest power of the variable number v dividing the poly nomial x int pvaluation GEN x GEN p GEN r applied to non zero integ
94. is created and returned In the generic case the remainder is created after the quotient and can be disposed of individually with a cgiv r The remainder is always of the sign of the dividend x GEN truedvmdii GEN x GEN y GEN r as dvmdii but with a non negative remainder void mpdvmdz GEN x GEN y GEN z GEN r assigns the Euclidean quotient of the integers x and y into the integer or real z putting the remainder into r unless r is equal to NULL or ONLY_REM as above void dvmdssz long s long t GEN z GEN r assigns the Euclidean quotient of the longs s and t into the integer or real z putting the remainder into r unless r is equal to NULL or ONLY_REM as above void dvmdsiz long s GEN x GEN z GEN r assigns the Euclidean quotient of the long s and the integer x into the integer or real z putting the remainder into r unless r is equal to NULL or ONLY_REM as above void dvmdisz GEN x long s GEN z GEN r assigns the Euclidean quotient of the integer x and the long s into the integer or real z putting the remainder into r unless r is equal to NULL or ONLY_REM as above void dvmdiiz GEN x GEN y GEN z GEN r assigns the Euclidean quotient of the integers x and y into the integer or real z putting the address of the remainder into r unless r is equal to NULL or ONLY_REM as above void diviiexact GEN x GEN y returns the Euclidean quotient x y assuming y divides x Uses Jebelean algorithm Jebelean Krand
95. is known to a small accuracy it is trivial to compute it to very high accuracy see the tutorial The library syntax is quadclassunit0 D flag tech Also available are buchimag D c1 c2 and buchreal D flag c1 c2 3 4 54 quaddisc 1 discriminant of the quadratic field Q z where x Q The library syntax is quaddisc z 3 4 55 quadhilbert D flag 0 relative equation defining the Hilbert class field of the quadratic field of discriminant D If flag is non zero and D lt 0 outputs form root form to be used for constructing subfields If flag is non zero and D gt 0 try hard to get the best modulus Uses complex multiplication in the imaginary case and Stark units in the real case The library syntax is quadhilbert D flag prec 3 4 56 quadgen D creates the quadratic number w a VD 2 where a 0 if x Omod 4 a 1 if D 1mod4 so that 1 w is an integral basis for the quadratic order of discriminant D D must be an integer congruent to 0 or 1 modulo 4 which is not a square The library syntax is quadgen z 87 3 4 57 quadpoly D v x creates the canonical quadratic polynomial in the variable v corresponding to the discriminant D i e the minimal polynomial of quadgen D D must be an integer congruent to 0 or 1 modulo 4 which is not a square The library syntax is quadpoly0 z v 3 4 58 quadray D f flag 0 relative equation for the ray class field of conductor f for the quadratic
96. it using long delete_var which deletes the latest temporary variable created and returns the variable number of the previous one or simply returns 0 if you try in vain to delete MAXVARN Of course you should make sure that the deleted variable does not appear anywhere in the objects you use later on Here is an example long first fetch_var long ni fetch_var long n2 fetch_var prepare three variables for internal use delete all variables before leaving do num delete_var while num amp amp num lt first The dangerous statement while delete_var empty removes all temporary variables that were in use except MAXVARN which cannot be deleted 192 4 7 Input and output Two important aspects have not yet been explained which are specific to library mode input and output of PARI objects 4 7 1 Input For input PARI provides you with two powerful high level functions which enables you to input your objects as if you were under GP In fact the second one is essentially the GP syntactical parser hence you can use it not only for input but for most computations that you can do under GP These functions are called flisexpr and flisseq The first one has the following syntax GEN flisexpr char s Its effect is to analyze the input string s and to compute the result as in GP However it is limited to one expression If you want to read and evaluate a sequence of expres
97. its output uses a lot of memory space and coping with the index shift is awkward The library syntax is bernvec z 3 3 17 besselh1 nu x H Bessel function of index nu and argument x The library syntax is hbessel1 nu x prec 3 3 18 besselh2 nu x H Bessel function of index nu and argument z The library syntax is hbessel2 nu x prec 3 3 19 besseli nu x I Bessel function of index nu and argument x If x converts to a power series the initial factor 1 2 TD v 1 is omitted since it cannot be represented in PARI when v is not integral The library syntax is ibessel nu x prec 3 3 20 besselj nu x J Bessel function of index nu and argument x If x converts to a power series the initial factor 1 2 D v 1 is omitted since it cannot be represented in PARI when v is not integral The library syntax is ibessel nu x prec 3 3 21 besseljh n x J Bessel function of half integral index More precisely besseljh n x computes Jn 1 2 x where n must be of type integer and x is any element of C In the present version 2 2 7 this function is not very accurate when x is small The library syntax is jbesselh n x prec 3 3 22 besselk nu x flag 0 K Bessel function of index nu which can be complex and argument x Only real and positive arguments x are allowed in the present version 2 2 7 If flag is equal to 1 uses another implementation of this function which is often faster The library synt
98. itself contains some amount of imprecision 1 2 5 Strings These contain objects just as they would be printed by the GP calculator 1 2 6 Notes 1 2 6 1 Exact and imprecise objects we have already said that integers and reals are called the leaves because they are ultimately at the end of every branch of a tree representing a PARI object Another important notion is that of an exact object by definition numbers of basic type real p adic or power series are imprecise and we will say that a PARI object having one of these imprecise types anywhere in its tree is not exact All other PARI objects will be called exact This is a very important notion since no numerical analysis is involved when dealing with exact objects 1 2 6 2 Scalar types the first nine basic types from t_INT to t_POLMOD will be called scalar types because they essentially occur as coefficients of other more complicated objects Note that type t_POLMOD is used to define algebraic extensions of a base ring and as such is a scalar type 1 2 6 3 What is zero This is a crucial question in all computer systems The answer we give in PARI is the following For exact types all zeros are equivalent and are exact and thus are usually represented as an integer zero The problem becomes non trivial for imprecise types For p adics the answer is as follows every p adic number including 0 has an exponent e and a mantissa a purist would say a significand u which is a
99. last expression of the sequence When using this instruction it is useful to prompt for the string by using the print1 function Note that in the present version 2 19 of pari el when using GP under GNU Emacs see Section 2 10 one must prompt for the string with a string which ends with the same prompt as any of the previous ones a will do for instance 3 11 2 13 install name code gpname lib loads from dynamic library lib the function name Assigns to it the name gpname in this GP session with argument code code see Section 4 9 2 for an explanation of those If lib is omitted uses libpari so If gpname is omitted uses name This function is useful for adding custom functions to the GP interpreter or picking useful functions from unrelated libraries For instance it makes the function system obsolete install system vs sys libc so sys ls gp gp c gp h gp_rl c But it also gives you access to all non static functions defined in the PARI library For instance the function GEN addii GEN x GEN y adds two PARI integers and is not directly accessible under GP it s eventually called by the operator of course install addii GG addii 1 2 hi 3 Re installing a function will print a Warning and update the prototype code if needed but will reload a symbol from the library even it the latter has been recompiled 166 Caution This function may not work on all systems especially when
100. lt TAB gt where lt TAB gt is the TAB key If there exists a unique command starting with the letters you have typed the command name will be completed If not 49 either the list of commands starting with the letters you typed will be displayed in a separate window which you can then kill by typing as usual C x 1 or by typing in more letters or no match found will be displayed in the Emacs command line If your GP was linked with the readline library read the section on completion in the section below the paragraph on online help is not relevant Note that if for some reason the session crashes due to a bug in your program or in the PARI system you will usually stay under Emacs but the GP buffer will be killed To recover it simply type again M x gp or C u M x gp and a new session of GP will be started after the old one so you can recover what you have typed Note that this will of course not work if for some reason you kill Emacs and start a new session You also have at your disposal a few other commands and many possible customizations colours prompt Read the file emacs pariemacs txt in standard distribution for details 2 11 Using GP with readline Thanks to the initial help of Ilya Zakharevich there is a possibility of line editing and command name completion outside of an Emacs buffer if you have compiled GP with the GNU readline library If you don t have Emacs available or can t stand using it we reall
101. m returns the smallest positive representative of x modulo m ulong powusmod ulong x long n ulong m as powuumod but n is allowed to be negative in which case it is assumed that x is invertible modulo m otherwise 0 is returned 5 2 Level 1 kernel operations on longs integers and reals In this section as elsewhere long denotes a BIL bit signed C integer integer denotes a PARI multiprecise integer type t_INT real denotes a PARI multiprecise real type t_REAL Refer to Chapters 1 2 and 4 for general background 208 Note Many functions consist of an elementary operation immediately followed by an assignment statement All such functions are obtained using macros see the file paricom h hence you can easily extend the list Below they will be introduced like in the following example GEN gadd z GEN x GEN y GEN z followed by the explicit description of the function GEN gadd GEN x GEN y which creates its result on the stack returning a GEN pointer to it and the parts in brackets indicate that there exists also a function void gaddz GEN x GEN y GEN z which assigns its result to the pre existing object z leaving the stack unchanged 5 2 1 Basic unit and subunit handling functions long typ GEN x returns the type number of x The header files included through pari h will give you access to the symbolic constants t_INT etc so you should never need to know the actual numerical values
102. next line if the current one is empty A sample gprc file called misc gprc dft is provided in the standard distribution It is a good idea to have a look at it and customize it to your needs Since this file does not use multiline constructs here is one note the terminating to separate the expressions if VERSION gt 2 2 3 48 read my_scripts syntax errors in older versions new_galois_format 1 default introduced in 2 2 4 if EMACS colors 9 5 no no 4 1 2 help gphelp detex ch 4 cb 0 cu 2 2 10 Using GP under GNU Emacs If GNU Emacs is installed on your machine it is possible to use GP as a subprocess in Emacs To use this you should include in your emacs file the following lines autoload gp mode pari nil t gp Pp autoload gp script mode pari nil t gp P p autoload gp pari nil t autoload gpman pari nil t setq auto mode alist cons gp gp script mode auto mode alist which autoloads functions from pari el See also pariemacs txt These files are included in the PARI distribution and are installed at the same time as GP Once this is done under GNU Emacs if you type M x gp where as usual M is the Meta key i e Escape or on SUN keyboards the Left key a special shell will be started which in particular launches GP with the default stack size prime limit and input buffer size If you type instead C u M x gp you will be asked for the na
103. not prefix operator amp amp and operator or operator exist giving as results 1 true or 0 false Note that and are also accepted as synonyms respectively for amp amp and However there is no bitwise and or or 1 3 3 Conversions and similar functions Many conversion functions are available to convert between different types For example floor ceiling rounding truncation etc Other simple functions are included like real and imaginary part conjugation norm absolute value changing precision or creating an integermod or a polmod 1 3 4 Transcendental functions They usually operate on any object in C and some also on p adics The list is everexpanding and of course contains all the elementary functions plus already a number of others Recall that by extension PARI usually allows a transcendental function to operate componentwise on vectors or matrices 1 3 5 Arithmetic functions Apart from a few like the factorial function or the Fibonacci numbers these are functions which explicitly use the prime factor decomposition of integers The standard functions are included In the present version 2 2 7 a primitive but useful version of Lenstra s Elliptic Curve Method ECM has been implemented There is now a very large package which enables the number theorist to work with ease in alge braic number fields All the usual operations on elements ideals prime ideals etc are available More soph
104. of gal pol see Section 2 6 2 Example G galoisinit x 4 1 galoisfixedfield G G group 2 2 x72 2 Mod x 3 x x74 1 x 2 y x 1 x72 y x 1 computes the factorization zt 1 x y 2x 1 x y 2x 1 The library syntax is galoisfixedfield gal perm flag v where v is a variable number an omitted v being coded by 1 3 6 34 galoisidentify gal gal being be a Galois field as output by galoisinit output the isomorphism class of the underlying abstract group as a two components vector o i where o is the group order and 7 is the group index in the GAP4 Small Group library by Hans Ulrich Besche Bettina Eick and Eamonn O Brien The current implementation is limited to degree less or equal to 127 Some larger easy orders are also supported The output is similar to the output of the function IdGroup in GAP4 Note that GAP4 IdGroup handles all groups of order less than 2000 except 1024 so you can use galoisexport and GAP4 to identify large Galois groups The library syntax is galoisidentif y gal 109 3 6 35 galoisinit pol den computes the Galois group and all necessary information for com puting the fixed fields of the Galois extension K Q where K is the number field defined by pol monic irreducible polynomial in Z X or a number field as output by nfinit The extension K Q must be Galois with Galois group weakly super solvable see nf galoisconj The output is a
105. of the polynomials always defines Q hence is equal to x 1 where n is the degree and another always defines the same order as x if x is irreducible The library syntax is ordred z 3 6 110 poltschirnhaus x applies a random Tschirnhausen transformation to the polynomial x which is assumed to be non constant and separable so as to obtain a new equation for the tale algebra defined by x This is for instance useful when computing resolvents hence is used by the polgalois function The library syntax is tschirnhaus z 3 6 111 rnfalgtobasis rnf x expresses x on the relative integral basis Here rnf is a relative number field extension L K as output by rnfinit and x an element of L in absolute form i e expressed as a polynomial or polmod with polmod coefficients not on the relative integral basis The library syntax is rnfalgtobasis rnf x 3 6 112 rnfbasis bnf L gives either a true bnf basis of Zz if it exists or an n 1 element generating set of L if not where n is the rank of L over bnf Here bnf is as output by bnfinit L is either a polynomial with coefficients in bnf defining a relative extension of bnf or a pseudo basis of such an extension as ouput by rnfpseudobasis The library syntax is rnfbasis bnf x 3 6 113 rnfbasistoalg rnf x computes the representation of x as a polmod with polmods coeffi cients Here rnf is a relative number field extension L K as output by rnfinit and x an element of
106. op to the long s and the real x GEN opis z GEN x long s GEN z applies op to the integer x and the long s GEN opiilz GEN x GEN y GEN z applies op to the integers x and y GEN opir z GEN x GEN y GEN z applies op to the integer x and the real y GEN oprs z GEN x long s GEN z applies op to the real x and the long s GEN opri z GEN x GEN y GEN z applies op to the real x and the integer y GEN oprr z GEN x GEN y GEN z applies op to the reals x and y Each of the above can be used with the following operators op add addition x y The result is real unless both x and y are integers or longs op sub subtraction x y The result is real unless both x and y are integers or longs op mul multiplication x y The result is real unless both x and y are integers or longs OR if x or y is the integer or long zero op div division x y In the case where x and y are both integers or longs the result is the Euclidean quotient where the remainder has the same sign as the dividend x If one of x or y is real the result is real unless x is the integer or long zero A division by zero error occurs if y is equal to zero op res remainder x y This operation is defined only when x and y are longs or integers The result is the Euclidean remainder corresponding to div i e its sign is that of the dividend x The result is always an integer op mod remainder x y This operation is
107. optional one flag whose default value is 0 The should not be typed it is just a convenient notation we will use throughout to denote optional arguments That is you can type foo x 2 or foo x which is then understood to mean foo x 0 As well a comma or closing parenthesis where an optional argument should have been signals to GP it should use the default Thus the syntax foo x is also accepted as a synonym for our last expression When a function has more than one optional argument the argument list is filled with user supplied values in order When none are left the defaults are used instead Thus assuming that foo s prototype had been foo x 1 y 2 2 3 typing in foo 6 4 would give you foo 6 4 3 In the rare case when you want to set some far away argument and leave the defaults in between as they stand you can use the empty arg trick alluded to above foo 6 1 would yield foo 6 2 1 By the way foo by itself yields foo 1 2 3 as was to be expected In this rather special case of a function having no mandatory argument you can even omit the a standalone foo would be enough though we don t really recommend it for your scripts for the sake of clarity In defining GP syntax we strove to put optional arguments at the end of the argument list of course since they would not make sense otherwise and in order of decreasing usefulness so that most of the time you will be able to ignore them
108. pari_timer T char format are equivalent to timer and msgtimer respectively except they use a unique timer T containing all the information needed so that no other function can mess with your timings They are used as follows pari_timer T void TIMER amp T initialize timer printf Total time ld n TIMER amp T or pari_timer T long i 196 GEN L void TIMER amp T initialize timer for i 1 i lt 10 i msgTIMER amp T for i hld L i 4Z i L il 4 8 A complete program Now that the preliminaries are out of the way the best way to learn how to use the library mode is to work through a detailed non trivial example of a main program We will write a program which computes the exponential of a square matrix x The complete listing is given in Appendix B but each part of the program will be produced and explained here We will use an algorithm which is not optimal but is not far from the one used for the PARI function gexp in fact embodied in the function mpexp1 This consists in calculating the sum of the series er 2 5 a k 0 for a suitable positive integer n and then computing e by repeated squarings First we will need to compute the L norm of the matrix z i e the quantity z lell2 yd 2 We will then choose the integer n such that the L norm of x 2 is less than or equal to 1 ice n In z In 2 if z gt 1 and n 0 otherw
109. recursion if n dive n 1 There s no way to increase the recursion limit which may be different on your machine from within since it would simply crash the GP process To increase it before launching GP you can use ulimit or limit depending on your shell to raise the process available stack space increase stacksize 38 Function which take functions as parameters This is easy in GP using the following trick neat example due to Bill Daly calc f x eval Str f GIO If you call this with calc sin 1 it will return sin 1 evaluated Restrictions on variable use it is not allowed to use the same variable name for different parameters of your function Or to use a given variable both as a formal parameter and a local variable in a given function Hence f x x 1 Hook user function f variable x declared twice Also the statement global x y z t see Section 3 11 2 11 declares the corresponding variables to be global It is then forbidden to use them as formal parameters or loop indexes as described above since the parameter would shadow the variable Implementation note For the curious reader here is how these stacks are handled a hashing function is computed from the identifier and used as an index in hashtable a table of pointers Each of these pointers begins a linked list of structures type entree The linked list is searched linearly for the identifier each list will typically have less
110. remainder into hiremainder An error occurs if the quotient cannot be represented by a ulong i e if hiremainder gt y initially The following routines are not part of the level 0 kernel per se but implement modular opera tions on words in terms of the above They are written so that no overflow may occur Let m gt 1 be the modulus all operands representing classes modulo m are assumed to belong to 0 m 1 the result may be wrong for a number of reasons otherwise it may not be reduced overflow can occur etc ulong adduumod ulong x ulong y ulong m returns the smallest positive representative of x y modulo m ulong subuumod ulong x ulong y ulong m returns the smallest positive representative of x y modulo m ulong muluumod ulong x ulong y ulong m returns the smallest positive representative of xy modulo m ulong invumod ulong x ulong m returns the smallest positive representative of c modulo m If x is not invertible mod m return 0 long invsmod long x long m returns the smallest positive representative of x7 modulo m If x is not invertible mod m return 0 In this routine no specific assumptions are made about the size or sign of x m is still assumed to be positive Consequently it is a little slower than invumod ulong divuumod ulong x ulong y ulong m returns the smallest positive representative of xy modulo m Ix y is not invertible mod m return 0 ulong powuumod ulong x ulong n ulong
111. solve discrete logarithms using ideallog you have to choose flag 2 The library syntax is idealstar0 nf J flag 3 6 62 idealtwoelt nf x a computes a two element representation of the ideal x in the number field nf using a straightforward exponential time search x can be an ideal in any form including perhaps an Archimedean part which is ignored and the result is a row vector a a with two components such that x aZkx aZx and a Z where a is the one passed as argument if any If x is given by at least two generators a is chosen to be the positive generator of z N Z Note that when an explicit a is given we use an asymptotically faster method however in practice it is usually slower The library syntax is ideal_two_elt0 nf x a where an omitted a is entered as NULL 3 6 63 idealval nf x vp gives the valuation of the ideal x at the prime ideal vp in the number field nf where vp must be a 5 component vector as given by idealprimedec The library syntax is idealval nf x vp and the result is a long integer 3 6 64 ideleprincipal nf x creates the principal idele generated by the algebraic number x which must be of type integer rational or polmod in the number field nf The result is a two component vector the first being a one column matrix representing the corresponding principal ideal and the second being the vector with r r2 components giving the complex logarithmic embedding of x The library
112. syntax is garg 2 prec 3 3 10 asin x principal branch of sin x ie such that Re asin 1 2 7 2 Ife e R and x gt 1 then asin x is complex The library syntax is gasin z prec 3 3 11 asinh x principal branch of sinh x i e such that Im asinh x 7 2 7 2 The library syntax is gash z prec 3 3 12 atan z principal branch of tan x i e such that Re atan x 7 2 7 2 The library syntax is gatan z prec 3 3 13 atanh x principal branch of tanh x ie such that Im atanh 7 2 7 2 If x R and z gt 1 then atanh x is complex The library syntax is gath x prec 70 3 3 14 bernfrac x Bernoulli number B where Bo 1 By 1 2 B2 1 6 expressed as a rational number The argument x should be of type integer The library syntax is bernfrac z 3 3 15 bernreal x Bernoulli number B as bernfrac but B is returned as a real number with the current precision The library syntax is bernreal z prec 3 3 16 bernvec creates a vector containing as rational numbers the Bernoulli numbers Bo Bo Baz This routine is obsolete Use bernfrac instead each time you need a Bernoulli number in exact form Note this routine is implemented using repeated independant calls to bernfrac which is faster than the standard recursion in exact arithmetic It is only kept for backward compatibility it is not faster than individual calls to bernfrac
113. syntax is principalidele nf x 3 6 65 matalgtobasis nf x nf being a number field in nfinit format and x a matrix whose coefficients are expressed as polmods in nf transforms this matrix into a matrix whose coefficients are expressed on the integral basis of nf This is the same as applying nfalgtobasis to each entry but it would be dangerous to use the same name The library syntax is matalgtobasis nf 1 3 6 66 matbasistoalg nf x nf being a number field in nfinit format and x a matrix whose coefficients are expressed as column vectors on the integral basis of nf transforms this matrix into a matrix whose coefficients are algebraic numbers expressed as polmods This is the same as applying nfbasistoalg to each entry but it would be dangerous to use the same name The library syntax is matbasistoalg nf x 3 6 67 modreverse a a being a polmod A X modulo T X finds the reverse polmod B X modulo Q X where Q is the minimal polynomial of a which must be equal to the degree of T and such that if 0 is a root of T then 9 B a for a certain root a of Q This is very useful when one changes the generating element in algebraic extensions The library syntax is polmodrecip z 116 3 6 68 newtonpoly z p gives the vector of the slopes of the Newton polygon of the polynomial x with respect to the prime number p The n components of the vector are in decreasing order where n is equal to the degree of x Vertical slopes o
114. than 7 components or so When the correct entree is found it points to the top of the stack of values for that identifier if it is a variable to the function itself if it is a predefined function and to a copy of the text of the function if it is a user defined function When an error occurs all of this maze rather a tree in fact is searched and hopefully restored to the state preceding the last call of the main evaluator Note The above syntax using the local keyword was introduced in version 2 0 13 The old syntax name list of true formal variables list of local variables seq is still recognized but is deprecated since genuine arguments and local variables become undistin guishable 2 6 5 Member functions Member functions use the dot notation to retrieve information from complicated structures by default types e11 nf bnf bnr and prime ideals The syntax structure member is taken to mean retrieve member from structure e g ell j returns the j invariant of the elliptic curve e11 or outputs an error message if e11 doesn t have the correct type To define your own member functions use the syntax structure member function text where function text is written as the seg in a standard user function without local variables whose only argument would be structure For instance the current implementation of the e11 type is simply an horizontal vector the j invariant being the thirteenth component This could
115. the S unit group modulo the unit group v 2 contains technical data needed by bnfissunit v 3 is an empty vector used to give the logarithmic embeddings of the generators in v 1 in version 2 0 16 v 4 is the S regulator this is the product of the regulator the determinant of v 2 and the natural logarithms of the norms of the ideals in S v 5 gives the S class group structure in the usual format a row vector whose three components give in order the S class number the cyclic components and the generators v 6 is a copy of S The library syntax is bnfsunit bnf S prec 3 6 16 bnfunit bnf bnf being a big number field as output by bnfinit outputs the vector of fundamental units of the number field This function is mostly useless since it will only succeed if bnf contains the units in which case bnf fu is recommanded instead or bnf was produced with bnfinit 2 which is itself deprecated The library syntax is buchfu bnf 3 6 17 bnrL1 bnr subgroup flag 0 bnr being the number field data which is output by bnrinit 1 and subgroup being a square matrix defining a congruence subgroup of the ray class group corresponding to bnr the trivial congruence subgroup if omitted returns for each character x of the ray class group which is trivial on this subgroup the value at s 1 or s 0 of the abelian L function associated to x For the value at s 0 the function returns in fact for each character x a
116. the file misc gprc dft or gprc dos if you re using GP EXE to HOME gprc Modify it to your liking For instance if you re not using an ANSI terminal remove control characters from the prompt variable You can also enable colors If desired read datadir misc gpalias from the gprc file which provides some common shortcuts to lengthy names fix the path in gprc first Unless you tampered with this via Configure datadir is prefix share pari If you have superuser privileges and want to provide system wide defaults copy your customized gprc file to etc gprc In older versions gphelp was hidden in pari lib directory and was not meant to be used from the shell prompt but not anymore If gp complains it cannot find gphelp check whether your gprc or the system wide gprc does contain explicit paths If so correct them according to the current misc gprc dft 4 Getting Started 4 1 Printable Documentation Building gp with make all also builds its documentation You can also type directly make doc In any case you need a working plain TX installation After that the doc directory contains various dvi files users dvi manual with a table of contents and an index tutorial dvi a short tutorial and refcard dvi a reference card for GP You can send these files to your favourite printer in the usual way probably via dvips The reference card is also provided as a PostScript document which may be easier to print than its dvi e
117. there is no solution Note that this functions return only one solution and not all the solutions The library syntax is qfbsolve Q n 3 4 53 quadclassunit D flag 0 tech Buchmann McCurley s sub exponential algo rithm for computing the class group of a quadratic order of discriminant D This function should be used instead of qfbclassno or quadregula when D lt 10 D gt 10 or when the structure is wanted If flag is non zero and D gt 0 computes the narrow class group and regulator instead of the ordinary or wide ones In the current version 2 2 7 this doesn t work at all use the general function bnfnarrow Optional parameter tech is a row vector of the form c c2 where c and cz are positive real numbers which control the execution time and the stack size To get maximum speed set ca c To get a rigorous result under GRH you must take cp 6 Reasonable values for c are between 0 1 and 2 The result of this function is a vector v with 3 components if D lt 0 and 4 otherwise The correspond respectively to e v 1 the class number e v 2 a vector giving the structure of the class group as a product of cyclic groups e v 3 a vector giving generators of those cyclic groups as binary quadratic forms e 4 omitted if D lt 0 the regulator computed to an accuracy which is the maximum of an internal accuracy determined by the program and the current default note that once the regulator
118. to assign an expression to the m th column of a matrix x use z m expr instead Similarly use z m expr to assign an expression to the m th row of x This process is recursive so if z is a matrix of matrices of an expression such as x 1 1 3 4 1 would be perfectly valid assuming of course that all matrices along the way have the correct dimensions Note We ll see in Section 2 6 4 that it is possible to restrict the use of a given variable by declaring it to be global or local This can be useful to enforce clean programming style but is in no way mandatory Not exactly since not all their arguments need be evaluated For instance it would be stupid to evaluate both branches of an if statement since only one will apply GP only expands this one An obvious but important exception are character strings which are evaluated essentially to themselves type t_STR Not exactly so though since we do some work to treat the quoted char acters correctly those preceded by a 33 Technical Note Each variable has a stack of values implemented as a linked list When a new scope is entered during a function call which uses it as a parameter or if the variable is used as a loop index see Section 2 6 4 and Section 3 11 the value of the actual parameter is pushed on the stack If the parameter is not supplied a special 0 value called gnil is pushed on the stack this value is not printed if it is returned as the resul
119. u v yields the empty 0x0 matrix The function Mat can be used to transform any object into a matrix see Chapter 3 Note that although the internal representation is essentially the same only the type number is different a row vector of column vectors is not a matrix for example multiplication will not work in the same way Note also that it is possible to create matrices by conversion of empty column vectors and concatenation or using the matrix function with a given positive number of columns each of which has zero rows It is not possible to create or represent matrices with zero columns and a nonzero number of rows 2 3 15 Lists type t_LIST lists cannot be input directly you have to use the function listcreate first then listput each time you want to append a new element but you can access the elements directly as with the vector types described above The function List can be used to transform row or column vectors into lists see Chapter 3 2 3 16 Strings type t_STR to enter a string just enclose it between double quotes like this this is a string The function Str can be used to transform any object into a string see Chapter 3 2 3 17 Small vectors type t_VECSMALL this is an internal type used to code in an efficient way vectors containing only small integers such as permutations Most GP functions will refuse to operate on these objects 2 4 GP operators Loosely speaking an operator
120. updated object q This means 1 we translate copy all the objects in the interval avma 1bot so that its right extremity abuts the address 1top Graphically bot avma lbot ltop top End of stack Start free memory garbage becomes bot avma ltop top End of stack 2 Start free memory where denote significant objects the unused part of the stack and the garbage we remove 180 2 The function then inspects all the PARI objects between avma and 1bot i e the ones that we want to keep and that have been translated and looks at every component of such an object which is not a codeword Each such component is a pointer to an object whose address is either between avma and 1bot in which case it will be suitably updated larger than or equal to 1top in which case it will not change or between 1bot and 1top in which case gerepile will scream an error message at you sig nificant pointers lost in gerepile 3 avma is updated we add 1top 1bot to the old value 4 We return the possibly updated object q if q initially pointed between avma and 1bot we return the translated address as in 2 If not the original address is still valid and we return it As stated above no component of the remaining objects in particular q should belong to the erased segment 1bot 1top and this is checked with
121. useless in library mode 3 8 10 matadjoint x adjoint matrix of x i e the matrix y of cofactors of x satisfying x y det x x Id x must be a non necessarily invertible square matrix The library syntax is adj x 3 8 11 matcompanion z the left companion matrix to the polynomial x The library syntax is assmat z 3 8 12 matdet x flag 0 determinant of x x must be a square matrix If flag 0 uses Gauss Bareiss If flag 1 uses classical Gaussian elimination which is better when the entries of the ma trix are reals or integers for example but usually much worse for more complicated entries like multivariate polynomials The library syntax is det x flag 0 and det2 x flag 1 3 8 13 matdetint x x being an m x n matrix with integer coefficients this function computes a multiple of the determinant of the lattice generated by the columns of zx if it is of rank m and returns zero otherwise This function can be useful in conjunction with the function mathnfmod which needs to know such a multiple To obtain the exact determinant assuming the rank is maximal you can compute matdet mathnfmod x matdetint x Note that as soon as one of the dimensions gets large m or n is larger than 20 say it will often be much faster to use mathnf x 1 or mathnf x 4 directly The library syntax is detint 3 8 14 matdiagonal x x being a vector creates the diagonal matrix whose diagonal entries are those of x
122. user defined functions see below and control statements It may be preceded by an assignment statement into a variable It always has a value which can be any PARI object Several expressions can be combined on a single line by separating them with semicolons and also with colons for those who are used to BASIC Such an expression sequence will be called simply a seg A seg also has a value which is the value of the last non empty expression in the sequence Under GP the value of the seg and only this last value is always put on the stack i e it will become the next object n The values of the other expressions in the seq are 35 discarded after the execution of the seq is complete except of course if they were assigned into variables In addition the value of the seg or of course of an expression if there is only one is printed if the line does not end with a semicolon 2 6 4 User defined functions It is very easy to define a new function under GP which can then be used like any other function The syntax is as follows name list of formal variables local list of local variables seq which looks better written on consecutive lines name zo zi local to ti local note that the first newline is disregarded due to the preceding sign and the others because of the enclosing braces Both lists of variables are comma separated and allowed to be empty
123. variables are unaffected We of course strongly encourage you to try and get used to the setting compatible 0 Note that the default new_galois_format is another compatibility setting which is completely independent of compatible 2 1 4 datapath default the location of installed precomputed data the name of directory containing the optional data files For now only the galdata package needed by polgalois in degrees 8 to 11 2 1 5 debug default 0 debugging level If it is non zero some extra messages may be printed some of it in French according to what is going on see g 2 1 6 debugfiles default 0 file usage debugging level If it is non zero GP will print information on file descriptors in use from PARI s point of view see gf 2 1 7 debugmem default 0 memory debugging level If it is non zero GP will regularly print information on memory usage If it s greater than 2 it will indicate any important garbage collecting and the function it is taking place in see gm Important Note As it noticeably slows down the performance and triggers bugs in some versions of a popular compiler the first functionality memory usage is disabled if you re not running a version compiled for debugging see Appendix A 2 1 8 echo default 0 this is a toggle which can be either 1 on or 0 off When echo mode is on each command is reprinted before being executed This can be useful when reading a file with the r o
124. vector r cy where r is the order of L s x at s 0 and cy the first non zero term in the expansion of L s x at s 0 in other words L s X ey sx O s x 1 near 0 flag is optional default value is 0 its binary digits mean 1 compute at s 1 if set to 1 or s 0 if set to 0 2 compute the primitive L functions associated to x if set to 0 or the L function with Euler factors at prime ideals dividing the modulus of bnr removed if set to 1 this is the so called Lg s x function where S is the set of infinite places of the number field together with the finite prime ideals dividing the modulus of bnr see the example below 3 returns also the character 104 Example bnf bnfinit x 2 229 bnr bnrinit bnf 1 1 bnrL1 bnr returns the order and the first non zero term of the abelian L functions L s x at s 0 where x runs through the characters of the class group of Q v229 Then bnr2 bnrinit bnf 2 1 bnrL1 bnr2 2 returns the order and the first non zero terms of the abelian L functions Lgs s x at s 0 where x runs through the characters of the class group of Q v229 and S is the set of infinite places of Q v229 together with the finite prime 2 note that the ray class group modulo 2 is in fact the class group so bnrL1 bnr2 0 returns exactly the same answer as bnrL1 bnr 0 The library syntax is bnrL1 bnr subgroup flag prec where an omitted subgroup is coded as NULL 3 6 18 bnrclass bnf
125. x prec 3 3 31 exp zx exponential of x p adic arguments with positive valuation are accepted The library syntax is gexp z prec 3 3 32 gammah x gamma function evaluated at the argument x 1 2 When z is an integer this is much faster than using gamma x 1 2 The library syntax is ggamd z prec 3 3 33 gamma x gamma function of x In the present version 2 2 7 the p adic gamma function is not implemented The library syntax is ggamma x prec 72 3 3 34 hyperu a b x U confluent hypergeometric function with parameters a and b The pa rameters a and b can be complex but the present implementation requires x to be positive The library syntax is hyperu a b x prec 3 3 35 incgam s x y incomplete gamma function x must be positive and s real The result returned is T e t 1 dt When y is given assume of course without checking that y T s For small x this will tremendously speed up the computation The library syntax is incgam s x prec and incgam0 s x y prec respectively an omitted y is coded as NULL There exist also the functions incgam1 and incgam2 which are used for internal purposes 3 3 36 incgamc s x complementary incomplete gamma function The arguments s and x must be positive The result returned is ik ett9 1 dt when z is not too large The library syntax is incgamc s x prec 3 3 37 log z flag 0 principal branch of the natural logarithm of x i e such that I
126. zero outputs a 3 component row vector z a k where z is the absolute equation of L over Q as in the default behaviour a expresses as an element of L a root a of the polynomial defining the base field nf and k is a small integer such that 0 8 ka where 0 is a root of z and b a root of pol The main variable of nf must be of lower priority than that of pol see Section 2 6 2 Note that for efficiency this does not check whether the relative equation is irreducible over nf but only if it is squarefree If it is reducible but squarefree the result will be the absolute equation of the tale algebra defined by pol If pol is not squarefree an error message will be issued The library syntax is rnfequation0 nf pol flag 3 6 124 rnfhnfbasis bnf x given a big number field bnf as output by bnfinit and either a polynomial x with coefficients in bnf defining a relative extension L of bnf or a pseudo basis x of such an extension gives either a true bnf basis of L in upper triangular Hermite normal form if it exists and returns O otherwise The library syntax is rnfhermitebasis nf x 3 6 125 rnfidealabstorel rnf x let rnf be a relative number field extension L K as output by rnfinit and x an ideal of the absolute extension L Q given by a Z basis of elements of L Returns the relative pseudo matrix in HNF giving the ideal x considered as an ideal of the relative extension L K If x is an ideal in HNF form associated to an nf
127. 0 PSUD sf ae a ee a a a 54 psubes tzl ama ld eye et 218 gs bsgl j at gS was Bo Ge Ned Ged 218 ESUDSE ia AA ele 140 220 ESOS O sy see ese ws Gee Ae ee es 140 EsublZ Ma pa Sol args Ws 218 ESUIMdIVE oos Bee ae ee Sa eS 88 GUAM aoe eo ae ie ee dee a 75 PtH salen iw alae eel Oe al ee ia 75 gtodouble 179 215 PCO ONG vec ke a ded eer ah he Edis 179 216 BEOMAt usada Ar 58 ELOPO cca Sets he E A a 59 216 gtopolyrev 59 216 tOSer eii eek ee Sosa nS 60 216 ELOSOLE ore bon Oe ek aE 60 PLOVEC cuca he ae Pe SE aes 61 216 gtovecsmall gs sra toponi e a es 61 BULTACO ma Ge de Tk ee E E 151 trais g ae Sg ia a sa 148 BELUNC aae an e OE E E ER 67 218 GUTS hot aside ache ot cdot waceat e S deg Gr 171 SUCIOS ee Ee aa dies 179 Ovals dr O Goh eres aes 4 217 OVA goers a SY 189 190 209 216 BZOTO eS Sos Gee ae id a 171 199 EZETA y a ib dick seek 76 gotak ac Ges Alaa ads 135 BED yao bh A a a 24 BAP att e ee ae ia 170 H Hadamard product 140 hashing function 39 hashtable Litro eye i eke S 39 hbesselldl o oo ooo ooo 71 HBESSEL2 rico tei ee ic a 71 NELASSNO sews ky Gad Ewe Oran A AEE Ses 85 heaping ate aes BE cde eat 24 231 herl tik ke Soe ra Bei ved Ae tides 91 DEP ds eas Bed Sek as tak ee 18 Hermite normal form 80 96 97 113 115 121 123 134 144 145 163 HSA Ah Sd Se AOS ca ba oe a a eed Bae a 144 hexadecimal tree 194 DAA ir ete A a
128. 00 GP keeps a history of the last histsize results computed so far which you can recover using the notation see Section 2 2 4 When this number is exceeded the oldest values are erased Tampering with this default is the only way to get rid of the ones you don t need anymore 2 1 12 lines default 0 if set to a positive value GP prints at most that many lines from each result terminating the last line shown with if further material has been suppressed The various print commands see Section 3 11 2 are unaffected so you can always type print a or b to view the full result If the actual screen width cannot be determined a line is assumed to be 80 characters long 2 1 13 log default 0 this can be either 0 off or 1 2 3 on see below for the various modes When logging mode is turned on GP opens a log file whose exact name is determined by the logfile default Subsequently all the commands and results will be written to that file see 1 In case a file with this precise name already existed it will not be erased your data will be appended at the end The specific positive values of log have the following meaning 1 plain logfile 2 emit color codes to the logfile if colors is set 3 write TeX output to the logfile can be further customized using TeXstyle 2 1 14 logfile default pari log name of the log file to be used when the log toggle is on Environment and time expansion are performe
129. 2 0 14 this wasn t guaranteed to return all the embeddings hence was triggered by a special flag This is no more the case The library syntax is nfisincl z y flag 3 6 96 nfisisom x y as nfisincl but tests for isomorphism If either x or y is a number field a much faster algorithm will be used The library syntax is nfisisom z y flag 3 6 97 nfnewprec nf transforms the number field nf into the corresponding data using current usually larger precision This function works as expected if nf is in fact a bnf update bnf to current precision but may be quite slow many generators of principal ideals have to be computed The library syntax is nfnewprec nf prec 3 6 98 nfkermodpr nf a pr kernel of the matrix a in Zx pr where pr is in modpr format see nfmodprinit The library syntax is nfkermodpr nf a pr 3 6 99 nfmodprinit nf pr transforms the prime ideal pr into modpr format necessary for all operations modulo pr in the number field nf Returns a two component vector P a where P is the Hermite normal form of pr and a is an integral element congruent to 1 modulo pr and congruent to 0 modulo p pr Here p ZN pr and e is the absolute ramification index The library syntax is nfmodprinit nf pr 3 6 100 nfsubfields pol d 0 finds all subfields of degree d of the number field defined by the monic integral polynomial pol all subfields if d is null or omitted The result is a vector of subfield
130. 2 carita 121 galolScOnjA emirato aio 121 galoisexport 108 109 galoisfixedfield 109 163 galoisidentify 109 galoisinit 108 109 110 galoisisabelian 110 galoispermtopol 110 111 galoissubcyclo 107 111 139 163 galoissubfields 111 112 123 galoissubgroups 112 PAM i se Ak a WE EOS ia 72 Gammahy ease sce Pa Pe GO ee es 72 PANG sh siete we Ae ok ae a ek ed ed ee 57 garbage collecting 179 CATS oth eel Ke wl a ete ere de 70 gasta Yas gh Baas Wo ed Wer 70 pasional Tk He a el eee a 70 Patan 6 ts vey a Sie Peace 70 Pathe aa Se at it re ak 70 gass ie ol SREY a es de ae A 148 gaussmodulo 148 gaussmodulo2 148 BHSZOUP es oe hk hte ae A Aa 77 220 gbitands si ay ee Pe SAS Bess 61 EDI ut Marie ea di 62 gbitnegimply 62 BDITOT ea o DS 62 gbittestiani ad ATE 62 gbittest3 uta rd o a 62 BDITXOL er a e e 62 gboundcf 2 2 4 ena Sa sia 78 gcarrecomplet 83 gcarreparfait 83 god ars Sel tees 82 geil eo hs ee A aAa 62 217 CCL A A AE a E 78 PCED a ae AO ate a eas ets top ed 78 BEA adi ate See lec alee oe 71 ELONE oo ais Ve eh ait be 178 179 215 BEMP Chie rd sae pede amp 57 217 BEMPO ia WY phe hal a ek we 57 216 Pempls ees aoe Ss Sy ee a Eh k 57 216 SCMPLS Lo ek Ae Seba ok Se S 217 238 BCMPSP iio Se Bae ee a es 217 BOMp st Pd ace i a al
131. 4 217 STAC e eed ae 64 217 gead forsor we A a ay ee Be ee 72 SSAMMA everest 72 Bed tE A ty 82 220 PO ir a ia sia 57 ggprecision o 66 BSTANdOCP e ta ar 135 OSU dir a ae ecw le a ee a 57 COVA coat a bro 68 217 ghalf a A o a id Lotte 171 grelos a e Mee dias o o Wee 91 pnella ii a dr ek di 91 SI E A E Be eh ae 171 PIMA Lia a nd din 64 PINV eo aaa a a 217 PINVMOd Ss iros FB eS 220 gisfundamental 82 gisirreducible 137 BISPTAMe cantata ee eh a eed eS 83 gispseudoprime 83 gissquarefree 83 glambdak ni doe ae dk Ae oes 135 A Ce ae ak ee a 84 220 ClO at aed Bek Shoe cele ge SER Eo ae 57 glength snog a it ye Ma 64 glngamma 73 elobal a so platina ee a Dial we hee ee 39 global mit cat SB Aes Fos 33 165 globalreduction 91 CVOR ie ek Ate E ae 73 SLOP ACM aoe paimt ds wR ee dde amp 73 BIE tle la a Ge aa ea 57 OMAR os ce a o a 58 emaxgs z vos a4 espa bk aed Noe aes 218 gmaxsg z oes eh ae as Soe a 218 max A eas nos odai Poe eon ete bated 218 BMA ys gS Boe ss ee here Oe Dect 58 mings zi oscilar 218 eminsg 2 vesa tae ir aa io 218 gminlz ioe aren E eae SES 218 GMO o he RN A ese d 56 smodgei2 ite snp Ba fe ek sees 219 gmodsglZl erratas hee oa 219 BMOdUICP e saa kai rra aA 59 216 gmodulesS ke ae rd ee 216 EMOL O xtc ch he 3 ade i rs a aba ded A 59 216 BMOdUISS 0 2 05 rd 216 Smod 21 2 8 e yates aE ete 219
132. 4 Euler totient function 76 79 E le r ii ached tvs ev SORA OE Ao a 155 Euler 2 45 a e Bho de ated 31 33 69 Euler Maclaurin 76 eulerpha ns caai a Bale che ae 76 79 A aG p e bh eg Beate dod dete 60 136 exact object 2 0 0 0 2 002000 8 A 194 SRP E A Aa 72 XP ai A Ea 209 OXPO 2 es 187 189 209 expression sequence 35 EXPIeESSION Lot ey a a 35 tern wn es els ee aie ee es 20 42 165 external prettyprint 0 19 OXtCTACH ek at eel eee Se u Be a 152 F Pact cantor acs see Bao Ge de 81 factmod ns bo a ee Ree ee 82 factmod9 e adeant aa eb A tee Ba pals 81 FACTOR 0 5 dy Hay AI Ae aloes 79 80 Tacto Ult gk Seg ee a ede Beds 80 Tactorback 2 nse thet ee aa 80 factorbackO 80 PAacCtorcantor isis eR EA 81 factoredbase si 5 5 25 fede ee e 117 factoredpolred 126 FACtOrEL 2 ehh a 4 Shae aii 79 81 factorial ss s eee ee ee a 81 factorint oso aeaa aa ee es 79 81 factormod slasi aun kaa Wines 79 81 82 factornf os a a aiia 79 80 108 factorpadic 2 sew Be oe a a A 136 factorpadic4 vide as 136 fetch_user_var ciao 191 fetch var in aa 191 LIDIA TS dd e aa As a es e 82 FIDO Ls Bs a Me BAS 82 237 fibonacci 42 ee Mae ee a 82 field discriminant 117 filename 15 filtera god we garde ena ae ok bee de 193 fineke pohst dui fhe det ee als 150 finite field osa ae 26 fixed floating point format
133. 8 For instance GP s postfix returns the new value like the prefix of C and the binary shifts lt lt gt gt have a priority which is different from higher than that of their C counterparts When in doubt just surround everything by parentheses besides your code will probably be more legible Here is the complete list in order of decreasing priority binary unless mentioned otherwise e Priority 10 and unary postfix x assigns the value x 1 to x then returns the new value of x This corresponds to the C statement x there is no prefix operator in GP x does the same with x 1 e Priority 9 op where op is any simple binary operator i e a binary operator with no side effects i e one of those defined below which is not a boolean operator comparison or logical x op y assigns x op y to x and returns the new value of x not a reference to the variable x Thus an assignment cannot occur on the left hand side of another assignment e Priority 8 is the assignment operator The result of x y is the value of the expression y which is also assigned to the variable x This is not the equality test operator Beware that a statement like x 1 is always true i e non zero and sets x to 1 The right hand side of the assignment operator is evaluated before the left hand side If the left hand side cannot be modified raise an error e Priority 7 is the selection operator x i retur
134. 8 53 setsearch z y flag 0 searches if y belongs to the set x If it does and flag is zero or omitted returns the index j such that x j y otherwise returns 0 If flag is non zero returns the index j where y should be inserted and 0 if it already belongs to x this is meant to be used in conjunction with listinsert This function works also if x is a sorted list see listsort The library syntax is setsearch z y flag which returns a long integer 3 8 54 setunion z y union of the two sets x and y The library syntax is setunion z y 3 8 55 trace x this applies to quite general x If x is not a matrix it is equal to the sum of x and its conjugate except for polmods where it is the trace as an algebraic number For x a square matrix it is the ordinary trace If x is a non square matrix but not a vector an error occurs The library syntax is gtrace z 3 8 56 vecextract z y z extraction of components of the vector or matrix x according to y In case x is a matrix its components are as usual the columns of x The parameter y is a component specifier which is either an integer a string describing a range or a vector If y is an integer it is considered as a mask the binary bits of y are read from right to left but correspond to taking the components from left to right For example if y 13 1101 2 then the components 1 3 and 4 are extracted If y is a vector which must have integer entries these en
135. AD and t_QFI and 5 for type t_PADIC and t_QFR However for the sake of efficiency no checking is done in the function cgetg so disasters will occur if you give an incorrect length 175 Notes 1 The main use of this function is to prepare for later assignments see Section 4 3 2 Most of the time you will use GEN objects as they are created and returned by PARI functions In this case you do not need to use cgetg to create space to hold them 2 For the creation of leaves i e integers or reals which is very common GEN cgeti long length GEN cgetr long length should be used instead of cgetg length t_INT and cgetg length t_REAL respectively 3 The macros lgetg lgeti lgetr are predefined as long cgetg long cgeti long cgetr respectively 4 Finally there are two low level routines to allocate space on the PARI stack GEN new_chunk size_t n allocates a GEN with n components without filling the required code words This is the low level constructor underlying cgetg which calls new_chunk then sets the first code word It works by simply returning the address GEN avma n after checking that it is larger than GEN bot char stackmalloc size_t n allocates memory on the stack for n chars not n GENs This is faster than using malloc and easier to use in most situations when temporary storage is needed In particular there is no need to free individually all variables thus allocated a simple avma oldavma
136. E must be a long vector of the type given by ellinit with flag 1 If flag 0 this computation is done using sigma and theta functions and a trick due to J Silverman If flag 1 use Tate s 4 algorithm which is much slower E is assumed to be integral given by a minimal model The library syntax is ellheight0 E z flag prec The Archimedean contribution alone is given by the library function hell F z prec Also available are ghell E z prec flag 0 and ghell2 E z prec flag 1 3 5 12 ellheightmatrix F x x being a vector of points this function outputs the Gram matrix of x with respect to the N ron Tate height in other words the i j component of the matrix is equal to ellbil E x i x j The rank of this matrix at least in some approximate sense gives the rank of the set of points and if x is a basis of the Mordell Weil group of E its determinant is equal to the regulator of E Note that this matrix should be divided by 2 to be in accordance with certain normalizations E is assumed to be integral given by a minimal model The library syntax is mathell E x prec 3 5 13 ellinit E flag 0 computes some fixed data concerning the elliptic curve given by the five component vector E which will be essential for most further computations on the curve The result is a 19 component vector E called a long vector in this section shortened to 13 components medium vector if flag 1 Both contain the followi
137. ETR 86 Riemann zeta function 37 76 MAA Bees tse AA ee R 96 rnfalgtobasis 127 TOTDaSIS 44 eis Sk ae Ye BA 127 rnfbasistoalg 127 rnfcharpoly a ae a A eee e 127 rnfconductor 127 128 rnfdedekind 2 2 ee ne ee ad x 128 PHPASLs aie a a RA 128 e A Ak eS Me G 128 wmnfdiscf a 2 20 4 m dike Bo ae ee 128 rnfelementabstorel 128 rnfelementdown 128 rnfelementreltoabs 128 rnfelementup 129 rnfeltabstorel 128 rnfeltdown 2 s com co aa ona ee 128 rnfeltreltoabs 128 TNfeltup na bbe ee ee ee bs 129 rnfequation cuido aa 129 rnfequationO 129 rnfhermitebasis 129 rnfhnfbasis s m as oa a naoide aa G 129 rnfidealabstorel 129 rnfidealdown 129 rnfidealhermite 130 rnfidealhnf 129 rntfidealmil olor MO Ee ns 130 rnfidealnormabS 130 rnfidealnormrel 130 rnfidealreltoabs 130 rnfidealtwoelement 130 rnfidealtwoelt 130 rnfidealup 130 131 wnfinite2 ena wo ae Bowes 131 FOIOS seals he doe ke Ri a 132 rnfisfr e on Soe She A 132 rnfisnorm eri a oe aa 132 rnfisnorminit 132 yrnfkummer 132 133 134 rnflllgram oo 133 THINOFMBYOUP 2008 ea ee a A 133 rnfpolred cotorra bes 133 rnfpolredabs 133 rnfp
138. GP has been compiled statically In that case the first use of an installed function will provoke a Segmentation Fault i e a major internal blunder this should never happen with a dynamically linked executable Hence if you intend to use this function please check first on some harmless example such as the ones above that it works properly on your machine 3 11 2 14 kill s kills the present value of the variable alias or user defined function s The corresponding identifier can now be used to name any GP object variable or function This is the only way to replace a variable by a function having the same name or the other way round as in the following example f 1 1 1 f x 0 Hook unused characters f x 0 a ki11 f x 0 710 12 0 When you kill a variable all objects that used it become invalid You can still display them even though the killed variable will be printed in a funny way following the same convention as used by the library function fetch_var see Section 4 6 For example a2 1 Jra a 1 kill a hi A2 lt 1 gt 2 1 If you simply want to restore a variable to its undefined value monomial of degree one use the quote operator a a Predefined symbols x and GP function names cannot be killed 3 11 2 15 print str outputs its string arguments in raw format ending with a newline 3 11 2 16 print1 str outputs its string arguments in raw
139. L expressed on the relative integral basis The library syntax is rnfbasistoalg rnf x 3 6 114 rnfcharpoly nf 7T a v x characteristic polynomial of a over nf where a belongs to the algebra defined by T over nf i e nf X T Returns a polynomial in variable v x by default The library syntax is rnfcharpoly nf 7 a v where v is a variable number 127 3 6 115 rnfconductor bnf pol flag 0 given bnf as output by bnfinit and pol a relative polynomial defining an Abelian extension computes the class field theory conductor of this Abelian extension The result is a 3 component vector conductor rayclgp subgroup where conductor is the conductor of the extension given as a 2 component row vector fo foo rayclgp is the full ray class group corresponding to the conductor given as a 3 component vector h cyc gen as usual for a group and subgroup is a matrix in HNF defining the subgroup of the ray class group on the given generators gen If flag is non zero check that pol indeed defines an Abelian extension return 0 if it does not The library syntax is rnfconductor rnf pol flag 3 6 116 rnfdedekind nf pol pr given a number field nf as output by nfinit and a polynomial pol with coefficients in nf defining a relative extension L of nf evaluates the relative Dedekind criterion over the order defined by a root of pol for the prime ideal pr and outputs a 3 component vector as the result The first component is a flag
140. La 7 PSL3 2 168 1 5 Az 2520 1 6 S7 5040 1 7 Warning The method used is that of resolvent polynomials and is sensitive to the current preci sion The precision is updated internally but in very rare cases a wrong result may be returned if the initial precision was not sufficient The library syntax is galois x prec To enable the new format in library mode set the global variable new_galois_format to 1 3 6 107 polred z flag 0 fa finds polynomials with reasonably small coefficients defining subfields of the number field defined by x One of the polynomials always defines Q hence is equal to x 1 and another always defines the same number field as x if x is irreducible All x accepted by nfinit are also allowed here e g non monic polynomials nf bnf x Z_K_basis The following binary digits of flag are significant 1 possibly use a suborder of the maximal order The primes dividing the index of the order chosen are larger than primelimit or divide integers stored in the addprimes table 2 gives also elements The result is a two column matrix the first column giving the elements defining these subfields the second giving the corresponding minimal polynomials If fa is given it is assumed that it is the two column matrix of the factorization of the discriminant of the polynomial x The library syntax is polredO z flag fa where an omitted fa is coded by NULL Also available are pol
141. MPES La A A See 212 CMPST ia Bee EA edith 212 CMPSE ii Stee ge PP hg e 212 COS fhe dot So AOR es a eh a dis E 216 code words 28 4 63 cC diff os a ely Pe i td a 98 COCTEL Gre an PP a 63 173 215 coefs to Col ad ee ee a 177 COETSUTOSINE ai Eek 177 coefs to pol sas dee ar ees 177 COSTS to ve aca Ha a hee A a a 177 COLOS aia tata a hed ee 16 column vector 7 27 190 comparison operators 57 compatible isis a es 16 completlo ena we aes aoe a 50 complex number 7 8 25 188 COMPO oe ae ai HL eh et oh Sal Ges 63 215 COMPONENT Vox gk tds E 63 components 63 composition 85 86 COMPOSTTUM a a p a e a a A a 124 COMP AW e e as BG fat Bh ho T a a 85 COMIPIESS Hina ae 24 COMCAE ii a A 40 142 143 conductor unica E a ed 106 COMTE AA Rs 64 ODJ VEC La sd 64 Corada se a nls a Be OR A 90 CODTEN G ipea To a 35 77 218 CONT FAC ia Wks ed hs Bees 78 GONT TAaCO da ara eR ee es 78 contfracpnqgn 78 continued fraction 78 235 Control statements 162 conversions 4 179 GONVOL no 8 as Ao es eee 140 Coordeh ais ls er ee Ane 91 COPY bei o rt Soke bd 178 COTE as AA DA eee ta 78 COTO a E AA A 78 eop E e A ee 78 COBOAISC oh ace o Lat o a 78 CorediseO ie Wd a ds ta Ey 78 GOTEdISG2 Muse ad 78 COS es os E Pe 71 GOSH tia A at ee ad 71 Coban aa as ale 72 CPU Mee sa is ee 21 CICLO Ea k
142. Matpascal ss core sora rorida s neo 147 Matqpascal Sy o e Sa a 147 matrank iie nea Pee a A 147 matric ooi sss sl aA 147 ee ne E o 7 8 27 41 190 matrik Lei eda a ok lle RE A 147 MACEUR GZ estos e a 147 matrixgzO Dana a A ata Ss Wako 147 MatSiZE iio a da 147 MAESODE tt a Bas a eee bes 147 matsnfO ito ds a fe a 148 MatSolLVe xa a ae oe ae as 148 matsolvemod fn ee aie ale te a e 148 matsolvemodO 148 matsupplement 148 Mattranspose 148 MA 8 Aiea dk Soe at 58 Maxprime 0 171 231 MAXVARN ce avin Goa elie e enha eri 172 191 MEDDEFAULTPREC 174 member functions 39 89 98 MAD Ye oh kee ely oye a eee ed eee oes 58 Mindanao Ghote ee 4 eS 114 pinin Lo a os aed ha 150 DIAM a a Ge dla o Bowe pi 150 minimal model 91 93 minimal polynomial 141 MOG ar ce nile ema Boke oS ae ded Been 59 MOO is eased o oR ies 59 MOOD ci a a eS 187 MO ii hE Re A A A 187 MOd64 ts Scan ai ee Gers 187 MODE ae Fok dd 123 MOATEVETSE as a Aceh ed 116 MmOdUulargCd ns gaa a o ooo 82 mod ler tira a Rea Be 97 Moebius 76 83 84 MOCDIUS ceci ES a pad bd des 76 84 Mordell Weil group 92 94 npada s so we ae ak Se aba body ad 173 Patio es POS a eis 210 dl asthe ANG ee LEE BO ee el tes 231 MP CMP 5 fers ie iN ey veh cae ee Gets 212 mpdivis 0 0 00 0 00 215 MPAVMAZ wedi bos a Aba ae PB aE eae A 214
143. N x GEN y same as gmodulcp except that the modulus y is copied onto the heap and not onto the PARI stack long gexpo GEN x returns the binary exponent of x or the maximal binary exponent of the coefficients of x Returns HIGHEXPOBIT if x has no components or is an exact zero long gsigne GEN x returns the sign of x 1 0 or 1 when x is an integer real or irreducible or reducible fraction Raises an error for all other types long gvar GEN x returns the main variable of x If no component of x is a polynomial or power series this returns BIGINT int precision GEN x If x is of type t_REAL returns the precision of x the length of x in BIL bit words if x is not zero and a reasonable quantity obtained from the exponent of x if x is numerically equal to zero If x is of type t_COMPLEX returns the minimum of the precisions of the real and imaginary part Otherwise returns 0 which stands in fact for infinite precision long sizedigit GEN x returns 0 if x is exactly 0 Otherwise returns gexpo x multiplied by log 9 2 This gives a crude estimate for the maximal number of decimal digits of the components of x 216 5 3 2 Comparison operators and valuations int gcmpO GEN x returns 1 true if x is equal to 0 0 false otherwise int isexactzero GEN x returns 1 true if x is exactly equal to 0 0 false otherwise Note that many PARI functions will return a pointer to gzero when they are aware that the result they return
144. O GEN P long flag GEN data long prec This function is in fact coded in basemath buch2 c and will in this case be completely identical to the GP function bnfinit but GP does not need to know about this only that it can be found somewhere in the shared library libpari so Important note You see in this example that it is the function s responsibility to correctly interpret its operands data NULL is interpreted by the function as an empty vector Note that since NULL is never a valid GEN pointer this trick always enables you to distinguish between a default value and actual input the user could explicitly supply an empty vector Note If install is not available we have to add a file functions number fields ClassGroupInit containing the following Function ClassGroupInit Section number_fields C Name bnfinit0 Prototype GDO L DGp Help ClassGroupInit P flag 0 tech this routine does 205 206 Chapter 5 Technical Reference Guide for Low Level Functions In this chapter we give a description all public low level functions of the PARI system These essentially include functions for handling all the PARI types Higher level functions such as arithmetic or transcendental functions are described fully in Chapter 3 of this manual Many other undocumented functions can be found throughout the source code These private functions are more efficient than the library functions that call them but much sloppier on
145. P will issue a warning then transform your polynomial so that it becomes monic Instead of the normal result say res you then get a vector res Mod a Q where Mod a Q Mod X P gives the change of variables The numbers c and c2 are positive real numbers which control the execution time and the stack size To get maximum speed set c2 c To get a rigorous result under GRH you must take c2 12 or c2 6 in the quadratic case but then you should use the much faster function quadclassunit Reasonable values for c are between 0 1 and 2 The defaults are c c2 0 3 nrpid is the maximal number of small norm relations associated to each ideal in the factor base Set it to 0 to disable the search for small norm relations Otherwise reasonable values are between 4 and 20 The default is 4 Remarks Apart from the polynomial P you don t need to supply any of the technical parameters under the library you still need to send at least an empty vector cgetg 1 t_VEC However should you choose to set some of them they must be given in the requested order For example if you want to specify a given value of nrel you must give some values as well for c and c2 and provide a vector c c2 nrel Note also that you can use an nf instead of P which avoids recomputing the integral basis and analogous quantities 3 6 1 bnfcertify bnf bnf being a big number field as output by bnfinit or bnfclassunit checks whether the result is co
146. PLEX However the components i e the real and imaginary part of such a complex number can be of any type The only sensible ones are integers we are then in Z i rational numbers Q i real numbers R i C or even elements of Z nZ Z nZ i when this makes sense or p adic numbers when p 3 mod 4 Q i This feature must of course not be used too rashly for example you are in principle allowed to create objects which are complex numbers of complex numbers but don t expect PARI to make sensible use of such objects you will mainly get nonsense On the other hand one thing which is allowed is to have components of different but com patible types For example taking again complex numbers the real part could be of type integer and the imaginary part of type rational By compatible we mean types which can be freely mixed in operations like or x For example if the real part is of type real the imaginary part cannot be of type integermod integers modulo a given number n Let us now describe the types As explained above they are built recursively from basic types which are as follows We use the letter T to designate any type the symbolic names correspond to the internal representations of the types type t_INT Z Integers with arbitrary precision type t_REAL R Real numbers with arbitrary precision type t_INTMOD Z nZ Integermods integers modulo n type t_FRAC Q Rational numbers in irreducible fo
147. Pe 161 pseudo basis o o 97 pseudo matrig 6 97 PS TLO Mala dd ee 20 157 PSI sh panged GEG tam ah he a ded Beds 74 PSPLOtH sc a eee Se ee hy es 161 psplothraw 0 162 pvaluation 217 Pythons A ete bial ae bea 46 Q QTD as eta oes eas ead ja a 60 QE DOM L552 ai Caylee ae Nae e 60 qfbclassno 4 84 85 qfbclassnoO 0 0 85 Q DCOMPTAV s a a e o 85 qfbhclassno 85 QEDNUCOMP ios ata o JR eS 86 QEDNUPOW us oe a ee A 86 qfbpowraw 2 ee en 86 qfbprimeform 86 M prod se et ee Eek i aa 86 ALDIEAO e Posie is Bee eda ae Be 86 qfbsolve aoa e E o o 86 87 ateval gup wg soa ta ds E 136 qfgaussred oaao oo o 148 aE aa E 60 Af jacobi sen eat eek Reed ark Sed 149 O sor aaae te a ee a 141 149 TITIO ayo hani fave mye fh atte op BA nde 3 149 Gftliilgram 02 ale ek ae ele 149 geL ram ene a nk 3 mias Gp dk sede r 150 QEMINIM 26200 a Pak ae Soy aA des 150 GL MIDIMO 9 0 hk hh we a ae 150 qfperfection 150 QE sc Boas sla ce eS Ee ie 60 o Yate bist ee th ae eG 150 GELCPO is os Sha a Ke ee eed oS 150 TSIP al e Ae 150 quadclassunit 87 quadclassunitO 87 GUaddTSG ce ck 6 teks rn ack 87 quadgen aa Be ed Ge A 26 87 quadhilbert ii ea he aa 87 Quad pol ye ted kh ak aoe eh ices 87 quadpoly0 irnos Bok ek wie es 87 quadratic number 7 8 26 188 quadray
148. QM ache Blake E es BS 55 239 PMUIAN aaea a a eg ele 57 199 pmul 2n Z gt s ee ee we a i aA 219 pMULSS LAY co seg eS eke ake eee eed ed 219 gm lselZli rias an SAGs 219 A 2 ees oe ete hee bao ee A 219 ghenr se ds a ody hi MOM 57 LOSE Dita Gon Y a ey haa ey 54 BNI ya Se ak a ook a ee ee ad 33 NOUNS td a O ake SNe es 65 AAA a eae ie See 65 199 ENOT i satin te e Nr ww alae ea a 57 BOLS Se LO tt ta Ay id da 57 GP e ee ee iaa 13 BPHE LP S20 oo hah AR ae a ae aS 23 Epi e a ale it ibe 231 EPOLVaL se ee Ole a ae eS By 68 BPOW iani i we a EA ase 57 69 220 BPOWBS o hae ee a aces 220 BPC sd wi uk hos Es Yh ap ar eyes ace 13 15 19 GPRG a Se aa te ae a a ad A 47 EPEC a e ipai a Toana aaa A E 47 PPro e Ba See a Ap D a ed 66 A NS 74 greal os essay o eS ae eS e Eos 66 Bred st soe nE ee oe ae 188 189 Bred Zi seir wk e ee wee 218 grofie mio deg eh aw heel e 215 BOS sec oe each as We OW a Bae dy Se a 220 GRH ias 87 98 99 102 132 141 BYNItOl tte ee e A es 67 218 Bround Lu soe Sets acy has 67 217 gscalmat 0 145 200 gscalsmat 145 200 SD a Mh A city a made Me edie ey igs 74 ESBITE nce senha ne gach veces Pie RA 57 200 ESHITTI 5 coche wee e Eos Boa a Ee 57 PshiftlZ osant eek a o 219 gsigne 58 187 209 216 EM al hai eee al wl ete 74 CSAT ss eit ane eee A 2 55 74 217 PSQPU hii n E ae eek Bias Bae A A 74 ESATI ti a ATA eee ke 75 GOTR its bos is Wf ak aay ae nh gt ah dt 19
149. RA da 210 a AN 210 ALLL hate ee eee BS e aa 210 e eon i E a a a a a 210 affs eta et soe Bie Dae a Re ths 210 o A So Bo Sess 210 atisr Vos eho Ae dab oe ie 210 ALESZ 2 oils ooo eH le ee ea ee ated 210 aB a eh ee a Re A 70 O behead we Rien wg alee e 90 algdep aS si hae eh a 141 algdepos esti Bae Wg OM shige de ale dk 141 algebraic dependence 141 algtobasis a ese athe be Sed ean as 117 alias Lleida hs Eh RR LEO OA hd 42 164 allocatemem 19 164 allocatemoremem 165 198 alternating series 155 233 ONC ys 675 Ga ko ea 57 M AA a ai A 61 Mel a 2 2 2 oe AS oe ee Wie a a Soe 2 90 apelli wisn ah ce eee es oh me Shs 90 BPS PIs ro wees Ske a AEE eds Bick 90 APPTEOT 6 4k ad Barras a PS ae 136 APPL gero fice Soke eee bac A A 136 ARCA s a o ties Ben oe BS hs 89 AE wee Soe eh ace e bet 70 Artin L function 107 Artin root number 107 ASIN ss sadio i See here Sh ROR A De 70 AS VINE ira a he ey ecco tee SG 70 assignment 177 ASSMA Gs sumada oe rs beeen tN wh te Mae Go 144 atan mir A cheese cas e Abe Ore 70 atanh 068 eo 4 See ee Ro wee Ph Eads 70 automatic simplification 20 available commands 23 VME 12s Slt eased creado Wan Bea te 178 179 B backslash character 32 Dase A eatin hema A ee 117 DASE a ee Say eb eee A Es 117 basistoal pen eo ky ol ae Ye 117 Berlekamp 2 81 Deti g
150. The library syntax is factormod z p flag Also available are factmod z p which is equiv alent to factormod z p 0 and simplefactmod z p factormod z p 1 3 4 25 fibonacci x xt Fibonacci number The library syntax is fibo x x must be a long 3 4 26 ffinit p n v x computes a monic polynomial of degree n which is irreducible over F For instance if P ffinit 3 2 y you can represent elements in F32 as polmods modulo P Starting with version 2 2 3 this function use a fast variant of Adleman Lenstra algorithm and is much faster than in earlier versions The library syntax is ffinit p n v where v is a variable number 3 4 27 gced z y flag 0 creates the greatest common divisor of x and y x and y can be of quite general types for instance both rational numbers Vector matrix types are also accepted in which case the GCD is taken recursively on each component Note that for these types gcd is not commutative flag is obsolete and should not be used If y is omitted and zx is a vector return the gcd of all components of x The algorithm used is a naive Euclid except for the following inputs e integers use modified right shift binary plus minus variant e univariate polynomials with coeffients in the same number field in particular rational use modular gcd algorithm e general polynomials use the subresultant algorithm if coefficient explosion is likely exact non modular coefficients
151. This copies the whole structure of x into y so many conditions must be met for the assignment to be possible For instance it is allowed to assign an integer into a real number but the converse is forbidden For that you must use the truncation or rounding function of your choice see section 3 2 It can also happen that y is not large enough or does not have the proper tree structure to receive the object x As an extreme example assume y is the zero integer with length equal to 2 Then all assignments of a non zero integer into y will result in an error message since y is not large enough to accommodate a non zero integer In general common sense will tell you what is possible keeping in mind the PARI philosophy which says that if it makes sense it is legal For instance the assignment of an imprecise object into a precise one does not make sense However a change in precision of imprecise objects is allowed All functions ending in z such as gaddz see Section 4 2 2 implicitly use this function In fact what they exactly do is record avma see Section 4 4 perform the required operation gaffect the result to the last operand then restore the initial avma You can assign ordinary C long integers into a PARI object not necessarily of type t_INT Use the function gaffsg with the following syntax void gaffsg long s GEN y Note due to the requirements mentioned above it is usually a bad idea to use gaffect statements Two exc
152. This is a quick and dirty way to check if ideals are principal without computing a full bnf structure but it s not a necessary condition hence a non trivial result doesn t prove the ideal is non trivial in the class group Note that this is not the same as the LLL reduction of the lattice J since ideal operations are involved The library syntax is ideallllred nf x vdir prec where an omitted vdir is coded as NULL 3 6 61 idealstar nf flag 1 nf being a number field and J either and ideal in any form or a row vector whose first component is an ideal and whose second component is a row vector of r 0 or 1 outputs necessary data for computing in the group Zx 1 If flag 2 the result is a 5 component vector w w 1 is the ideal or module I itself w 2 is the structure of the group The other components are difficult to describe and are used only in conjunction with the function ideallog If flag 1 default as flag 2 but do not compute explicit generators for the cyclic components which saves time If flag 0 computes the structure of Zx I as a 3 component vector v v 1 is the order v 2 is the vector of SNF cyclic components and v 3 the corresponding generators When the row vector is explicitly included the non zero elements of this vector are considered as real embeddings of nf 115 in the order given by polroots i e in nf 6 nf roots and then J is a module with components at infinity To
153. User s Guide to PARI GP C Batut K Belabas D Bernardi H Cohen M Olivier Laboratoire A2X U M R 9936 du C N R S Universit Bordeaux I 351 Cours de la Lib ration 33405 TALENCE Cedex FRANCE e mail pari math u bordeaux fr Home Page http pari math u bordeaux fr Primary ftp site ftp pari math u bordeaux fr pub pari last updated 11 December 2003 for version 2 2 7 Copyright 2000 2003 The PARI Group Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies Permission is granted to copy and distribute modified versions or translations of this manual under the conditions for verbatim copying provided also that the entire resulting derived work is distributed under the terms of a permission notice identical to this one PARI GP is Copyright 2000 2003 The PARI Group PARI GP is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation It is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY WHATSOEVER Table of Contents Chapter 1 Overview of the PARI system 1 1 Introduction 1 2 The PARI types 1 3 Operations and functions Chapter 2 Specific Use of the GP Caleuletor 2 1 Defaults and output formats 2 2 Simple metacommands 2 3 Input formats for the PARI types 2
154. Weierstrass form x y represents the Weierstrass wp function and its derivative If z is in the lattice defining E over C the result is the point at infinity 0 The library syntax is pointell E z prec 3 6 Functions related to general number fields In this section can be found functions which are used almost exclusively for working in general number fields Other less specific functions can be found in the next section on polynomials Functions related to quadratic number fields can be found in the section Section 3 4 Arithmetic functions We shall use the following conventions e nf denotes a number field i e a 9 component vector in the format output by nfinit This contains the basic arithmetic data associated to the number field signature maximal order discriminant etc e bnf denotes a big number field i e a 10 component vector in the format output by bnfinit This contains nf and the deeper invariants of the field units class groups as well as a lot of technical data necessary for some complex functions like bnfisprincipal e bnr denotes a big ray number field i e some data structure output by bnrinit even more complicated than bnf corresponding to the ray class group structure of the field for some modulus e rnf denotes a relative number field see below e ideal can mean any of the following a Z basis in Hermite normal form HNF or not In this case x is a square matrix an idel
155. a 58 vecmin obier e254 as E Bare 58 Vecsmall sti i mora dhiana a ee de e 61 VOCSO hb aa sa 152 VMEGSOTGO ehh as Pet a o a 152 VOCbCUL ira a d hee Sha a 153 vecteursmall ares in gota tsia Sateen 153 VECUOR Sosa deve ce See ts Ae eee eee Dee S 8 Vetor iii ates ye Aa a SE ee 153 vectorsmall esla Ge a Wee amp A 153 VECTOR sirel alas de tw ao isl ws tele td 153 version number 24 NAS fais aes Seva wea eas Set ete A ee 50 VOTE eet ea So cota rane ee 194 VVECECUM Gert ace a 2s As Beats eS Les 153 W W got vas Se By aa Be ee oe Sad OD A A E 89 weber iaa a 75 Neber sree o a EA 76 Weierstrass g function 95 Weierstrass equation o oo 89 Weill curve so Sec Ha SRG ee oe 95 WOipell oon La wa ale de ee nd ia 95 WES poa Fy nha eee ire Gok iIa 76 WEL ck Ayan de r Bat Be A 76 WEA Ss caren ace ere te eee Mca et Kean 76 whatnow 42 169 UNELE a a e AN 164 Wales os dba a ta 90 Write la as 21 24 42 169 WE COLD aia Bec a a 169 WYILEDIN 0 A A 168 Writebin bestias wo aE ea es 169 WHItETEX ea ele eee Se Oe a 170 X s AE E ES A E E oe han Sat Maron 63 Slims aeii a a n a ee x 63 AS E E TE E E 63 MVA e a tad tees ra dd E 63 Z Zassenhaus 81 136 Zborent ida ass FA Yas 155 ZORO hha Ete I te Ne SE dh 94 DOOR YL ttt Ate ath ae ee Be od 8 DOLO ehh sale Hoe ho Be hel BRA ES ae gS 173 zeta function 37 DOLAR a oh Set Bob eae id pes eat 76 wetak oo Wise ges k
156. a PARI format is It is a character string similar to the one printf uses where characters have a special meaning It describes the format to use when printing the remaining operands But in addition to the standard format types you can use 4Z to denote a GEN object we would have liked to pick G but it was already in use For instance you could write err talker x d Z is not invertible i x i since the err function accepts PARI formats Here i is an int x a GEN which is not a leaf and this would insert in raw format the value of the GEN x i 4 7 5 Timers and timing output To profile your functions you can use the PARI timer The functions long timer and long timer2 return the elapsed time since the last call of the same function in milliseconds Two different functions identical except for their independent time of last call memories are provided so you can have both global timing and fine tuned profiling You can also use void msgtimer char format which prints prints Time then the remaining arguments as specified by format which is a PARI format then the output of timer2 This mechanism is simple to use but not foolproof If some other function uses these timers and many PARI functions do use timer2 when DEBUGLEVEL is high enough the timings will be meaningless To handle timing in a reentrant way PARI defines a dedicated datatype pari_timer The functions long TIMER pari_timer T long msgTIMER
157. a matrix between vector matrices of incompatible sizes and between an integermod and a real number The library syntax is gadd z y x y gsub z y for x y 3 1 3 The expression x y is the product of x and y Among the prominent impossibilities are multiplication between vector matrices of incompatible sizes between an integermod and a real number Note that because of vector and matrix operations is not necessarily commutative Note also that since multiplication between two column or two row vectors is not allowed to obtain the scalar product of two vectors of the same length you must multiply a line vector by a column vector if necessary by transposing one of the vectors using the operator or the function mattranspose see Section 3 8 If x and y are binary quadratic forms compose them See also gfbnucomp and qfbnupow The library syntax is gmul z y for x y Also available is gsqr x for x x faster of course 3 1 4 The expression x y is the quotient of x and y In addition to the impossibilities for multiplication note that if the divisor is a matrix it must be an invertible square matrix and in that case the result is y Furthermore note that the result is as exact as possible in particular division of two integers always gives a rational number which may be an integer if the quotient is exact and not the Euclidean quotient see x y for that and similarly the quotient of two polynomials is a ra
158. able number occur in the components of a polynomial whose main variable has a higher number lower priority even though there is nothing PARI can do to prevent you from doing it see Section 2 6 2 for a discussion of possible problems in a similar situation 4 6 2 Creating variables A basic difficulty is to create a variable As we have seen in Sec tion 4 1 a plethora of objects is associated to variable number v Here is the complete list polun v and polx v which you can use in library mode and which represent respectively the monic mono mials of degrees 0 and 1 in v varentries v and polvar v The latter two are only meaningful to GP but they have to be set nevertheless All of them must be properly defined before you can use a given integer as a variable number Initially this is done for 0 the variable x under GP and MAXVARN which is there to address the need for a temporary new variable in library mode and cannot be input under GP No documented library function can create from scratch an object involving MAXVARN of course if the operands originally involve MAXVARN the function will abide We call the latter type a temporary variable The regular variables meant to be used in regular objects are called user variables 4 6 2 1 User variables When the program starts x is the only user variable number 0 To define new ones use long fetch_user_var char s which inspects the user variable named s
159. actorback 2 2 3 1 1 12 factorback 2 2 3 1 2 12 factorback 5 2 3 3 30 factorback 2 2 3 1 nfinit x 3 2 44 16 0 0 Lo 16 0 o O 16 nf nfinit x 2 1 fa idealfactor nf 10 15 2 1 11 2 1 1 11 21 5 2 1 ale 1 2 11 1 5 2 11 1 1 2 11 1 factorback fa eK forbidden multiplication t_VEC t_VEC factorback fa nf 46 10 0 o 10 In the fourth example 2 and 3 are interpreted as principal ideals in a cubic field In the fifth one factorback fa is meaningless since we forgot to indicate the number field and the entries in the first column of fa can t be multiplied 80 The library syntax is factorback0 f e nf where an omitted nf or e is entered as NULL Also available is factorback f nf case e NULL where an omitted nf is entered as NULL 3 4 20 factorcantor x p factors the polynomial x modulo the prime p using distinct degree plus Cantor Zassenhaus The coefficients of x must be operation compatible with Z pZ The result is a two column matrix the first column being the irreducible polynomials dividing x and the second the exponents If you want only the degrees of the irreducible polynomials for example for computing an L function use factormod z p 1 Note that the factormod algorithm is usually faster than factorcantor The library syntax is factcantor z p 3 4 21 factorff x p a factors the po
160. ad 62 bit_accuracy 174 199 ONF sects Pee Se ek Duan a E tuck tak Be Lea 3 96 bot A Be eto a 39 98 bnitcertity lt a waren ee koe AES 99 bnfclassgrouponly 100 bnfclassunit 99 bniclassunitO iia ea es 100 bofelgp i id BAH el aes 100 bnfdecodemodule 100 A ee ee 96 100 DOFIDTLO sai hs eo RA 102 bnfisintnorm 102 bnfisnorm oh 2 ad BO toe Vee A 102 bnfisprincipal 101 102 bnfissunit lt 2 mo rss asa 102 pb fis nit Least al a 103 bufmak8 alo a bod ta ees 103 bnfnarrow 87 103 pifre ta aches hae AA eas 104 bnfsignunit 103 BON SUNIT s 66s iaa Ba ae eS 104 DO UD o e a nte ee le HR Ba ae 104 O oD 96 BYE tues O O 39 b rc rass aniti i a E A 105 bnviclassO vato misii a n SO Ee tc 105 bnrclassno 105 234 bnrclassnolist 105 bnrconductor 105 bnrconductorofchar 106 PALAS ox 00 S es Gh Bs eh ee a A 106 DOFATSCO curan ae ge ket wee Fan fe ahh ede lied 106 bnrdisclist vacia ee ak Bae 106 bnrdisclist0 107 Prinits op ake rta os 105 106 107 DOYINICO tf es eet y ces das Bad Eek ia 107 bnrisconductor 107 bnrisprincipal 101 107 DUPLE ina ee Bes Re a 104 105 bnrrootnumber 107 bnrstark 88 107 108 134 boolean operators 57 brace characters 32 break loop o
161. ae Be le ee a 217 BEMPE L aa od ter Sah we dol him de Cel 57 CONIA ia Bo he a Seca Ek eye res ee aa 64 ECOPY ida a 178 179 215 PCOS 2 a o ei 71 BCOtaN mida Gwe Petey aoe ad lard 72 BCVtOl ch a Ba A ok ee es 67 218 POVOD ir A De a a ats 216 PIEU a a io 219 BdOUX Aii ic tn aa S 171 I a te e de Ne A alah TE 55 BdIVENt 2 Lb er ee a we 55 gdiventgs z 219 gdiventres 0 0 56 219 gdiventsg z 219 gdivent z cia paa 219 pdives 2 ii a A re 219 gdivis y a a ad oes 220 gdivmod 0 0 000 219 gdivround 55 219 edivselZl osa it ees 219 di o a oe ada ee 219 peral a a a a BUG 57 217 gegaleS ida it a A ayes 217 gegalsg o oo o 217 gen member function 98 CEN o co taa da 7 171 PONOT 08 Wi Gn Pod E AA 89 generic matrix o 41 ao A Meta acy hae ees 66 GENtOStr o e aonda pa ee es 60 194 BOQ a aap Ge ale ee a eee 57 gerepile 177 180 182 200 210 gerepileall 182 185 gerepilecopy 182 185 gerepilemany 184 gerepilemany 182 gerepilemanysp 182 gerepileupto 177 181 182 201 Betheap swe vn eee hee Fa we 165 getrand 00 be a ee es 165 GBetstack an ko tala ad heads 165 Be Time o o 165 goular oy wits is o ee eS 231 goval lid o mee ds e bed ied 136 BORDA a 72 197 expo mico Oki ce o e 187 209 216 A a a aa E 6
162. ailable commands This might not be available for all architectures Whether extended on line help and line editing are available or not is indicated in the GP banner between the version number and the copyright message If you type you will get a short description of the metacommands keyboard shortcuts Finally typing will return the list of available pre defined member functions These are functions attached to specific kind of objects used to retrieve easily some information from complicated structures you can define your own but they won t be shown here We will soon describe these commands in more detail As a general rule under GP commands starting with or with some other symbols like or are not computing commands but are metacommands which allow the user to exchange information with GP The available metacommands can be divided into default setting commands explained below and simple commands or keyboard shortcuts to be dealt with in Section 2 2 14 2 1 Defaults and output formats There are many internal variables in GP defining how the system will behave in certain situations unless a specific override has been given Most of them are a matter of basic customization colors prompt and will be set once and for all in your preferences file see Section 2 9 but some of them are useful interactively set timer on increase precision etc The function used to manipulate these values is called defau
163. ak return or C d EOF any of which will let GP perform its usual cleanup and send you back to the GP prompt If the error is not fatal inputing an empty line i e hitting the lt Return gt key at the break gt prompt will continue the temporarily interrupted computation An empty line has no effect in case of a fatal error to ensure you do not get out of the loop prematurely thus losing most debugging data during the cleanup since user variables will be restored to their former values In current version 2 2 7 an error is non fatal if and only if it was initiated by a C c typed by the user Break loops are useful as a debugging tool to inspect the values of GP variables to understand why an error occurred or to change GP state in the middle of a computation increase debugging level start storing results in a logfile set variables to different values hit C c type in your modifications then let the computation go on as explained above A break loop looks like this for v 2 2 print 1 v 1 2 i division by zero in gdiv gdivgs or ginv kk Starting break loop type break to go back to GP kK for v 2 2 print 1 v break gt So the standard error message is printed first except now we always have context whether the error comes from the library or the parser The break gt at the bottow is a prompt and hitting v then lt Return gt we see break gt v 44 0 explaining the problem We could
164. aken into account by the gnuplot interface 160 3 10 20 plotrbox w dz dy draw in the rectwindow w the outline of the rectangle which is such that the points 1 y1 and x1 dx yl dy are opposite corners where x1 yl is the current position of the cursor Only the part of the rectangle which is in w is drawn The virtual cursor does not move 3 10 21 plotrecth w X a b expr flag 0 n 0 writes to rectwindow w the curve output of ploth w X a b expr flag n 3 10 22 plotrecthraw w data flag 0 plot graph s for data in rectwindow w flag has the same significance here as in ploth though recursive plot is no more significant data is a vector of vectors each corresponding to a list a coordinates If parametric plot is set there must be an even number of vectors each successive pair corresponding to a curve Otherwise the first one contains the x coordinates and the other ones contain the y coordinates of curves to plot 3 10 23 plotrline w dx dy draw in the rectwindow w the part of the segment x1 yl v1 dx yl dy which is inside w where x1 y1 is the current position of the virtual cursor and move the virtual cursor to x1 dz yl dy even if it is outside the window 3 10 24 plotrmove w dx dy move the virtual cursor of the rectwindow w to position x1 dx y1 dy where x1 yl is the initial position of the cursor i e to position dx dy relative to the initial curso
165. al 138 Legendre symbol 83 legendre oo ooo 138 E A ea ahi ae Maly Sle 64 Lenstrai e a4 bee Pus wae oy ards 81 136 LO tn ds 57 LEXEMP gt at E O o 58 217 LEXSOE b aa da aes a 152 MA e Sa Bk 2S 186 199 209 BCR A tty ie tnd 189 190 209 Tgefint p Ate dy 8s e ak wake 2 8 187 209 Let no as Gk d we eae tie 175 DT BSTAS cda gids Seat Skat die ea bed 175 TEETr ete RO eM A A 175 library mod viso seek Bee oe ao bot 171 LEIDA eka Gina Sea us Yee eh 81 TIEG oo 8 esas e as Fe ee 62 64 65 TEE CO foe te Qe ass OO A ea eee A ae 65 imit ait em als ee a E a 38 TANCE rico o BS 143 LINASPO siisii cd Ee ded Gk 143 line editor o o 50 linear dependence 143 Fineste hea qe Ht ete a ee Slt Se 4 18 MAUR 2 Bots Ded Mot os e a Gk ee 13 204 TUSOX Pes io o Ge Se 193 TISGEN ston Son as Wels as Pee eee 193 198 Lisp 22 2h gd eee a ee as 46 DT SSSQ ia A een ite Ra ac 193 listn a Bashy Ap oan oye Etat Ghee 7 28 190 LISE fesse e ar ah A oe ded ideas Le 58 LiStcrFeate Tara pee eke big des ged 143 listilisert e ruina Sa as 143 TESTA a as A 143 LUESEPUL ike sty a eS ee ees 143 LESESOLC A is a ta Does 143 LLD ves 115 120 141 143 145 146 149 LUL tr he we Sh E S 149 DEV AM ca ete tee Gila a a te 150 DTV er AMIN ae ee ete Bho ee Be 150 lllgramkerim 150 PLEIN ti er AA e PY oR br 149 ICM Sins a td Was des 149 lngamma o 73 local nuts sima
166. al to r Multiple roots are not repeated p is assumed to be a prime and pol to be non zero modulo p The library syntax is rootpadic pol p r where r is a long 3 7 22 polsturm pol a b number of real roots of the real polynomial pol in the interval Ja b using Sturm s algorithm a resp b is taken to be oo resp 00 if omitted The library syntax is sturmpart pol a b Use NULL to omit an argument sturm pol is equivalent to sturmpart pol NULL NULL The result is a long 3 7 23 polsubcyclo n d v x gives polynomials in variable v defining the sub Abelian extensions of degree d of the cyclotomic field Q where d n If there is exactly one such extension the output is a polynomial else it is a vector of polyno mials eventually empty To be sure to get a vector you can use concat polsubcyclo n d The function galoissubcyclo allows to specify more closely which sub Abelian extension should be computed The library syntax is polsubcyclo n d v where n d and v are long and v is a variable number When Z nZ is cyclic you can use subcyclo n d v where n d and v are long and v is a variable number 3 7 24 polsylvestermatrix z y forms the Sylvester matrix corresponding to the two polynomi als x and y where the coefficients of the polynomials are put in the columns of the matrix which is the natural direction for solving equations afterwards The use of this matrix can be essential whe
167. alnorm rnfidealnormrel rnf x but faster The library syntax is rnfidealnormabs rnf x 3 6 130 rnfidealnormrel rnf x rnf being a relative number field extension L K as output by rnfinit and x being a relative ideal which can be as in the absolute case of many different types including of course elements computes the relative norm of x as a ideal of K in HNF The library syntax is rnfidealnormrel rnf x 3 6 131 rnfidealreltoabs rnf x rnf being a relative number field extension L K as output by rnfinit and x being a relative ideal which can be as in the absolute case of many different types including of course elements gives the ideal zZz as an absolute ideal of L Q in the form of a Z basis given by a vector of polynomials modulo rnf po1 The following routine might be useful return y rnfidealreltoabs rnf as an ideal in HNF form associated to nf nfinit rnf pol idealgentoHNF nf y local z z nfalgtobasis nf y z 1 Mat z 1 mathnf concat z The library syntax is rnfidealreltoabs rnf x 3 6 132 rnfidealtwoelt rnf x rnf being a relative number field extension L K as output by rnfinit and x being an ideal of the relative extension L K given by a pseudo matrix gives a vector of two generators of x over Zr expressed as polmods with polmod coefficients The library syntax is rnfidealtwoelement rnf x 130 3 6 133 rnfidealup rnf x rnf being a relative number field
168. alred 91 ellhelight e rgen ao ee 91 ellheight0 91 ellheightmatriX 92 CLIMA aa e a 89 92 Tlinit ni eaa a n aa a A 93 ellisoncurve 0 93 BT pie ahs and a as tee tad 93 elllocalred 93 elllseries nes acs ee ee eo Weeks 93 ellminimalmodel 91 93 94 6llorder iea rora da ee ee ea A 94 6llordinate s a c orsa ea ey esie ales 94 ellpointtoz a ci 4 Sit aw a da 94 OL POW ici a da PA A 94 eLITOO NO aame is a Be ets 94 eblsipma T o a ado 94 6llsub sr ennai pak Os Re Me DA 94 elltaniyama 95 ell torss e Asura Gain ek ted 95 ClLtorsO metio eek ee ee ee e 95 CLUWP oc geo ace nce a Boa ROR ts 95 Llp ase yey a oe et 95 A a e Ae a A a 95 ellztopoint 2 2 86 ea e eh oe 95 Emacs ace eana le Ps a 49 EMX abai oa A Be Gt A Aa A ADA n 13 eontred its a aay 39 54 203 environment expansion 61 environment expansion 15 environment variable 61 OVP Ce gee heats Sep des Eke E N A oy 72 OLE A le dees o 195 196 198 Srrfil sev ese Pe Bee AS a Es 195 error handler 04 4 45 error recovery o 43 error trapping oa earr e ee ee 44 EPOR foe a wel ee ee lee ae 42 195 SELON ki oe dee piel ub WR Le es 42 44 165 Oba gus iras rare db 72 89 Euelid 3 4r aa ad ee dd 82 Euclidean quotient 55 Euclidean remainder 55 Euler product 78 85 15
169. and history objects can not be saved via this function Just as a regular input file a binary file can be compressed using gzip provided the file name has the standard gz extension In the present implementation the binary files are architecture dependent and compatibility with future versions of GP is not guaranteed Hence binary files should not be used for long term storage also they are larger and harder to compress than text files 3 11 2 31 writetex filename str x as write in TEX format 170 Chapter 4 Programming PARI in Library Mode 4 1 Introduction initializations universal objects To be able to use PARI in library mode you must write a C program and link it to the PARI library See the installation guide in Appendix A on how to create and install the library and include files A sample Makefile is presented in Appendix B Probably the best way to understand how programming is done is to work through a complete example We will write such a program in Section 4 8 Before doing this a few explanations are in order First one must explain to the outside world what kind of objects and routines we are going to use This is done simply with the statement include lt pari h gt This file pari h imports all the necessary constants variables and functions defines some important macros and also defines the fundamental type for all PARI objects the type GEN which is simply a pointer to long Technical note
170. and you can set flag 1 if the character is known to be primitive Example bnf bnfinit x 2 145 bnr bnrinit bnf 7 1 bnrrootnumber bnr 5 returns the root number of the character x of Cl7 Q V145 such that x g where g is the generator of the ray class field and e 7 N where N is the order of g N 12 as bnr cyc readily tells us The library syntax is bnrrootnumber bnf chi flag 107 3 6 29 bnrstark bnr subgroup flag 0 bnr being as output by bnrinit 1 finds a rel ative equation for the class field corresponding to the modulus in bnr and the given congruence subgroup using Stark units omit subgroup 0 if you want the whole ray class group The main variable of bnr must not be x and the ground field and the class field must be totally real When the base field is Q the vastly simpler galoissubcyclo is used instead flag is optional and may be set to 0 default to obtain a reduced relative polynomial 1 to be satisfied with any relative polynomial 2 to obtain an absolute polynomial and 3 to obtain the irreducible relative polynomial of the Stark unit Example bnf bnfinit y 2 3 bnr bnrinit bnf 5 1 bnrstark bnr 0 returns the ray class field of Q 3 modulo 5 Remark The result of the computation depends on the choice of a modulus verifying special conditions By default the function will try few moduli choosing the one giving the smallest result In some cases where the r
171. andard minimal integral model of the rational elliptic curve E If present sets v to the corresponding change of variables which is a vector lu r s t with rational components The return value is identical to that of ellchangecurve E v The resulting model has integral coefficients is everywhere minimal a is 0 or 1 az is 0 1 or 1 and az is 0 or 1 Such a model is unique and the vector v is unique if we specify that u is positive which we do The library syntax is ellminimalmodel E amp v where an omitted v is coded as NULL 3 5 19 ellorder F z gives the order of the point z on the elliptic curve E if it is a torsion point zero otherwise In the present version 2 2 7 this is implemented only for elliptic curves defined over Q The library syntax is orderell z 3 5 20 ellordinate E x gives a 0 1 or 2 component vector containing the y coordinates of the points of the curve E having x as x coordinate The library syntax is ordell F x 3 5 21 ellpointtox E z if E is an elliptic curve with coefficients in R this computes a complex number t modulo the lattice defining E corresponding to the point z i e such that in the standard Weierstrass model g t z 1 p t 2 2 In other words this is the inverse function of ellztopoint If E has coefficients in Qp then either Tate s u is in Qp in which case the output is a p adic number corresponding to the point z under the Tate parametrization or
172. ands for name none but gmp stand for auto gmp e The default kernel is auto none 2 2 Troubleshooting and fine tuning Decide whether you agree with what Configure printed on your screen in particular the architecture compiler and optimization flags Look for messages prepended by which probably report genuine problems If anything should have been found and was not consider that Configure failed and follow the instructions below Look especially for the readline and X11 libraries and the perl and gunzip or zcat binaries In case the default Configure run fails miserably try Configure a interactive mode and answer all the questions there aren t that many Of course Configure will still provide defaults for each answer but if you accept them all it will fail just the same so be wary In any case we would appreciate a bug report including the complete output from Configure and the file Oxxx pari cfg that was produced in the process 2 3 Problems related to readline Configure does not try very hard to find the readline library and include files If they are not in a standard place it won t find them Nonetheless it first searches the distribution toplevel for a readline directory Thus if you just want to give readline a try as you probably should you can get the source and compile it there you do not need to install it You can also use this feature together with a symbolic link named readline in the PARI topleve
173. arametrization 92 15 is equal to the square of the u value in the notation of Tate 16 is the u value itself if it belongs to Q otherwise zero 17 is the value of Tate s q for the curve E tate will yield the three component vector u u q See amp amp y 18 E w is the value of Mestre s w this is technical and E 19 is arbitrarily set equal to zero For all other base fields or rings the last six components are arbitrarily set equal to zero See also the description of member functions related to elliptic curves at the beginning of this section The library syntax is ellinitO E flag prec Also available are initell F prec flag 0 and smallinitell E prec flag 1 3 5 14 ellisoncurve F z gives 1 i e true if the point z is on the elliptic curve E 0 otherwise If E or z have imprecise coefficients an attempt is made to take this into account i e an imprecise equality is checked not a precise one The library syntax is oncurve F z and the result is a long 3 5 15 ellj x elliptic j invariant x must be a complex number with positive imaginary part or convertible into a power series or a p adic number with positive valuation The library syntax is jell x prec 3 5 16 elllocalred E p calculates the Kodaira type of the local fiber of the elliptic curve E at the prime p E must be given by a medium or long vector of the type given by ellinit and is assumed to have all its coe
174. argument checking and damage control Use them at your own risk 5 1 Level 0 kernel operations on unsigned longs Level 0 operations simulate basic operations of the 68020 processor on which PARI was originally implemented The type ulong is defined in the file parigen h as unsigned long Note that in the prototypes below a ulong is sometimes implicitly typecast to int or long The global ulong variables overflow which will contain only 0 or 1 and hiremainder used to be declared in the file pariinl h However for certain architectures they are no longer needed and or have been replaced with local variables for efficiency and the functions mentioned below are really chunks of assembler code which will be inlined at each invocation by the compiler If you really need to use these lowest level operations directly make sure you know your way through the PARI kernel sources and understand the architecture dependencies To make the following descriptions valid both for 32 bit and 64 bit machines we will set BIL to be equal to 32 resp 64 an abbreviation of BITS_IN_LONG which is what is actually used in the source code ulong addll ulong x ulong y adds the ulongs x and y returns the lower BIL bits and puts the carry bit into overflow ulong addllx int x ulong y adds overflow to the sum of the ulongs x and y returns the lower BIL bits and puts the carry bit into overflow ulong subll ulong x ulong y subtracts the ulongs x a
175. arrow class group and the third is a vector giving the generators of the corresponding v gen cyclic groups Note that this function is a special case of bnrclass The library syntax is buchnarrow bnf 3 6 13 bnfsignunit bnf bnf being a big number field output by bnfinit this computes an r X r 12 1 matrix having 1 components giving the signs of the real embeddings of the fundamental units The following functions compute generators for the totally positive units exponents of totally positive units generators on bnf tufu tpuexpo bnf local S d K S bnfsignunit bnf d matsize S S matrix d 1 da 2 i j if S i j lt 0 1 0 S concat S vectorv d 1 i 1 sign 1 K lift matker S Mod 1 2 if K mathnfmodid K 2 2 matid d 1 totally positive units tpu bnf 1 local vu ex tpuexpo bnf vu nfbasistoalg bnf bnf tufu vector length ex 1 i factorback vu ex i 1 Al ex 1 is 1 103 The library syntax is signunits bnf 3 6 14 bnfreg bnf bnf being a big number field output by bnfinit computes its regulator The library syntax is regulator bnf tech prec where tech is as in bnfclassunit 3 6 15 bnfsunit bnf S computes the fundamental S units of the number field bnf output by bnfinit where S is a list of prime ideals output by idealprimedec The output is a vector v with 6 components v 1 gives a minimal system of integral generators of
176. ary exponent of the difference between the original and the rounded value the fractional part If the exponent of x is too large compared to its precision i e e gt 0 the result is undefined and an error occurs if e was not given 66 Important remark note that contrary to the other truncation functions this function operates on every coefficient at every level of a PARI object For example 2 4 X 1 7 t t runcate Xx 24 x whereas 3 24x X 1 7 2x X 2 roun A X X An important use of round is to get exact results after a long approximate computation when theory tells you that the coefficients must be integers The library syntax is grndtoi z e where e is a long integer Also available is ground z 3 2 44 simplify x this function tries to simplify the object x as much as it can The simplifi cations do not concern rational functions which PARI automatically tries to simplify but type changes Specifically a complex or quadratic number whose imaginary part is exactly equal to 0 i e not a real zero is converted to its real part and a polynomial of degree zero is converted to its constant term For all types this of course occurs recursively This function is useful in any case but in particular before the use of arithmetic functions which expect integer arguments and not for example a complex number of 0 imaginary part and integer real part which is however printed as an integer The
177. asting will be necessary a priori V i for i 1 2 is a long but we will want to use it as a GEN The following two constructions will be exceedingly frequent x and V are GENs V il long x x GEN Vil Note that a typecast is not a valid lvalue cannot be put on the left side of an assignment so GEN V i x would be incorrect though some compilers may accept it Due to this annoyance the PARI functions and variables that occur most frequently have analogues which are macros including the typecast The complete list can be found in the file paricast h which is included by pari h and can be found at the same place For instance you can abbreviate long gzero gt zero long gun gt un 172 long polx v gt lpolx v long gadd x y gt ladd x y In general replacing a leading g by an 1 in the name of a PARI function will typecast the result to long Note that ldiv is an ANSI C function which is hidden in PARI by a macro of the same name representing long gdiv The macro coeff x m n exists with exactly the meaning of x m n under GP when x is a matrix This is a purely syntactical trick to reduce the number of typecasts and thus does not create a copy of the coefficient contrary to all the library functions It can be put on the left side of an assignment statement and its value of type long integer is a pointer to the desired coefficient object The macro gcoeff is a synonym fo
178. at precise type respectively output by rnfinit nfinit bnfinit and bnrinit However and even though it may not be specified in the descriptions of the functions below it is permissible if the function expects a nf to use a bnf instead which contains much more information The program will make the effort of converting to what it needs On the other hand if the program requires a big number field the program will not launch bnfinit for you which is a costly operation Instead it will give you a specific error message The data types corresponding to the structures described above are rather complicated Thus as we already have seen it with elliptic curves GP provides you with some member functions to retrieve the data you need from these structures once they have been initialized of course The relevant types of number fields are indicated between parentheses bnf bnr bnf big number field clgp bnr bnf classgroup This one admits the following three subclasses cyc cyclic decomposition SNF gen generators no number of elements diff bnr bnf nf the different ideal codiff bnr bnf nf the codifferent inverse of the different in the ideal group disc bnr bnf nf discriminant fu bnr bnf nf fundamental units futu bnr baf u w u is a vector of fundamental units w generates the torsion nf bnr bnf nf number field reg bnr bnf regulator roots bnr bnf nf
179. ath u bordeaux fr Implementation notes You can skip this section and switch to Section 1 2 if you re not inter ested in hardware technicalities You won t miss anything that would be mentioned here The PARI package contains essentially three variants See Appendix A for how to set up one of these on your system In a first version some time critical parts of the multiprecision kernel are written in assembler a few hundred lines at most At present there exists three versions for the Sparc architecture one for Sparc version 7 e g Sparcstation 1 1 IPC IPX or 2 one for Sparc version 8 with SuperSparc processors e g SparcStation 10 and 20 and one for Sparc version 8 with MicroSparc I or II processors e g SparcClassic or SparcStation 4 and 5 No specific version is written for the UltraSparc since it can use the MicroSparc II version In addition versions exist for the DEC Alpha 64 bit processors and finally the whole ix86 family Intel AMD Cyrix starting at the 386 up to the Xbox gaming console Finally there are unmaintained much less optimized and not really usable anymore micro kernels for the HP PA architecture only for HPUX 9 does not work with HPUX 10 and for the PowerPC architecture only for the 601 does not compile anymore A second version uses the GNU MP library to implement its multiprecision kernel and should run on any platform where GMP could be installed The GMP kernel is about as fast as the na
180. ation at pr of the element xz The same result could be obtained using idealval nf x pr since x would then be converted to a principal ideal but it would be less efficient The library syntax is element_val nf x pr and the result is a long 3 6 86 nffactor nf x factorization of the univariate polynomial x over the number field nf given by nfinit x has coefficients in nf i e either scalar polmod polynomial or column vector The main variable of nf must be of lower priority than that of x see Section 2 6 2 However if the polynomial defining the number field occurs explicitly in the coefficients of x as modulus of a t_POLMOD its main variable must be the same as the main variable of x For example nf nfinit y 2 1 nffactor at x 2 y OK nffactor nf x 2 Mod y y 2 1 OK nffactor nf x 2 Mod z z 2 1 AN WRONG NN NN The library syntax is nffactor nf x 3 6 87 nffactormod nf x pr factorization of the univariate polynomial x modulo the prime ideal pr in the number field nf x can have coefficients in the number field scalar polmod polynomial column vector or modulo the prime ideal integermod modulo the rational prime under pr polmod or polynomial with integermod coefficients column vector of integermod The prime ideal pr must be in the format output by idealprimedec The main variable of nf must be of lower priority than that of x see Section 2 6 2 However if the coefficients of
181. ax is kbessel nu x prec and kbessel2 nu x prec respectively 71 3 3 23 besseln nu x N Bessel function of index nu and argument zx The library syntax is nbessel nu x prec 3 3 24 cos x cosine of x The library syntax is gcos z prec 3 3 25 cosh x hyperbolic cosine of x The library syntax is gch z prec 3 3 26 cotan x cotangent of x The library syntax is gcotan z prec 3 3 27 dilog x principal branch of the dilogarithm of x i e analytic continuation of the power series log x do ns1 r n2 The library syntax is dilog z prec 3 3 28 eint1 x n exponential integral 2 2 dt x R If n is present outputs the n dimensional vector eint1 x eint1 nx x gt 0 This is faster than repeatedly calling eint1 i x The library syntax is veceint1 x n prec Also available is eint1 z prec 3 3 29 erfc x complementary error function 2 yr f7 e dt x R The library syntax is erfe x prec 3 3 30 eta x flag 0 Dedekind s 7 function without the q 2 This means the following if x is a complex number with positive imaginary part the result is 1 q where q e 7 If x is a power series or can be converted to a power series with positive valuation the result is I 1 2 If flag 1 and x can be converted to a complex number i e is not a power series computes the true 7 function including the leading q 2 The library syntax is eta
182. bcomponents of the components of k can also be a vector in which case the sorting is done lexicographically according to the components listed in the vector k For example if k 2 1 3 sorting will be done with respect to the second component and when these are equal with respect to the first and when these are equal with respect to the third The binary digits of flag mean e 1 indirect sorting of the vector zx i e if x is an n component vector returns a permutation of 1 2 n which applied to the components of x sorts x in increasing order For example vecextract x vecsort x 1 is equivalent to vecsort x e 2 sorts x by ascending lexicographic order as per the lex comparison function e 4 use descending instead of ascending order The library syntax is vecsort0 x k flag To omit k use NULL instead You can also use the simpler functions sort 1 vecsort0 x NULL 0 indexsort x vecsort0 1 NULL 1 lexsort x vecsort0 x NULL 2 152 Also available are sindexsort and sindexlexsort which return a vector of C long integers private type t_VECSMALL v where v 1 v n contain the indices 3 8 58 vector n X expr 0 creates a row vector type t_VEC with n components whose components are the expression expr evaluated at the integer points between 1 and n If one of the last two arguments is omitted fill the vector with zeroes The library syntax is vecteur GEN nmax entree ep c
183. be created on the stack but on the heap The function which does this is called gclone with the predefined macro Iclone as a synonym for long gclone Its syntax is GEN gclone GEN x A clone can be removed from the heap thus destroyed using void gunclone GEN x No PARI object should keep references to a clone which has been destroyed If you want to copy a clone back to the stack then delete it use forcecopy and not gcopy otherwise some components might not be copied moduli of t_INTMODs and t_POLMODs for instance 4 3 5 Conversions The following functions convert C objects to PARI objects creating them on the stack as usual GEN stoi long s C long integer small to PARI integer t_INT GEN dbltor double s C double to PARI real t_REAL The accuracy of the result is 19 decimal digits i e a type t_REAL of length DEFAULTPREC although on 32 bit machines only 16 of them will be significant We also have the converse functions long itos GEN x x must be of type t_INT double rtodbl GEN x x must be of type t_REAL as well as the more general ones long gtolong GEN x double gtodouble GEN x 4 4 Garbage collection 4 4 1 Why and how As we have seen the pari_init routine allocates a big range of addresses the stack that are going to be used throughout Recall that all PARI objects are pointers Except for a few universal objects they will all point at some part of the stack The stack starts at the add
184. be implemented as gt i if type x t_VEC length x lt 14 error this is not a proper elliptic curve x x 13 39 You can redefine one of your own member functions simply by typing a new definition for it On the other hand as a safety measure you can t redefine the built in member functions so typing the above text would in fact produce an error you d have to call it e g x j2 in order for GP to accept it Warning contrary to user functions arguments the structure accessed by a member function is not copied before being used Any modification to the structure s components will be permanent Note Member functions were not meant to be too complicated or to depend on any data that wouldn t be global Hence they do no have parameters besides the implicit structure or local variables Of course if you need some preprocessing work in there there s nothing to prevent you from calling your own functions using freely their local variables from a member function For instance one could implement a dreadful idea as far as efficiency goes correct_ell_if_needed x local tmp if type x t_VEC tmp ellinit x some further checks tmp x j correct_ell_if_needed x 13 Typing um will output the list of user defined member functions 2 6 6 Strings and Keywords GP variables can now hold values of type character string internal type t_STR This section describes ho
185. ble If your machine does not have one we strongly suggest that you obtain the gcc g compiler from the Free Software Foundation or by anonymous ftp As for all GNU software mentioned afterwards you can find the most convenient site to fetch gcc at the address http www gnu ai mit edu order ftp html You can certainly compile PARI with a different compiler but the PARI kernel takes advantage of some optimizations provided by gcc if it is available This results in at least 20 speedup on most architectures 1 1 Optional packages The following programs and libraries are useful in conjunction with GP but not mandatory They re probably already installed somewhere on your system with the possible exception of readline which we think is really worth a try In any case get them before proceeding if you want the functionalities they provide All of them are free though you ought to make a small donation to the FSF if you use and like GNU wares e GNU MP library This provides an alternative multiprecision kernel which is faster than PARI s native one To enable detection of GMP use Configure with gmp This is an experi mental feature use at your own risk e GNU readline library This provides line editing under GP an automatic context dependent completion and an editable history of commands Note that it is incompatible with SUN command tools yet another reason to dump Suntools for X Windows A recent readline version number
186. ble to compute the Dedekind zeta and lambda functions respectively zetak x and zetak x 1 This function calls in particular the bnfinit program The result is a 9 component vector v whose components are very technical and cannot really be used by the user except through the zetak function The only component which can be used if it has not been computed already is v 1 4 which is the result of the bnfinit call This function is very inefficient and should be rewritten It needs to computes millions of coef ficients of the corresponding Dirichlet series if the precision is big Unless the discriminant is small it will not be able to handle more than 9 digits of relative precision For instance zetakinit x78 2 needs 440MB of memory at default precision The library syntax is initzeta z 135 3 7 Polynomials and power series We group here all functions which are specific to polynomials or power series Many other functions which can be applied on these objects are described in the other sections Also some of the functions described here can be applied to other types 3 7 1 O a b p adic if a is an integer greater or equal to 2 or power series zero in all other cases with precision given by b The library syntax is ggrandocp a b where bis a long 3 7 2 deriv x v derivative of x with respect to the main variable if v is omitted and with respect to v otherwise x can be any type except polmod The derivative of a sca
187. ble x into a PARI real double gtodouble GEN x if x is a real number but not necessarily of type t_REAL converts x into a C double if possible long gtolong GEN x if x is an integer not a C long but not necessarily of type t_INT converts x into a C long if possible GEN gtopoly GEN x long v converts or truncates the object x into a polynomial with main variable number v A common application would be the conversion of coefficient vectors GEN gtopolyrev GEN x long v converts or truncates the object x into a polynomial with main variable number v but vectors are converted in reverse order GEN gtoser GEN x long v converts the object x into a power series with main variable number v GEN gtovec GEN x converts the object x into a row vector GEN co8 GEN x long 1 applied to a quadratic number x type t_QUAD converts x into a real or complex number depending on the sign of the discriminant of x to precision 1 BIL bit words GEN gcevtop GEN x GEN p long 1 converts x into a p adic number of precision 1 GEN gmodulcp GEN x GEN y creates the object Mod x y on the PARI stack where x and y are either both integers and the result is an integermod type t_INTMOD or x is a scalar or a polynomial and y a polynomial and the result is a polymod type t_POLMOD GEN gmodulgs GEN x long y same as gmodulcp except y is a long GEN gmodulss long x long y same as gmodulcp except both x and y are longs GEN gmodulo GE
188. bnf ideal flag 1 The library syntax is bnrinitO bnf ideal flag 3 6 26 bnrisconductor al a2 a3 al a2 a3 represent an extension of the base field given by class field theory for some modulus encoded in the parameters Outputs 1 if this modulus is the conductor and 0 otherwise This is slightly faster than bnrconductor The library syntax is bnrisconductor al a2 a3 and the result is a long 3 6 27 bnrisprincipal bnr x flag 1 bnr being the number field data which is output by bnrinit 1 and z being an ideal in any form outputs the components of x on the ray class group generators in a way similar to bnfisprincipal That is a 2 component vector v where v 1 is the vector of components of x on the ray class group generators v 2 gives on the integral basis an element a such that a 97 If flag 0 outputs only v1 In that case bnr need not contain the ray class group generators i e it may be created with bnrinit 0 The library syntax is isprincipalrayall bnr x flag 3 6 28 bnrrootnumber bnr chi flag 0 if x chi is a not necessarily primitive character over bnr let L s x gt y x id N id be the associated Artin L function Returns the so called Artin root number i e the complex number W x of modulus 1 such that A 8 x W x A s X where A s x A x 7 s L s X is the enlarged L function associated to L The generators of the ray class group are needed
189. ccur iff the constant coefficient of x is zero and are denoted by VERYBIGINT the biggest single precision integer representable on the machine 231 1 resp 263 1 on 32 bit resp 64 bit machines see Section 3 2 48 The library syntax is newtonpoly z p 3 6 69 nfalgtobasis nf x this is the inverse function of nfbasistoalg Given an object x whose entries are expressed as algebraic numbers in the number field nf transforms it so that the entries are expressed as a column vector on the integral basis nf zk The library syntax is algtobasis nf x 3 6 70 nfbasis x flag 0 fa integral basis of the number field defined by the irreducible preferably monic polynomial x using a modified version of the round 4 algorithm by default due to David Ford Sebastian Pauli and Xavier Roblot The binary digits of flag have the following meaning 1 assume that no square of a prime greater than the default primelimit divides the discrim inant of x i e that the index of x has only small prime divisors 2 use round 2 algorithm For small degrees and coefficient size this is sometimes a little faster This program is the translation into C of a program written by David Ford in Algeb Thus for instance if flag 3 this uses the round 2 algorithm and outputs an order which will be maximal at all the small primes If fa is present we assume without checking that it is the two column matrix of the factorization of the di
190. chinese is applied recursively to the components of zx yielding a residue belonging to the same class as all components of x Finally chinese x x x regardless of the type of x this allows vector arguments to contain other data so long as they are identical in both vectors The library syntax is chinois z y 77 3 4 8 content 1 computes the gcd of all the coefficients of x when this gcd makes sense If x is a scalar this simply returns the absolute value of x if x is rationnal t_INT t_FRAC or t_FRACN and x otherwise If x is a polynomial and by extension a power series it gives the usual content of x If x is a rational function it gives the ratio of the contents of the numerator and the denominator Finally if x is a vector or a matrix it gives the gcd of the contents of all entries The library syntax is content z 3 4 9 contfrac z b nmazx creates the row vector whose components are the partial quotients of the continued fraction expansion of x That is a result ao n means that x ay 1 a1 1 a The output is normalized so that a 4 1 unless we also have n 0 The number of partial quotients n is limited to nmax If x is a real number the expansion stops at the last significant partial quotient if nmaz is omitted x can also be a rational function or a power series If a vector b is supplied the numerators will be equal to the coefficients of b instead of all equal to 1 as above T
191. chitectures the odd ones have no precise equivalents on 64 bit machines 4 3 Creation of PARI objects assignments conversions 4 3 1 Creation of PARI objects The basic function which creates a PARI object is the function cgetg whose prototype is GEN cgetg long length long type Here length specifies the number of longwords to be allocated to the object and type is the type number of the object preferably in symbolic form see Section 4 5 for the list of these The precise effect of this function is as follows it first creates on the PARI stack a chunk of memory of size length longwords and saves the address of the chunk which it will in the end return If the stack has been used up a message to the effect that the PARI stack overflows will be printed and an error raised Otherwise it sets the type and length of the PARI object In effect it fills correctly and completely its first codeword z 0 or z Many PARI objects also have a second codeword types t_INT t_REAL t_PADIC t_POL and t_SER In case you want to produce one of those from scratch this should be exceedingly rare it is your responsibility to fill this second codeword either explicitly using the macros described in Section 4 5 or implicitly using an assignment statement using gaffect Note that the argument length is predetermined for a number of types 3 for types t_INTMOD t_FRAC t_FRACN t_COMPLEX t_POLMOD t_RFRAC and t_RFRACN 4 for type t_QU
192. command see above For user defined functions member functions see Nu and un 2 2 7 Md prints the defaults as described in the previous section shortcut for default see Section 3 11 2 4 2 2 8 Ne n switches the echo mode on 1 or off 0 If n is explicitly given set echo to n 2 2 9 g n sets the debugging level debug to the non negative integer n 2 2 10 gf n sets the file usage debugging level debugfiles to the non negative integer n 2 2 11 gm n sets the memory debugging level debugmem to the non negative integer n 2 2 12 h m n outputs some debugging info about the hashtable If the argument is a number n outputs the contents of cell n Ranges can be given in the form m n from cell m to cell n last cell If a function name is given instead of a number or range outputs info on the internal structure of the hash cell this function occupies a struct entree in C If the range is reduced to a dash outputs statistics about hash cell usage 23 UNIX 2 2 13 1 logfile switches log mode on and off If a logfile argument is given change the default logfile name to logfile and switch log mode on 2 2 14 m as a but using prettymatrix format 2 2 15 o n sets output mode to n 0 raw 1 prettymatrix 2 prettyprint 3 external prettyprint 2 2 16 p n sets realprecision to n decimal digits Prints its current value if n is omitted 2 2 17 ps n sets seriesprecision to
193. ctionalities are provided by an external perl script that you are free to use outside any GP session and modify to your liking if you are perl knowledgeable It is called gphelp lies in the doc subdirectory of your distribution just make sure you run Configure first see Appendix A and is really two programs in one The one which is used from within GP is gphelp which runs TFX on a selected part of this manual then opens a previewer gphelp detex is a text mode equivalent which looks often nicer especially on a colour capable terminal see misc gprc dft for examples The default help selects which help program will be used from within GP You are welcome to improve this help script or write new ones and we really would like to know about it so that we may include them in future distributions By the way outside of GP you can give more than one keyword as argument to gphelp 2 2 2 comment Everything between the stars is ignored by GP These comments can span any number of lines 2 2 3 one line comment The rest of the line is ignored by GP 2 2 4 a n prints the object number n n in raw format If the number n is omitted print the latest computed object 2 2 5 b n Same as a in prettyprint i e beautified format 2 2 6 c prints the list of all available hardcoded functions under GP not including operators written as special symbols see Section 2 4 More information can be obtained using the meta
194. cts We now consider variables and formal computations and give the technical details corresponding to the general discussion in Section 2 6 2 As we have seen in Section 4 5 the codewords for types t_POL and t_SER encode a variable number This is an integer ranging from 0 to MAXVARN The lower it is the higher the variable priority In fact the way an object will be considered in formal computations depends entirely on its principal variable number which is given by the function long gvar GEN z which returns a variable number for z even if z is not a polynomial or power series The variable number of a scalar type is set by definition equal to BIGINT which is bigger than any legal variable number The variable number of a recursive type which is not a polynomial or power series is the minimal variable number of its components But for polynomials and power series only the 190 outermost number counts we directly access varn x in the codewords the representation is not symmetrical at all Under GP one need not worry too much since the interpreter will define the variables as it sees them and do the right thing with the polynomials produced however have a look at the remark in Section 2 3 8 But in library mode they are tricky objects if you intend to build polynomials yourself and not just let PARI functions produce them which is usually less efficient For instance it does not make sense to have a vari
195. d 2 1 15 new galois format default 0 if this is set the polgalois command will use a different more consistent naming scheme for Galois groups This default is provided to ensure that scripts can control this behaviour and do not break unexpectedly Note that the default value of 0 unset will change to 1 set in the next major version 18 UNIX UNIX 2 1 16 output default 1 there are four possible values 0 raw 1 prettymatrix 2 prettyprint or 3 external prettyprint This means that independently of the default format for reals which we explained above you can print results in four ways either in raw format i e a format which is equivalent to what you input including explicit multiplication signs and everything typed on a line instead of two dimensional boxes This can have several advantages for instance it allows you to pick the result with a mouse or an editor and to paste it somewhere else The second format is the prettymatrix format The only difference to raw format is that matrices are printed as boxes instead of horizontally This is prettier but takes more space and cannot be used for input Column vectors are still printed horizontally The third format is the prettyprint format or beautified format In the present version 2 2 7 this is not beautiful at all The fourth format is external prettyprint which pipes all GP output in TeX format to an external prettyprinter according to t
196. d you would have to rely on bnrstark or rnfkummer The library syntax is subgrouplistO0 bnr bound flag where flag is a long integer and an omitted bound is coded by NULL 3 6 146 zetak znf x flag 0 znf being a number field initialized by zetakinit not by nfinit computes the value of the Dedekind zeta function of the number field at the complex number x If flag 1 computes Dedekind A function instead i e the product of the Dedekind zeta function by its gamma and exponential factors The accuracy of the result depends in an essential way on the accuracy of both the zetakinit program and the current accuracy Be wary in particular that x of large imaginary part or on the contrary very close to an ordinary integer will suffer from precision loss yielding less significant digits than expected Computing with 28 eight digits of relative accuracy we have zeta 3 1 1 202056903159594285399738161 zeta 3 1e 20 2 1 202056903159594285401719424 zetak zetakinit x 3 1e 20 3 1 202056903159594285401720529 the last 4 digits are wrong zetak zetakinit x 3 1e 28 4 1 202056914058477276496200278 MAN the last 20 digits are wrong zetak zetakinit x 3 1e 40 15 1989629366932171 633904021690 junk The library syntax is glambdak znf x prec or gzetak znf x prec 3 6 147 zetakinit x computes a number of initialization data concerning the number field de fined by the polynomial x so as to be a
197. d bottom row otherwise The empty matrix is considered to have a number of rows compatible with any operation in particular concatenation Note that this is definitely not the case for empty vectors or If y is omitted x has to be a row vector or a list in which case its elements are concatenated from left to right using the above rules concat 1 2 3 4 1 1 2 3 4 a 1 2 3 4 concat a 12 1 2 3 4 a i Mat a 1 concat a 43 1 3 2 4 concat 1 2 3 4 5 6 142 44 1 2 5 3 4 6 concat 7 8 1 2 3 4 5 1 5 7 2 3 4 6 8 1 2 3 4 The library syntax is concat z y 3 8 4 lindep z flag 0 x being a vector with real or complex coefficients finds a small integral linear combination among these coefficients If flag 0 uses a variant of the LLL algorithm due to Hastad Lagarias and Schnorr STACS 1986 If flag gt 0 uses the LLL algorithm flag is a parameter which should be between one half the number of decimal digits of precision and that number see algdep If flag lt 0 x is allowed to be and in any case interpreted as a matrix Returns a non trivial element of the kernel of x or 0 if x has trivial kernel The library syntax is lindepO z flag prec Also available is lindep x prec flag 0 3 8 5 listcreate n creates an empty list of maximal length n This function is useless in library mode
198. diately and then a gerepile for each coefficient of the result vector to get rid of the garbage which has accumulated while this particular coefficient was computed We leave the details to the reader who can look at the answer in the file basemath gen1 c in the function gmul case t_VEC times case t_MAT It would theoretically be possible to have a single connected piece of garbage but it would be a much less natural and unnecessarily complicated program which would in fact be slower 4 4 2 2 gerepilemany Let us now see why we may need the gerepilemany variants Although it is not an infrequent occurrence we will not give a specific example but a general one suppose that we want to do a computation usually inside a larger function producing more than one PARI object as a result say two for instance Then even if we set up the work properly before cleaning up we will have a stack which has the desired results z1 z2 say and then connected garbage from lbot to ltop If we write zi gerepile ltop lbot z1 then the stack will be cleaned the pointers fixed up but we will have lost the address of z2 This is where we need one of the gerepilemany functions we declare GEN gptr 2 Array of pointers to GENs gptr 0 amp z1 gptr 1 amp z2 and now the call gerepilemany ltop gptr 2 copies z1 and z2 to new locations cleans the stack from ltop to the old avma and updates the pointers z1 and z2 An equivalent slighl
199. ds the sum of the long s and the object x 218 GEN gaddgs z GEN x long s GEN z yields the sum of the object x and the long s GEN gsub z GEN x GEN y GEN z yields the difference of the objects x and y GEN gsubgs z GEN x long s GEN z yields the difference of the object x and the long s GEN gsubsg z long s GEN x GEN z yields the difference of the long s and the object x GEN gmulj z GEN x GEN y GEN z yields the product of the objects x and y GEN gmulsg z long s GEN x GEN z yields the product of the long s with the object x GEN gmulgs z GEN x long s GEN z yields the product of the object x with the long s GEN gshift z GEN x long n GEN z yields the result of shifting the components of x left by n if n is non negative or right by n if n is negative Applies only to integers reals and vectors matrices of such For other types it is simply multiplication by 2 GEN gmul2n z GEN x long n GEN z yields the product of x and 2 This is different from gshift when n is negative and x is of type t_INT gshift truncates while gmul2n creates a fraction if necessary GEN gdiv z GEN x GEN y GEN z yields the quotient of the objects x and y GEN gdivgs z GEN x long s GEN z yields the quotient of the object x and the long s GEN gdivsg z long s GEN x GEN z yields the quotient of the long s and the object x GEN gdivent z GEN x GEN y GEN z yields the true Euc
200. e cs 8 sa a Kaa 139 PostScript e n 2 08 8 fk ee a A 157 pomelo o a lke e Bie 94 power series 7 8 27 189 powering 0 0 56 69 PONA Aik top shee St A eet Ge 86 POWUSMOd Lic ee a a 208 POWUUMOG ac as a a ee 208 precdl 20h aie tee e dale eel ees 69 PYECISION 4 a PA Gas eS Be 68 precision on Son hue Sas E A 66 216 precisiomM 66 PESEP E ea 188 209 precprime o o 84 preferences file 13 14 47 204 prettymatriz format o 19 prettyprint format n o naaa aaa 19 prettyprinter 19 PAM s 6 alae o eA elas 84 PEIMEAECE ao as a a oe ES 115 primeform 2 sd eee ee ee ee 86 PrimeLimit penea Ae Bett 19 126 Primer 40 6 404 2h hee Paw NG 232 primes fees Ao ye ed ea a ee 84 principal ideal 115 principalideal 115 principalidele 116 PrE aA OAT ds 40 42 167 PLING oct cepa E sk A eee 167 printi rua ds ey Hoe Boa Bee bee 195 PEANED sha he Bye ae Ae ee eg as 167 PEINCPL forces d anita tn lees ec en A 167 PEINttER ob te ee tale ied Mas 167 PUOIS sn a E Roe Pye 28 PIO th ee Se RAS SS A a 154 prodeuler 0040 154 Prodinf o ia E 154 pr rodinf i ue o a aa aoe a e A R A 154 product eod nee dae eNe ahh ese de 55 produit fie Rechte o sees bn ir 154 programming 32 162 PHOMPE iaa es as Ae ey Ae ess 19 prompt_cont 20 psdrawW 2 5 25 ew Be PRG Sod ay
201. e For matrices a common mistake is to think that z 176 cgetg 4 t MAT for example will create the root of the matrix one needs also to create the column vectors of the matrix obviously since we specified only one dimension in the first cgetg This is because a matrix is really just a row vector of column vectors hence a priori not a basic type but it has been given a special type number so that operations with matrices become possible Finally to facilitate input of constant objects when speed is not paramount there are four varargs functions GEN coefs_to_int long n returns the non negative t_INT whose development in base 232 is given by the following n words unsigned long It is assumed that all such arguments are less than 2 the actual word size is irrelevant the behaviour is also as above on 64 bit machines coefs_to_int 3 a2 al a0 returns a224 a123 ao GEN coefs_to_pol long n Returns the t_POL whose n coefficients GEN follow in order of decreasing degree coefs_to_pol 3 gun gdeux gzero returns the polynomial X 2X in variable 0 use setvarn if you want other variable numbers Beware that n is the number of coefficients not the degree GEN coefs_to_vec long n returns the t_VEC whose n coefficients GEN follow GEN coefs_to_col long n returns the t_COL whose n coefficients GEN follow Warning Contrary to the policy of general PARI functions the latter three
202. e i e a 2 component vector the first being an ideal given as a Z basis the second being a r r2 component row vector giving the complex logarithmic Archimedean information a Zk generating system for an ideal a column vector x expressing an element of the number field on the integral basis in which case the ideal is treated as being the principal idele or ideal generated by zx a prime ideal i e a 5 component vector in the format output by idealprimedec a polmod zx i e an algebraic integer in which case the ideal is treated as being the principal idele generated by z an integer or a rational number also treated as a principal idele ea character on the Abelian group Z N Z g is given by a row vector x la a such that x 9 exp Qir Y ain Ni 96 Warnings 1 An element in nf can be expressed either as a polmod or as a column vector of components on the integral basis nf zk A row vector will not be recognized 2 When giving an ideal by a Zg generating system to a function expecting an ideal it must be ensured that the function understands that it is a Z generating system and not a Z generating system When the number of generators is strictly less than the degree of the field there is no ambiguity and the program assumes that one is giving a Zx generating set When the number of generators is greater than or equal to the degree of the field however the program assumes on the c
203. e or a matrix taken to be a HNF left divisor of the SNF for Z nZ of type N cyc giving the generators of H in terms of N gen e N the output of bnrinit bnfinit y m 1 where m is a module H as in the first case or a matrix taken to be a HNF left divisor of the SNF for the ray class group modulo m of type N cyc giving the generators of H in terms of N gen In this last case beware that H is understood relatively to N in particular if the infinite place does not divide the module e g if m is an integer then it is not a subgroup of Z nZ but of its quotient by 1 If fl 0 compute a polynomial in the variable v defining the the subfield of Q fixed by the subgroup H of Z nZ If fl 1 compute only the conductor of the abelian extension as a module If fl 2 output pol N where pol is the polynomial as output when fl 0 and N the conductor as output when fl 1 The following function can be used to compute all subfields of Q C of exact degree d if d is set subcyclo n d 1 local bnr L IndexBound IndexBound if d lt 0 n d bnr bnrinit bnfinit y n 11 1 1 L subgrouplist bnr IndexBound 1 vector L i galoissubcyclo bnr L i Setting L subgrouplist bnr d would produce subfields of exact conductor noo The library syntax is galoissubcyclo N H fl v where fl is a C long integer and v a variable number 111 3 6 39 galoissubfields G
204. e O p The library syntax is gevtoi x amp e where e is a long integer Also available is gtrunc z 67 3 2 48 valuation x p computes the highest exponent of p dividing x If p is of type integer x must be an integer an integermod whose modulus is divisible by p a fraction a q adic number with q p or a polynomial or power series in which case the valuation is the minimum of the valuation of the coefficients If p is of type polynomial must be of type polynomial or rational function and also a power series if x is a monomial Finally the valuation of a vector complex or quadratic number is the minimum of the component valuations If x 0 the result is VERYBIGINT 2 1 for 32 bit machines or 2 1 for 64 bit machines if x is an exact object If x is a p adic numbers or power series the result is the exponent of the zero Any other type combinations gives an error The library syntax is ggval x p and the result is a long 3 2 49 variable x gives the main variable of the object x and p if x is a p adic number Gives an error if x has no variable associated to it Note that this function is useful only in GP since in library mode the function gvar is more appropriate The library syntax is gpolvar x However in library mode this function should not be used Instead test whether x is a p adic type t_PADIC in which case p is in x 2 or call the function gvar x which returns the variable number of x
205. e Priority 3 x multiplication exact division 3 2 3 2 not 1 5 4 Euclidean quotient and remainder i e if x qy r with 0 lt r lt y if x and y are polynomials assume instead that degr lt degy and that the leading terms of r and x have the same sign then x y q x y r rounded Euclidean quotient for integers rounded towards 00 when the exact quotient would be a half integer lt lt gt gt left and right binary shift x lt lt n 2x2 ifn gt 0 and x 2 otherwise Right shift is defined by x gt gt n x lt lt n e Priority 2 addition subtraction e Priority 1 lt gt lt gt the usual comparison operators returning 1 for true and 0 for false For instance x lt 1 returns 1 if x lt 1 and 0 otherwise lt gt test for exact inequality test for exact equality e Priority 0 amp amp amp logical and logical inclusive or Any sequence of logical or and and operations is evaluated from left to right and aborted as soon as the final truth value is known Thus for instance x amp amp 1 x or type p t_INT amp amp isprime p will never produce an error since the second argument need not and will not be processed when the first is already zero false 30 Remark For optimal efficiency you should use the and op operators whenever possible a 200000 i 0 while i lt a i i 1 time 4 919 ms i 0
206. e an actual value to each of them the empty arg trick won t work here Careful use of these parameters may speed up your computations considerably The library syntax is bnfclassunitO P flag tech prec 3 6 3 bnfclgp P tech as bnfclassunit but only outputs v 5 i e the class group The library syntax is bnfclassgrouponly P tech prec where tech is as described under bnfclassunit 3 6 4 bnfdecodemodule nf m if m is a module as output in the first component of an extension given by bnrdisclist outputs the true module The library syntax is decodemodule nf m 3 6 5 bnfinit P flag 0 tech essentially identical to bnfclassunit except that the output contains a lot of technical data and should not be printed out explicitly in general The result of bnfinit is used in programs such as bnfisprincipal bnfisunit or bnfnarrow The result is a 10 component vector bnf e The first 6 and last 2 components are technical and in principle are not used by the casual user However for the sake of completeness their description is as follows We use the notations explained in the book by H Cohen A Course in Computational Algebraic Number Theory Graduate Texts in Maths 138 Springer Verlag 1993 Section 6 5 and subsection 6 5 5 in particular bnf 1 contains the matrix W i e the matrix in Hermite normal form giving relations for the class group on prime ideal generators 1 lt i lt r bnf 2 conta
207. e filename strx writes appends to filename the remaining arguments and appends a newline same output as print 3 11 2 29 writel filename strx writes appends to filename the remaining arguments with out a trailing newline same output as print1 169 3 11 2 30 writebin filename x writes appends to filename the object x in binary format This format is not human readable but contains the exact internal structure of x and is much faster to save load than a string expression as would be produced by write The binary file format includes a magic number so that such a file can be recognized and correctly input by the regular read or r function If saved objects refer to polynomial variables that are not defined in the new session they will be displayed in a funny way see Section 3 11 2 14 If x is omitted saves all user variables from the session together with their names Reading such a named object back in a GP session will set the corresponding user variable to the saved value E g after x 1 writebin log reading log into a clean session will set x to 1 The relative variables priorities see Section 2 6 2 of new variables set in this way remain the same preset variables retain their former priority but are set to the new value In particular reading such a session log into a clean session will restore all variables exactly as they were in the original one User functions installed functions
208. e is a multiple of the conductor is checked The result is undefined if the assumption is not correct The library syntax is rnfnormgroup bnr pol 3 6 141 rnfpolred nf pol relative version of polred Given a monic polynomial pol with co efficients in nf finds a list of relative polynomials defining some subfields hopefully simpler and containing the original field In the present version 2 2 7 this is slower and less efficient than rnfpolredabs The library syntax is rnfpolred nf pol prec 3 6 142 rnfpolredabs nf pol flag 0 relative version of polredabs Given a monic polyno mial pol with coefficients in nf finds a simpler relative polynomial defining the same field The binary digits of flag mean 1 returns P a where P is the default output and a is an element expressed on a root of P whose characteristic polynomial is pol 2 returns an absolute polynomial same as rnfequation nf rnfpolredabs nf pol but faster 16 possibly use a suborder of the maximal order This is slower than the default when the relative discriminant is smooth and much faster otherwise See Section 3 6 108 133 Remark In the present implementation this is both faster and much more efficient than rnf polred the difference being more dramatic than in the absolute case This is because the imple mentation of rnfpolred is based on a partial implementation of an incomplete reduction theory of lattices over number fields the function r
209. e precision of the number which is in fact redundant with p but is included for the sake of efficiency 1 2 3 Complex numbers and quadratic numbers quadratic numbers are numbers of the form a bw where w is such that Z w Z 2 and more precisely w Vd 2 when d 0mod 4 and w 1 Vd 2 when d 1mod 4 where d is the discriminant of a quadratic order Complex numbers correspond to the very important special case w y 1 Complex and quadratic numbers are partially recursive the two components a and b can be of type integer real rational integermod or p adic and can be mixed subject to the limitations mentioned above For example a bi with a and b p adic is in Q i but this is equal to Qp when p 1mod4 hence we must exclude these p when one explicitly uses a complex p adic type 1 2 4 Polynomials power series vectors matrices and lists they are completely recur sive their components can be of any type and types can be mixed however beware when doing operations Note in particular that a polynomial in two variables is simply a polynomial with polynomial coefficients Note that in the present version 2 2 7 of PARI there is a bug in the handling of power series of power series i e power series in several variables However power series of polynomials which are power series in several variables of a special type are OK The reason for this bug is known but it is difficult to correct because the mathematical problem
210. e result in type real evaluate 1 3 or add 0 to 1 3 The result of operations between imprecise objects will be as precise as possible Consider for example one of the most difficult cases that is the addition of two real numbers x and y The accuracy of the result is a priori unpredictable it depends on the precisions of x and y on their sizes i e their exponents and also on the size of x y PARI works out automatically the right precision for the result even when it is working in calculator mode GP where there is a default precision In particular this means that if an operation involves objects of different accuracies some digits will be disregarded by PARI It is a common source of errors to forget for instance that a real number is given as r 2 where r is a rational approximation e a binary exponent and e is a nondescript real number less than 1 in absolute value Hence any number less than 2 may be treated as an exact zero 0 E 28 1 E 100 1 0 E 28 0 E100 1 92 0 E100 As an exercise if a 27 100 why do a 0 anda 1 differ this is actually not quite true internally the format is 2 a where a and b are integers 9 The second part of the PARI philosophy is that PARI operations are in general quite permissive For instance taking the exponential of a vector should not make sense However it frequently happens that a computation comes out with a result which is a vector wi
211. e test compat runs produce a Postscript file pari ps in Orxx which you can send to a Postscript printer The output should bear some similarity to the screen images Finally make test kernel is only useful to developpers and should only be applied to an optimized gp build not a debugging one it checks whether the inline assembler kernel seems to work and provides simple diagnostics if it does not 3 Installation When everything looks fine type make install You may have to do this with superuser privileges depending on the target directories Tip for MacOS X beginners use sudo make install In this case it is advised to type make all first to avoid running unnecessary commands as root Beware that if you chose the same installation directory as before in the Configure process this will wipe out any files from version 1 39 15 and below that might already be there Libraries and executable files from newer versions starting with version 1 900 are not removed since they are only links to files bearing the version number beware of that as well if you re an avid GP fan don t forget to delete the old pari libraries once in a while This installs in the directories chosen at Configure time the default GP executable probably gp dyn under the name gp the default PARI library probably libpari so the necessary include files the manual pages the documentation and help scripts and emacs macros To save on disk space you ca
212. e used to compute the intersection of any two Z modules The result is given in HNF The library syntax is idealintersect nf x y 3 6 50 idealinv nf x inverse of the ideal x in the number field nf The result is the Hermite normal form of the inverse of the ideal together with the opposite of the Archimedean information if it is given The library syntax is idealinv nf x 3 6 51 ideallist nf bound flag 4 computes the list of all ideals of norm less or equal to bound in the number field nf The result is a row vector with exactly bound components Each component is itself a row vector containing the information about ideals of a given norm in no specific order This information can be either the HNF of the ideal or the idealstar with possibly some additional information If flag is present its binary digits are toggles meaning 1 give also the generators in the idealstar 2 output L U where L is as before and U is a vector of zinternallogs of the units 4 give only the ideals and not the idealstar or the ideallog of the units The library syntax is ideallist0 nf bound flag where bound must be a C long integer Also available is ideallist nf bound corresponding to the case flag 0 113 3 6 52 ideallistarch nf list arch flag 0 vector of vectors of all idealstarinit see idealstar of all modules in list with Archimedean part arch added void if omitted list is a vector of big ideals as output
213. eae 7 25 186 E INTMOD ee A 7 25 188 tULIST ect scp ek fod es 7 28 190 MATS A e te a lee od 7 27 190 GLPADIC cose eo gee A es 7 25 188 ELPOL hte hoe Sr tomate aay aes 7 27 189 t_POLMOD 7 26 188 A TE ye Ge he ty Heel a Ge ae dete 7 27 190 TONER iaa Rm wear oe te nd 7 27 189 QUAD cur ee ee 7 26 188 tAREAL cabare sio he be ee 7 25 187 GREER AC avec tee BH ie ahah oc ee 7 27 189 t_RFRACN ic a eo 7 27 189 USER Ve A ie ale de 7 27 189 STR iP ace ke e AN tte e ds 7 28 190 VEG oan ee la ees 7 27 190 t VECSMAELL e 90 a a 28 U ULLOA E a a e E 177 timit a e Sh 38 249 WT ONG aci a tied a ee 207 MDs a ab ete doe OI Oe eet a eae ee n 173 universal object 231 TOD cio or he eas Bias AA O 164 user defined functions 36 V vall Arta Lada e 211 Wal pire tale Ls eas 188 189 209 MAL Sass tee hy Sele ete ee OS A E 211 val ation ora a dean See Pea 67 van Hoeij o 79 108 VALLES Stok Me A ES ah lest Oh Wh a oe 177 vVarentries 2 22 6 saw da 191 variable priority 26 34 190 variable temporary 191 variable user Dr Goes 191 variable number 189 190 202 variable 26 27 29 32 172 variable tc as ees 68 MAD a Bese es 189 190 209 VECES O Ge Glen dee 27 61 vecbezout 0 0 000 77 vecbezoutres 77 veceintl israel a 72 VOCcextIacO imac alan Ea a 146 151 wecmax Ati nd a d
214. ecomposition of x on the fundamental units and the roots of unity if x is a unit the empty vector otherwise More precisely if u1 uy are the fundamental units and is the generator of the group of roots of unity found by bnfclassunit or bnfinit the output is a vector 11 tr amp r 1 such that z uj u r C rtt The x are integers for i lt r and is an integer modulo the order of fori r 1 The library syntax is isunit bnf x 3 6 11 bnfmake sbnf sbnf being a small bnf as output by bnfinit x 3 computes the com plete bnfinit information The result is not identical to what bnfinit would yield but is func tionally identical The execution time is very small compared to a complete bnfinit Note that if the default precision in GP or prec in library mode is greater than the precision of the roots sbnf 5 these are recomputed so as to get a result with greater accuracy Note that the member functions are not available for sbnf you have to use bnfmake explicitly first The library syntax is makebigbnf sbnf prec where prec is a C long integer 3 6 12 bnfnarrow bnf bnf being a big number field as output by bnfinit computes the narrow class group of bnf The output is a 3 component row vector v analogous to the corresponding class group component bnf clgp bnf 8 1 the first component is the narrow class number v no the second component is a vector containing the SNF cyclic components v cyc of the n
215. ection Of course break loops are no longer available the new handler has replaced the default one Besides user defined handlers as above there are two special handlers you can use in trap which are e trap do nothing handler to disable the trapping mechanism and let errors propa gate which is the default situation on startup e trap omitted argument default handler to trap errors by a break loop 2 7 5 Protecting code Finally trap can define a temporary handler used within the scope of a code frament protecting it from errors by providing replacement code should the trap be activated The expression trap recovery statements evaluates and returns the value of statements unless an error occurs during the evaluation in which case the value of recovery is returned As in an if else clause with the difference that statements has been partially evaluated with possible side effects For instance one could define a fault tolerant inversion function as follows inv x trap oo 1 x for i 1 1 print inv i 1 00 1 Protected codes can be nested without adverse effect the last trap seen being the first to spring 45 UNIX 2 7 6 Trapping specific exceptions We have not yet seen the use of the first argument of trap which has been omitted in all previous examples It simply indicates that only errors of a specific type should be intercepted to be chosen among accurer accuracy problem gdi
216. ector which compares very favorably with much more sophisticated methods used in other systems Our benchmarks indicate that the price paid for using gerepile when properly used is usually around 1 or 2 percents of the total running time which is quite acceptable Secondly in many cases in particular when the tree structure and the size of the PARI objects which will appear in a computation are under control one can avoid gerepile altogether by creating sufficiently large objects at the beginning using cgetg and then using assignment statements and operations ending with z such as gaddz Coming back to our first example note that if we know that x and y are of type real and of length less than or equal to 5 we can program without using gerepile at all z cgetr 5 ltop avma pl gsqr x p2 gsqr y gaddz p1 p2 z avma ltop This practice will usually be slower than a craftily used gerepile though and is certainly more cumbersome to use As a rule assignment statements should generally be avoided Thirdly the philosophy of gerepile is the following keep the value of the stack pointer avma at the beginning and just before the last operation Afterwards it would be too late since the lower 185 end address of the garbage zone would have been lost Of course you can always use gerepileupto but you will have to assume that the object was created before its components Lastly if all seems lost just use gerepilecopy
217. ed and then the additional primes in this table If x is empty or omitted just returns the current list of extra primes The entries in x are not checked for primality They need only be positive integers not divisible by any of the pre computed primes It s in fact a nice trick to add composite numbers which for example the function factor x 0 was not able to factor In case the message impossible inverse modulo some INTMOD shows up afterwards you have just stumbled over a non trivial factor Note that the arithmetic functions in the narrow sense like eulerphi do not use this extra table To remove primes from the list use removeprimes The library syntax is addprimes z 76 3 4 2 bestappr z A B if B is omitted finds the best rational approximation to x R or R X or R with denominator at most equal to A using continued fractions If B is present x is assumed to be of type t_INTMOD modulo M or a recursive combination of those and the routine returns the unique fraction a b in coprime integers a lt A and b lt B which is congruent to x modulo M If M lt 2AB uniqueness is not guaranteed and the function fails with an error message If rational reconstruction is not possible no such a b exists for at least one component of x returns 1 The library syntax is bestapprO z A B Also available is bestappr z A corresponding to an omitted B 3 4 3 bezout x y finds u and v minimal in a natu
218. ed in the COPYING file modified versions of the PARI package can be distributed under the conditions of the GNU General Public License If you do modify PARI however it is certainly for a good reason hence we would like to know about it so that everyone can benefit from it There is then a good chance that the modifications that you have made will be incorporated into the next release Recall the e mail address pari math u bordeaux fr or use the mailing lists Roughly four types of modifications can be made The first type includes all improvements to the documentation in a broad sense This includes correcting typos or inaccurracies of course but also items which are not really covered in this document e g if you happen to write a tutorial or pieces of code exemplifying some fine points that you think were unduly omitted The second type is to expand or modify the configuration routines and skeleton files the Con figure script and anything in the config subdirectory so that compilation is possible or easier or more efficient on an operating system previously not catered for This includes discovering and removing idiosyncrasies in the code that would hinder its portability The third type is to modify existing mathematical code either to correct bugs to add new functionalities to existing functions or to improve their efficiency Finally the last type is to add new functions to PARI We explain here how to do this so that
219. ee Be ee Woe Bs 135 Zetakinite ios ly eke Fos ts eles 135 zidal log oho a ee a eo i 114 ZE saa ce Do he a De ee BS et 98 ZKS fei oes dd we cher fo Gb oe ae eA eS 98 ZN LOG Fined ahs ean ete ane Bed Ate 88 ZNOV ACR toa A tee ei a 89 zgnprimroot 40 89 20star Lic we Biko Po Pe a ai 89 250
220. ementary divisors of Z a f a Zla where a is a root of the polynomial f The components of the result are all positive and their product is equal to the absolute value of the discriminant of f The library syntax is reduceddiscsmith z 3 7 12 polhensellift x y p e given a prime p an integral polynomial x whose leading coefficient is a p unit a vector y of integral polynomials that are pairwise relatively prime modulo p and whose product is congruent to x modulo p lift the elements of y to polynomials whose product is congruent to x modulo p The library syntax is polhensellift x y p e where e must be a long 3 7 13 polinterpolate xa ya v xj amp e given the data vectors za and ya of the same length n za containing the x coordinates and ya the corresponding y coordinates this function finds the interpolating polynomial passing through these points and evaluates it at v If ya is omitted return the polynomial interpolating the i xa i If present e will contain an error estimate on the returned value The library syntax is polint za ya v amp e where e will contain an error estimate on the returned value 137 3 7 14 polisirreducible pol pol being a polynomial univariate in the present version 2 2 7 returns 1 if pol is non constant and irreducible 0 otherwise Irreducibility is checked over the smallest base field over which pol seems to be defined The library syntax is gisirreducible pol
221. ental functions here 3 is only taken into account when the arguments are exact objects and thus no a priori precision can be determined from the objects themselves To cater for this possibility if s is of type t_REAL we use the function setlg which effectively sets the precision of s to the required value Note that here since we are using a numeric value for a cget function the program will run slightly differently on 32 bit and 64 bit machines we want to use the smallest possible bit accuracy and this is equal to BITS_IN_LONG Note that the matrix x is allowed to have complex entries but the function gnorm12 guarantees that s is a non negative real number not necessarily of type t_REAL of course If we had not known this fact we would simply have added the instruction s greal s just after the for loop Note also that the function gnorm12 works as desired on matrices so we really did not need this loop at all s gnorm12 x would have been enough but we wanted to give examples of function usage Similarly it is of course not necessary to take the square root for testing whether the norm exceeds 1 In the fifth place note that we initialized the sum s to gzero which is an exact zero This is logical but has some disadvantages if all the entries of the matrix are integers or rational numbers the computation will take rather long about twice as long as with real numbers of the same length It would be better to initialize s t
222. ents with the following special rules for y 0 1 or 2 e x 0 1 if x 1 and 0 otherwise e x 1 1 if x gt 0 and 1 otherwise e 1 2 0 if x is even and 1 if x 1 1mod8 and 1 if z 3 3 mod 8 The library syntax is kronecker z y the result 0 or 1 is a long 3 4 35 Iem z y least common multiple of x and y i e such that lem x y ged x y abs a y If y is omitted and x is a vector return the lcm of all components of x The library syntax is glem z y 3 4 36 moebius z Moebius y function of x x must be of type integer The library syntax is mu zx the result 0 or 1 is a long 3 4 37 nextprime z finds the smallest pseudoprime see ispseudoprime greater than or equal to x x can be of any real type Note that if x is a pseudoprime this function returns x and not the smallest pseudoprime strictly larger than x To rigorously prove that the result is prime use isprime The library syntax is nextprime z 3 4 38 numdiv x number of divisors of x must be of type integer The library syntax is numbdiv z 3 4 39 numbpart n gives the number of unrestricted partitions of n usually called p n in the litterature in other words the number of nonnegative integer solutions to a 2b 3c n n must be of type integer and 1 lt n lt 1015 The algorithm uses the Hardy Ramanujan Rademacher formula The library syntax is numbpart n 3 4 40 omega x numbe
223. eptions e for simple objects e g leaves whose size is controlled they can be easier to use than gerepile and about as efficient e to coerce an inexact object to a given precision For instance gaffect x tmp cgetr 3 x tmp at the beginning of a routine where precision can be kept to a minimum otherwise the precision of x will be used in all subsequent computations which will be a disaster if x is known to thousands of digits 4 3 3 Copy It is also very useful to copy a PARI object not just by moving around a pointer as in the y x example but by creating a copy of the whole tree structure without pre allocating a possibly complicated y to use with gaffect The function which does this is called gcopy with the predefined macro lcopy as a synonym for long gcopy Its syntax is GEN gcopy GEN x and the effect is to create a new copy of x on the PARI stack Beware that universal objects which occur in specific components of certain types mainly moduli for types t_INTMOD and t_PADIC are not copied as they are assumed to be permanent In this case gcopy only copies the pointer Use GEN forcecopy GEN x if you want a complete copy Please be sure at this point that you really understand the difference between y x y gcopy x and gaffect x y this will save you from many obvious mistakes later on 178 4 3 4 Clones Sometimes it may be more efficient to create a permanent copy of a PARI object This will not
224. equal to 1 if the enlarged order could be proven to be pr maximal and to 0 otherwise it may be maximal in the latter case if pr is ramified in L the second component is a pseudo basis of the enlarged order and the third component is the valuation at pr of the order discriminant The library syntax is rnfdedekind nf pol pr 3 6 117 rnfdet nf M given a pseudo matrix M over the maximal order of nf computes its determinant The library syntax is rnfdet nf M 3 6 118 rnfdisc nf pol given a number field nf as output by nfinit and a polynomial pol with coefficients in nf defining a relative extension L of nf computes the relative discriminant of L This is a two element row vector D d where D is the relative ideal discriminant and d is the relative discriminant considered as an element of nf nf The main variable of nf must be of lower priority than that of pol see Section 2 6 2 The library syntax is rnfdiscf bnf pol 3 6 119 rnfeltabstorel rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of L expressed as a polynomial modulo the absolute equation rnf pol computes x as an element of the relative extension L K as a polmod with polmod coefficients The library syntax is rnfelementabstorel rnf x 3 6 120 rnfeltdown rnf x rnf being a relative number field extension L K as output by rn finit and x being an element of L expressed as a polynomial or polmod with polmod coe
225. er CC C compiler DLLD Dynamic library linker LD Static linker The contents of the following variables are appended to the values computed by Configure CFLAGS Flags for CC LDFLAGS Flags for LD For instance Configure may avoid bin cc on some architectures due to various problems which may have been fixed in your version of the compiler You can try env CC cc Configure and compare the benches Also if you insist on using a C compiler and run into trouble with a fussy g try to use g fpermissive 222 Technical note 3 Configure accepts many other flags besides the ones mentionned above See Configure help for a complete list In particular there are sets of flags related to GNU MP with gmp and GNU readline library with readline Note that autodetection of GMP is disabled by default Technical note 4 The multiprecision kernel can be fully specified via the kernel fgkn switch The PARI kernel is build from two kernels called level 0 LO operation on words and level 1 L1 operation on multi precision integer and real Available kernels LO auto none and alpha hppa ix86 ppc sparcvY sparcv8_micro sparcv8_super L1 auto none and gmp auto means to use the auto detected value LO none means to use the portable C kernel no assembler Ll none means to use the PARI L1 kernel e A fully qualified kernel name fgkn is of the form Lo L e A name not containing a dash is an alias An alias st
226. ere When doing large computations unwanted intermediate results clutter up memory very fast so some kind of garbage collecting is needed Most large systems do garbage collecting when the memory is getting scarce and this slows down the performance In PARI we have taken a different approach you must do your own cleaning up when the intermediate results are not needed anymore Special purpose routines have been written to do this we will see later how and when if at all you should use them 171 e the following universal objects by definition objects which do not belong on the stack the integers 0 1 and 2 respectively called gzero gun and gdeux the fraction gt ghalf the complex number gi All of these are of type GEN In addition space is reserved for the polynomials x polx v and the polynomials 1 polun v Here x is the name of variable number v where 0 lt v lt MAXVARN the exact value of which depends on your machine at least 16383 in any case Both polun and polx are arrays of GENs the index being the polynomial variable number However except for the ones corresponding to variables 0 and MAXVARN these polynomials are not created upon initialization It is the programmer s responsibility to fill them before use We will see how this is done in Section 4 6 never through direct assignment e a heap which is just a linked list of permanent universal objects For now it contains exactly the ones
227. eries Here precdl is not an argument of the function but a global variable Then the Taylor series expansion of the function around X 0 where X is the main variable is computed to a number of terms depending on the number of terms of the argument and the function being computed e If the argument is a vector or a matrix the result is the componentwise evaluation of the function In particular transcendental functions on square matrices which are not implemented in the present version 2 2 7 see Appendix B however will have a slightly different name if they are implemented some day 3 3 1 If y is not of type integer x y has the same effect as exp y 1n x It can be applied to p adic numbers as well as to the more usual types The library syntax is gpow x y prec 3 3 2 Euler Euler s constant 0 57721 Note that Euler is one of the few special reserved names which cannot be used for variables the others are I and Pi as well as all function names The library syntax is mpeuler prec where prec must be given Note that this creates y on the PARI stack but a copy is also created on the heap for quicker computations next time the function is called 3 3 3 I the complex number y 1 The library syntax is the global variable gi of type GEN 3 3 4 Pi the constant 7 3 14159 The library syntax is mppi prec where prec must be given Note that this creates 7 on the PARI stack but a copy is also created on the hea
228. ers is also at most 80807123 significant decimal digits and the binary exponent must be in absolute value less than 273 8388608 Note that PARI has been optimized so that it works as fast as possible on numbers with at most a few thousand decimal digits In particular not too much effort has been put into fancy multiplication techniques only the Karatsuba multiplication is implemented Hence although it is possible to use PARI to do computations with 10 decimal digits much better programs can be written for such huge numbers Integers and real numbers are completely non recursive types and are sometimes called the leaves 1 2 2 Integermods rational numbers irreducible or not p adic numbers polmods and rational functions these are recursive but in a restricted way For integermods or polmods there are two components the modulus which must be of type integer resp polynomial and the representative number resp polynomial For rational numbers or rational functions there are also only two components the numerator and the denominator which must both be of type integer resp polynomial Finally p adic numbers have three components the prime p the modulus p and an approx imation to the p adic number Here Z is considered as lim Z pFZ and Q as its field of fractions Like real numbers the codewords contain an exponent giving essentially the p adic valuation of the number and also the information on th
229. ers x and p returns the highest exponent e such that p divides x creates the quotient x p and returns its address in r In particular if p is a prime this returns the valuation at p of x and r will obtain the prime to p part of x 5 3 3 Assignment statements void gaffsg long s GEN x assigns the long s into the object x void gaffect GEN x GEN y assigns the object x into the object y 217 5 3 4 Unary operators GEN gneg z GEN x GEN z yields x GEN gabs z GEN x GEN z yields x GEN gsqr GEN x creates the square of x GEN ginv GEN x creates the inverse of x GEN gfloor GEN x creates the floor of x i e the true integral part GEN gfrac GEN x creates the fractional part of x i e x minus the floor of x GEN gceil GEN x creates the ceiling of x GEN ground GEN x rounds the components of x to the nearest integers Exact half integers are rounded towards 00 GEN grndtoi GEN x long e same as round but in addition puts minus the number of signif icant binary bits left after rounding into e If e is positive all significant bits have been lost This kind of situation raises an error message in ground but not in grndtoi GEN gtrunc GEN x truncates x This is the false integer part if x is an integer i e the unique integer closest to x among those between 0 and x If x is a series it will be truncated to a polynomial if x is a rational function this takes the polynomial part GEN
230. ery rapidly but which is much smaller and hence easy to store and print It is supposed to be used in conjunction with bnfmake The output is a 12 component vector v as follows Let bnf be the result of a full bnfinit complete with units Then v 1 is the polynomial P v 2 is the number of real embeddings rj v 3 is the field discriminant v 4 is the integral basis v 5 is the list of roots as in the sixth component of nfinit v 6 is the matrix MD of nfinit giving a Z basis of the different v 7 is the matrix W bnf 1 v S is the matrix matalpha bnf 2 v 9 is the prime ideal factor base bnf 5 coded in a compact way and ordered according to the permutation bnf 6 v 10 is the 2 component vector giving the number of roots of unity and a generator expressed on the integral basis v 11 is the list of fundamental units expressed on the integral basis v 12 is a vector containing the algebraic numbers alpha corresponding to the columns of the matrix matalpha expressed on the integral basis 101 Note that all the components are exact integral or rational except for the roots in v 5 In practice this is the only component which a user is allowed to modify by recomputing the roots to a higher accuracy if desired Note also that the member functions will not work on sbnf you have to use bnfmake explicitly first The library syntax is bnfinitO P flag tech prec 3 6 6 bnfisintnorm bnf x computes a complete system of solu
231. estriction of flag 2 but a warning is issued when it is not proven complete If flag 1 use nfroots require a number field If flag 2 use complex approximations to the roots and an integral LLL The result is not guaranteed to be complete some conjugates may be missing no warning issued especially so if the corresponding polynomial has a huge index In that case increasing the default precision may help If flag 4 use Allombert s algorithm and permutation testing If the field is Galois with weakly super solvable Galois group return the complete list of automorphisms else only the identity element If present d is assumed to be a multiple of the least common denominator of the conjugates expressed as polynomial in a root of pol A group G is weakly super solvable WKSS if it contains a super solvable normal subgroup H such that G H or G H Ay or G H S4 Abelian and nilpotent groups are WKSS In practice almost all groups of small order are WKSS the exceptions having order 36 1 exception 48 2 56 1 60 1 72 5 75 1 80 1 96 10 and gt 108 Hence flag 4 permits to quickly check whether a polynomial of order strictly less than 36 is Galois or not This method is much faster than nfroots and can be applied to polynomials of degree larger than 50 This routine can only compute Q automorphisms but it may be used to get K automorphism for any base field K as follows rnfgaloisconj nfK R
232. esult is however very large you can tell the function to try more moduli by adding 4 to the value of flag Whether this flag is set or not the function may fail in some extreme cases returning the error message Cannot find a suitable modulus in FindModule In this case the corresponding congruence group is a product of cyclic groups and for the time being the class field has to be obtained by splitting this group into its cyclic components The library syntax is bnrstark bnr subgroup flag where an omitted subgroup is coded by NULL 3 6 30 dirzetak nf b gives as a vector the first b coefficients of the Dedekind zeta function of the number field nf considered as a Dirichlet series The library syntax is dirzetak nf b 3 6 31 factornf x t factorization of the univariate polynomial x over the number field defined by the univariate polynomial t x may have coefficients in Q or in the number field The algorithm reduces to factorization over Q Trager s trick The direct approach of nffactor which uses van Hoeij s method in a relative setting is in general faster The main variable of t must be of lower priority than that of x see Section 2 6 2 However if the coefficients of the number field occur explicitly as polmods as coefficients of x the variable of these polmods must be the same as the main variable of t For example factornf x 2 Mod y y 2t1 y 2 1 factornf x 2 1 y 2 1 these two are OK
233. et op be some unary operation of type GEN GEN The names and prototypes of the low level functions corresponding to op will be as follows GEN mpop GEN x creates the result of op applied to the integer or real x GEN ops long s creates the result of op applied to the long s GEN opi GEN x creates the result of op applied to the integer x GEN opr GEN x creates the result of op applied to the real x GEN mpopz GEN x GEN z assigns the result of applying op to the integer or real x into the integer or real z 211 Remark it has not been considered useful to include the functions void opsz long GEN void opiz GEN GEN and void oprz GEN GEN The above prototype schemes apply to the following operators op neg negation x The result is of the same type as x op abs absolute value x The result is of the same type as x In addition there exist the following special unary functions with assignment void mpinvz GEN x GEN z assigns the inverse of the integer or real x into the real z The inverse is computed as a quotient of real numbers not as a Euclidean division void mpinvsr long s GEN z assigns the inverse of the long s into the real z void mpinvir GEN x GEN z assigns the inverse of the integer x into the real z void mpinvrr GEN x GEN z assigns the inverse of the real x into the real z 5 2 6 Comparison operators int mpcmp GEN x GEN y compares the integer or real x to the integer or real y
234. etc in the input and can do computations e g matid 2 or 1 0 0 1 are equally valid inputs Finally sor is the general output routine We have chosen to give d significant digits since this is what was asked for Note that there is a trick hidden here if a negative d was input then the computation will be done in precision 3 i e about 9 7 decimal digits for 32 bit machines and 19 4 for 64 bit machines and in the function sor giving a negative third argument outputs all the significant digits which is entirely appropriate Now let us attack the main course the function matexp GEN matexp GEN x long prec pari_sp lbot ltop avma long 1x lg x i k n GEN y r s p1 p2 check that x is a square matrix if typ x t_MAT err talker this expression is not a matrix if 1x 1 return cgetg 1 t_MAT if 1x lg x 1 err talker not a square matrix compute the Ly norm of x Ss gzero for i 1 i lt lx i s gadd s gnorml2 GEN x i if typ s t_REAL setlg s 3 s gsqrt s 3 we do not need much precision on s if s lt 1 we are happy k expo s if k lt 0 n 0 pi x else n k 1 pl gmul2n x n setexpo s 1 Before continuing several remarks are in order First before starting this computation which will produce garbage on the stack we have carefully saved the value of the stack pointer avma in ltop Note that we are going to assume through
235. ethod and can fail miserably if expr is not defined in the whole of a b try solve x 1 2 tan x The library syntax is zbrent void E GEN eval GEN void GEN a GEN b long prec Where eval x E returns the value of the function at x You may store any additional information required by eval in EF or set it to NULL 3 9 6 sum X a b expr x 0 sum of expression expr initialized at x the formal parameter going from a to b As for prod the initialization parameter x may be given to force the type of the operations being performed As an extreme example compare sum i 1 5000 1 i rational number denominator has 2166 digits time 1 241 ms sum i 1 5000 1 i 0 time 158 ms 2 9 094508852984436967 261245533 The library syntax is somme entree ep GEN a GEN b char expr GEN x This is to be used as follows ep represents the dummy variable used in the expression expr compute a72 b 2 define the dummy variable i entree ep is_entry i sum for a lt i lt b return somme ep a b i72 gzero 3 9 7 sumalt X a expr flag 0 numerical summation of the series expr which should be an alternating series the formal variable X starting at a If flag 0 use an algorithm of F Villegas as modified by D Zagier This is much better than Euler Van Wijngaarden s method which was used formerly If flag 1 use a variant with slightly different poly
236. ex e g acos 2 0 or acosh 0 0 Note also that the principal branch is always chosen e If the argument is an integermod or a p adic at present only a few functions like sqrt square root sqr square log exp powering teichmuller Teichmiiller character and agm arithmetic geometric mean are implemented 68 Note that in the case of a 2 adic number sqr x may not be identical to x for example if x 1 0O 2 and y 1 0O 2 then zxy 1 O 2 while sqr x 1 O 2 Here xxx yields the same result as sqr x since the two operands are known to be identical The same statement holds true for p adics raised to the power n where v n gt 0 Remark note that if we wanted to be strictly consistent with the PARI philosophy we should have x x y 4mod8 and sqr x 4mod 32 when both x and y are congruent to 2 modulo 4 However since integermod is an exact object PARI assumes that the modulus must not change and the result is hence 0 mod 4 in both cases On the other hand p adics are not exact objects hence are treated differently e If the argument is a polynomial power series or rational function it is if necessary first converted to a power series using the current precision held in the variable precd1 Under GP this again is transparent to the user When programming in library mode however the global variable precdl must be set before calling the function if the argument has an exact type i e not a power s
237. explosion detz can be computed using the function nfdetint The library syntax is nfhermitemod nf x detz 3 6 93 nfinit pol flag 0 pol being a non constant preferably monic irreducible polynomial in Z X initializes a number field structure nf associated to the field K defined by pol As such it s a technical object passed as the first argument to most nfzzxx functions but it contains some information which may be directly useful Access to this information via member functions is preferred since the specific data organization specified below may change in the future Currently nf is a row vector with 9 components nf 1 contains the polynomial pol nf pol nf 2 contains r1 r2 nf sign the number of real and complex places of K nf 3 contains the discriminant d K nf disc of K nf 4 contains the index of nf 1 i e Zx Z 0 where 0 is any root of nf 1 nf 5 is a vector containing 7 matrices M G T2 T MD TI MDI useful for certain com putations in the number field K e M is the r1 72 xn matrix whose columns represent the numerical values of the conjugates of the elements of the integral basis e G is such that T2 GG where T2 is the quadratic form T2 2 Y lo 2 1 o running over the embeddings of K into C e The T2 component is deprecated and currently unused 121 e T is the n x n matrix whose coefficients are Tr wjw where the w are the elements of the integral basis
238. f for instance Configure pg will create an Oxxx pr directory where a suitable version of PARI can be built 2 5 Compilation and tests To compile the GP binary and build the documentation type make all To only compile the GP binary type make gp in the distribution directory If your make program supports parallel make you can speed up the process by going to the Oxaaz directory that Configure created and doing a parallel make here for instance make j4 with GNU make It may even work from the toplevel directory The GP binary built above is optimized If you have run Configure g or pg and want to build a special purpose binary you can cd to the dbg or prf directory and type make gp there You can also invoke make gp dbg or make gp prf directly from the toplevel 2 5 1 Testing To test the binary type make bench This will build a static executable the default built by make gp is probably dynamic and run a series of comparative tests on those two To test only the default binary use make dobench which starts the bench immediately The static binary should be slightly faster In any case this should not take more than a few seconds user time on modern machines See the file MACHINES to get an idea of how much time comparable systems need We would appreciate a short note in the same format in case your system is not listed and you nevertheless have a working GP executable If a BUG message shows up something
239. f the polynomial pol given as a column vector where each root is repeated according to its multiplicity The precision is given as for transcendental functions under GP it is kept in the variable realprecision and is transparent to the user but it must be explicitly given as a second argument in library mode The algorithm used is a modification of A Sch nhage s root finding algorithm due to and implemented by X Gourdon Barring bugs it is guaranteed to converge and to give the roots to the required accuracy If flag 1 use a variant of the Newton Raphson method which is not guaranteed to converge but is rather fast If you get the messages too many iterations in roots or INTERNAL ERROR incorrect result in roots use the default algorithm This used to be the default root finding function in PARI until version 1 39 06 The library syntax is roots pol prec or rootsold pol prec 138 3 7 20 polrootsmod pol p flag 0 row vector of roots modulo p of the polynomial pol The particular non prime value p 4 is accepted mainly for 2 adic computations Multiple roots are not repeated If p lt 100 you may try setting flag 1 which uses a naive search In this case multiple roots are repeated with their order of multiplicity The library syntax is rootmod pol p flag 0 or rootmod2 pol p flag 1 3 7 21 polrootspadic pol p r row vector of p adic roots of the polynomial pol with p adic precision equ
240. factornf x 2 Mod z z7 2 1 y 2 1 eK incorrect type in gmulsg The library syntax is polfnf z t 108 3 6 32 galoisexport gal flag 0 gal being be a Galois field as output by galoisinit export the underlying permutation group as a string suitable for no flags or flag 0 GAP or flag 1 Magma The following example compute the index of the underlying abstract group in the GAP library G galoisinit x 6 108 s galoisexport G 2 Group Cts 2 3 4s 5 6 1 4 2 6 8 5 extern echo IdGroup s gap q 43 6 1 galoisidentify G 4 6 1 The library syntax is galoisexport gal flag 3 6 33 galoisfixedfield gal perm flag 0 v y gal being be a Galois field as output by galoisinit and perm an element of gal group or a vector of such elements computes the fixed field of gal by the automorphism defined by the permutations perm of the roots gal roots P is guaranteed to be squarefree modulo gal p If no flags or flag 0 output format is the same as for nfsubfield returning P x such that P is a polynomial defining the fixed field and x is a root of P expressed as a polmod in gal pol If flag 1 return only the polynomial P If flag 2 return P x F where P and x are as above and F is the factorization of gal pol over the field defined by P where variable v y by default stands for a root of P The priority of v must be less than the priority of the variable
241. ff Note In recent versions of readline 2 1 for instance the Alt or Meta key can give funny re sults output 8 bit accented characters for instance If you don t want to fall back to the Esc combination put the following two lines in your inputrc set convert meta on set output meta off 2 11 2 Command completion and online help As in the Emacs shell lt TAB gt will complete words for you But under readline this mechanism will be context dependent GP will strive to only give you meaningful completions in a given context it will fail sometimes but only under rare and restricted conditions For instance shortly after a we expect a user name then a path to some file Directly after default has been typed we would expect one of the default keywords After whatnow we expect the name of an old function which may well have disappeared from this version After a we expect a member keyword And generally of course we expect any GP symbol which may be found in the hashing lists functions both yours and GP s and variables If at any time only one completion is meaningful GP will provide it together with e an ending comma if we re completing a default e a pair of parentheses if we re completing a function name In that case hitting lt TAB gt again will provide the argument list as given by the online help Otherwise hitting lt TAB gt once more will give you the list of possible completions Just ex
242. fficients computes x as an element of K as a polmod assuming z is in K otherwise an error will occur If x is given on the relative integral basis apply rnfbasistoalg first otherwise PARI will believe you are dealing with a vector The library syntax is rnfelementdown rnf x 128 3 6 121 rnfeltreltoabs rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of L expressed as a polynomial or polmod with polmod coefficients computes x as an element of the absolute extension L Q as a polynomial modulo the absolute equation rnf pol If x is given on the relative integral basis apply rnfbasistoalg first otherwise PARI will believe you are dealing with a vector The library syntax is rnfelementreltoabs rnf x 3 6 122 rnfeltup rnf x rnf being a relative number field extension L K as output by rnfinit and x being an element of K expressed as a polynomial or polmod computes x as an element of the absolute extension L Q as a polynomial modulo the absolute equation rnf pol If x is given on the integral basis of K apply nfbasistoalg first otherwise PARI will believe you are dealing with a vector The library syntax is rnfelementup rnf x 3 6 123 rnfequation nf pol flag 0 given a number field nf as output by nfinit or simply a polynomial and a polynomial pol with coefficients in nf defining a relative extension L of nf computes the absolute equation of L over Q If flag is non
243. fficients a in Z The result is a 4 component vector f kod v c Here f is the exponent of p in the arithmetic conductor of E and kod is the Kodaira type which is coded as follows 1 means good reduction type Ip 2 3 and 4 mean types II III and IV respectively 4 v with y gt 0 means type I finally the opposite values 1 2 etc refer to the starred types Ig H etc The third component v is itself a vector u r s t giving the coordinate changes done during the local reduction Normally this has no use if u is 1 that is if the given equation was already minimal Finally the last component c is the local Tamagawa number cp The library syntax is localreduction p 3 5 17 elllseries E s A 1 E being a medium or long vector given by ellinit this computes the value of the L series of E at s In the present version 2 2 7 s must be a real number It is assumed that E is a minimal model over Z The optional parameter A is a cutoff point for the integral which must be chosen close to 1 for best speed The result must be independent of A so this allows some internal checking of the function Note that if the conductor of the curve is large say greater than 10 this function will take an unreasonable amount of time since it uses an O N algorithm The library syntax is lseriesell E s A prec where prec is a long and an omitted A is coded as NULL 93 3 5 18 ellminimalmodel amp v return the st
244. file default pari ps if you did not tamper with it Each time a new PostScript output is asked for the PostScript output is appended to that file Hence the user must remove this file or change the value of psfile first if he does not want unnecessary drawings from preceding sessions to appear On the other hand in this manner as many plots as desired can be kept in a single file None of the graphic functions are available within the PARI library you must be under GP to use them The reason for that is that you really should not use PARI for heavy duty graphical work there are much better specialized alternatives around This whole set of routines was only meant as a convenient but simple minded visual aid If you really insist on using these in your program we warned you the source plot c should be readable enough for you to achieve something 3 10 1 plot X a b expr Ymin Ymax crude ASCII plot of the function represented by expression expr from a to b with Y ranging from Ymin to Ymaz If Ymin resp Ymar is not given the minima resp the maxima of the computed values of the expression is used instead 3 10 2 plotbox w x2 y2 let x1 y1 be the current position of the virtual cursor Draw in the rectwindow w the outline of the rectangle which is such that the points xl yl and 2 y2 are opposite corners Only the part of the rectangle which is in w is drawn The virtual cursor does not move 3 10 3 plotc
245. fl 0 v Output all the subfields of the Galois group G as a vector This works by applying galoisfixedfield to all subgroups The meaning of the flag ff is the same as for galoisfixedfield The library syntax is galoissubfields G fl v where fl is a long and v a variable number 3 6 40 galoissubgroups G Output all the subgroups of the Galois group G A subgroup is a vector gen orders with the same meaning as for gal gen and gal orders Hence gen is a vector of permutations generating the subgroup and orders is the relatives orders of the generators The cardinal of a subgroup is the product of the relative orders The library syntax is galoissubgroups G 3 6 41 idealadd nf x y sum of the two ideals x and y in the number field nf When x and y are given by Z bases this does not depend on nf and can be used to compute the sum of any two Z modules The result is given in HNF The library syntax is idealadd nf x y 3 6 42 idealaddtoone nf x y x and y being two co prime integral ideals given in any form this gives a two component row vector a b such that a x b y and a b 1 The alternative syntax idealaddtoone nf v is supported where v is a k component vector of ideals given in any form which sum to Zg This outputs a k component vector e such that el z i for 1 lt i lt k and gt 2 2 eli 1 The library syntax is idealaddtoone0 nf x y where an omitted y is coded as NULL 3 6 43 ideala
246. fordiv n X seq the formal variable X ranging through the positive divisors of n the sequence seq is evaluated n must be of type integer 3 11 1 4 forprime X a b seq the formal variable X ranging over the prime numbers between a to b including a and b if they are prime the seq is evaluated More precisely the value of X is incremented to the smallest prime strictly larger than X at the end of each iteration Nothing is done if a gt b Note that a and b must be in R forprime p 2 12 print p if p 3 p 6 2 3 7 11 162 3 11 1 5 forstep X a b s seq the formal variable X going from a to b in increments of s the seg is evaluated Nothing is done if s gt 0 and a gt b or ifs lt 0 and a lt b s must be in R or a vector of steps s1 5n In the latter case the successive steps are used in the order they appear in s forstep x 5 20 2 4 print x 5 7 11 13 17 19 3 11 1 6 forsubgroup H G B seq executes seg for each subgroup H of the abelian group G given in SNF form or as a vector of elementary divisors whose index is bounded by B The subgroups are not ordered in any obvious way unless G is a p group in which case Birkhoff s algorithm produces them by decreasing index A subgroup is given as a matrix whose columns give its generators on the implicit generators of G For example the following prints all subgroups of index less than 2 in G Z 2Zg1 x Z 2Zga G
247. fou Bi Seg ee hd E a 40 42 60 DUECHAE eee A anta A 60 61 Strexpand 66 86 bbe eh es 61 Stritime lt errat oe ate Fe 15 19 Strictmatch cesa Yaw Bes ee 21 string contett 2 2 eee 40 202 SULTING xf bho ee Bek ee ae ae 7 28 40 SUrtex cimas keeles ne Oa ea 61 STECOGEN 5 iy he sa RT 60 SEUL ty hse cole es ode Fees Oke bl da Gd 139 sturmpatta a Gs wah Sh ee da 139 SUDCY CLO sic Baty Sede cs SER his 139 Subell o o a G 94 248 subfield ceci ae e 123 SUbfieldS s neg a is e AS 123 SUDETOUD ss ur a A ee 163 subgrouplist 134 163 subgrouplist0 134 SUBL A cate OR Geng OTS Ras 207 SUBD ate en ee see Ee A E AR N A A 207 SUDEES Di g pca en Ha we Seas 138 220 SUPTOSOXE io Be os Boe A Ese 77 subresultant algorithm 82 137 138 SUBS a a 140 SUBUUMOG Meos Da at Dros So A g ees 208 SUD gaui atio a a ae 54 SUM ais eaa ee e 153 155 suma lto niyin ei 155 SUMAIV ende a Take i SEE Ms 88 155 SUMING iA Gad aca aye A Lene d 155 SUMPOS ica Behe ees hu Eo IE A 155 156 SUppL IA ee ed GO ey ds 148 SWIGCGHEN fee ed vce Oe Bee ASE ta 195 SWitChout feo se ad Soe ee Be 194 195 sylvestermatrix 139 symmetric powers 139 system Gove ol hae Soe el see 20 42 166 168 T Lt ed e E ld es o in 98 tallennat ae ee eek hE 67 215 TILLIE Eg aii Galas bee aa Ree ae as 67 TALKOT ais hoe te 195 Tamagawa number 91 93 Gan generis te eh et Sie eee ee Ged 75
248. functions do not copy of its arguments nor do they produce an object a priori suitable for gerepileupto For instance gerepile safe components are universal objects z coefs_to_vec 3 gun gzero gdeux NOT OK for gerepileupto stoi 1 creates component before vector root z coefs_to_vec 3 stoi 1 gzero gdeux NO First vector component is destroyed x gclone gun z coefs_to_vec 3 x gzero gdeux gunclone x The following function is also available as a special case of coefs_to_int GEN u2toi ulong a ulong b Returns the GEN equal to 2 7a b assuming that a b lt 2 This does not depend on sizeof long the behaviour is as above on both 32 and 64 bit machines 4 3 2 Assignments Firstly if x and y are both declared as GEN i e pointers to something the ordinary C assignment y x makes perfect sense we are just moving a pointer around However physically modifying either x or y for instance x 1 0 will also change the other one which is usually not desirable 177 Very important note Using the functions described in this paragraph is very inefficient and often awkward one of the gerepile functions see Section 4 4 should be preferred See the paragraph end for some exceptions to this rule The general PARI assignment function is the function gaffect with the following syntax void gaffect GEN x GEN y Its effect is to assign the PARI object x into the preexisting object y
249. g The library syntax is gtovecsmall z 3 2 15 binary x outputs the vector of the binary digits of x Here x can be an integer a real number in which case the result has two components one for the integer part one for the fractional part or a vector matrix The library syntax is binaire 3 2 16 bitand z y bitwise and of two integers x and y that is the integer Se and y 2 Negative numbers behave as if modulo a huge power of 2 The library syntax is gbitand z y 61 3 2 17 bitneg x n 1 bitwise negation of an integer x truncated to n bits that is the integer S not x 2 i 0 The special case n 1 means no truncation an infinite sequence of leading 1 is then represented as a negative number Negative numbers behave as if modulo a huge power of 2 The library syntax is gbitneg z 3 2 18 bitnegimply x y bitwise negated imply of two integers x and y or not x gt y that is the integer So 2 andnot y 2 Negative numbers behave as if modulo a huge power of 2 The library syntax is gbitnegimply z y 3 2 19 bitor x y bitwise inclusive or of two integers x and y that is the integer Ao or yi 2 Negative numbers behave as if modulo a huge power of 2 The library syntax is gbitor z y 3 2 20 bittest x n c 1 extracts c bits starting from number n from the right in the development of x i e the coefficient of 2 in the binary expansion of x returning the bits
250. g importance Killed variables see kill will be shown as 0 If x is non empty it must be a permutation of variable names and this permutation gives a new order of importance of the variables for output only For example if the existing order is x y z then after reorder z x the order of importance of the variables with respect to output will be z y x The internal representation is unaffected 3 11 2 23 setrand n reseeds the random number generator to the value n The initial seed is n l The library syntax is setrand n where n is a long Returns n 3 11 2 24 system str str is a string representing a system command This command is executed its output written to the standard output this won t get into your logfile and control returns to the PARI system This simply calls the C system command 3 11 2 25 trap e rec seq tries to execute seq trapping error e that is effectively pre venting it from aborting computations in the usual way the recovery sequence rec is executed if the error occurs and the evaluation of rec becomes the result of the command If e is omitted all exceptions are trapped Note in particular that hitting C Control C raises an exception See Section 2 7 2 for an introduction to error recovery under GP trap division by 0 inv x trap gdiver INFINITY 1 x inv 2 1 1 2 inv 0 2 INFINITY If seg is omitted defines rec as a default action when catching excep
251. g obtained by simply typing 7 These names are in general not informative enough More details can be obtained by typing function which gives a short explanation of the function s calling convention and effects Of course to have complete information read Chapter 3 of this manual the source code is at your disposal as well though a trifle less readable Much better help can be obtained through the extended help system see below If the line before the copyright message indicates that extended help is available this means perl is installed on your system GP was told about it at compile time and the whole PARI distribution was correctly installed you can add more signs for extended functionalities 7 keyword yields the functions description as it stands in this manual usually in Chapter 2 or 3 If you re not satisfied with the default chapter chosen you can impose a given chapter by ending the keyword with followed by the chapter number e g Hello 2 will look in Chapter 2 for section heading Hello which doesn t exist by the way All operators e g amp amp etc are accepted by this extended help as well as a few other keywords describing key GP concepts e g readline the line editor integer nf number field as used in most algebraic number theory computations e11 elliptic curves etc In case of conflicts between function and default names e g log simplify the function has higher prior
252. g typ GEN z returns the type number of z void settyp GEN z long n sets the type number of z to n you should not have to use this function if you use cgetg long lg GEN z returns the length in longwords of the root of z long setlg GEN z long 1 sets the length of z to 1 you should not have to use this function if you use cgetg however see an advanced example in Section 4 8 If you know enough about PARI to need to access the clone bit then you will be able to find usage hints in the code It is technical after all These macros are written in such a way that you do not need to worry about type casts when using them i e if z is a GEN typ z 2 will be accepted by your compiler without your having to explicitly type typ GEN z 2 Note that for the sake of efficiency none of the codeword handling macros check the types of their arguments even when there are stringent restrictions on their use The possible second codeword is used differently by the different types and we will describe it as we now consider each of them in turn 186 4 5 1 Type t_INT integer this type has a second codeword z 1 which contains the following information the sign of z coded as 1 0 or 1 if z gt 0 z 0 z lt 0 respectively the effective length of z i e the total number of significant longwords This means the following apart from the integer 0 every integer is normalized meaning that the first mantissa lon
253. ger satisfying the inequality 0 lt z 2 lt z 1 It is good practice to keep the modulus object on the heap so that new integermods resulting from operations can point at this common object instead of carrying along their own copies of it on the stack The library functions implement this practice almost by default 4 5 4 Type t_FRAC and t_FRACN rational number z 1 points to the numerator and z 2 to the denominator Both must be of type integer In principle z 2 gt 0 but this rule does not have to be strictly obeyed Note that a type t_FRACN rational number can be converted to irreducible form using the function GEN gred GEN x 4 5 5 Type t_COMPLEX complex number z 1 points to the real part and z 2 to the imaginary part A priori z 1 and z 2 can be of any type but only certain types are useful and make sense 4 5 6 Type t_PADIC p adic numbers this type has a second codeword 1 which contains the following information the p adic precision the exponent of p modulo which the p adic unit corresponding to z is defined if z is not 0 i e one less than the number of significant p adic digits and the exponent of z This information can be handled using the following functions long precp GEN z returns the p adic precision of z void setprecp GEN z long 1 sets the p adic precision of z to 1 long valp GEN z returns the p adic valuation of z i e the exponent This is defined even if z is equal to 0 see Section 1
254. gn is 0 However its effective length may be equal to 2 or greater than 2 If it is greater than 2 this means that all the coefficients of the polynomial are equal to zero as they should for a zero polynomial but not all of these zeros are exact zeros and more precisely the leading term z 1gef z 1 is not an exact zero 4 5 10 Type t_SER power series This type also has a second codeword which encodes a sign i e O if the power series is 0 and 1 if not a variable number as for polynomials and an exponent This information can be handled with the following functions signe setsigne varn setvarn as for polynomials and valp setvalp for the exponent as for p adic numbers Beware do not use expo and setexpo on power series If the power series is non zero z 2 z 3 z 1g z 1 point to the coefficients of z in ascending order z 2 being the first non zero coefficient Note that the exponent of a power series can be negative i e we are then dealing with a Laurent series with a finite number of negative terms 189 4 5 11 Type t_RFRAC and t_RFRACN rational function z 1 points to the numerator and z 2 on the denominator The denominator must be of type polynomial Note that a type t_RFRACN rational function can be converted to irreducible form using the function gred 4 5 12 Type t_QFR indefinite binary quadratic form z 1 z 2 z 3 point to the three coefficients of the form and should be of type integer z
255. gword i e z 2 is non zero However the integer may have been created with a longer length Hence the length which is in z 0 can be larger than the effective length which is in z 1 Accessing z i for i larger than or equal to the effective length will yield random results This information is handled using the following macros long signe GEN z returns the sign of z void setsigne GEN z long s sets the sign of z to s long lgefint GEN z returns the effective length of z void setlgefint GEN z long 1 sets the effective length of z to 1 The integer 0 can be recognized either by its sign being 0 or by its effective length being equal to 2 When z 0 the word z 2 exists and is non zero and the absolute value of z is z 2 z 3 z 1gefint z 1 in base 2 BITS_IN_LONG where as usual in this notation z 2 is the highest order longword The following further macros are available long mpodd GEN x which is 1 if x is odd and 0 otherwise long mod2 GEN x mod4 x and so on up to mod64 x which give the residue class of x modulo the corresponding power of 2 for positive x By definition modn a modn z for x lt 0 the macros disregard the sign and the result is undefined if z 0 These macros directly access the binary data and are thus much faster than the generic modulo functions Besides they return long integers instead of GENs so they do not clutter up the stack 4 5 2 Type t_REAL rea
256. h that d pol Z g f 0 L K rnf 5 is currently unused rnf 6 is currently unused rnf T is a two component row vector where the first component is the relative integral pseudo basis expressed as polynomials in the variable of pol with polmod coefficients in nf and the second component is the ideal list of the pseudobasis in HNF rnf 8 is the inverse matrix of the integral basis matrix with coefficients polmods in nf rnf 9 is currently unused rnf 10 is nf rnf 11 is the output of rnfequation nf pol 1 Namely a vector vabs with 3 entries describing the absolute extension L Q vabs 1 is an absolute equation more conveniently obtained as rnf pol vabs 2 expresses the generator a of the number field nf as a polynomial modulo the absolute equation vabs 1 vabs 3 is a small integer k such that if 8 is an abstract root of pol and a the generator of nf the generator whose root is vabs will be 8 ka Note that one must be very careful if k 4 0 when dealing simultaneously with absolute and relative quantities since the generator chosen for the absolute extension is not the same as for the relative one If this happens 131 one can of course go on working but we strongly advise to change the relative polynomial so that its root will be G ka Typically the GP instruction would be pol subst pol x x k Mod y nf pol rnf 12 is by default unused and set equal to 0 This field is used to store further inf
257. har expr 3 8 59 vectorsmall n X expr 0 creates a row vector of small integers type t_VECSMALL with n components whose components are the expression expr evaluated at the integer points between 1 and n If one of the last two arguments is omitted fill the vector with zeroes The library syntax is vecteursmall GEN nmax entree ep char expr 3 8 60 vectorv n X expr as vector but returns a column vector type t_COL The library syntax is vvecteur GEN nmax entree ep char expr 3 9 Sums products integrals and similar functions Although the GP calculator is programmable it is useful to have preprogrammed a number of loops including sums products and a certain number of recursions Also a number of functions from numerical analysis like numerical integration and summation of series will be described here One of the parameters in these loops must be the control variable hence a simple variable name The last parameter can be any legal PARI expression including of course expressions using loops Since it is much easier to program directly the loops in library mode these functions are mainly useful for GP programming The use of these functions in library mode is a little tricky and its explanation will be mostly omitted although the reader can try and figure it out by himself by checking the example given for the sum function In this section we only give the library syntax with no semantic explanation The
258. hat we have lost the address just before the final result note that the loop is not executed if n is 0 It is safe to use gerepileupto here as y will have been created by either gsqr or gadd both of which are guaranteed to return suitable objects Remarks As such the program should work most of the time if x is a square matrix with real or complex entries Indeed since essentially the first thing that we do is to multiply by the real number 1 the program should work for integer real rational complex or quadratic entries This is in accordance with the behavior of transcendental functions Furthermore since this program is intended to be only an illustrative example it has been written a little sloppily In particular many error checks have been omitted and the efficiency is far from optimal An evident improvement would be the use of gerepileupto mentioned above Another improvement is to multiply the matrix x by the real number 1 right at the beginning speeding up the computation of the L norm in many cases These improvements are included in the version given in Appendix B Still another improvement would come from a better choice of n If the reader takes a look at the implementation of the function mpexp1 in the file basemath trans1 c he can make the necessary changes himself Finally there exist other algorithms of a different nature to compute the exponential of a matrix 4 9 Adding functions to PARI 4 9 1 Nota Bene As mention
259. he 108 iSO 3 oy aoe Lae bie er Ae A 89 98 GASCE siete a De doh ded ee 118 AUS CST A he ot odes hone iva i Sh lip a eye 137 diviiexactis arrenar fan ea eh 214 divise hls bees ON we eis 214 AVUI ie sony ae ad NS i OS 214 ATVISOES cacr si goede oe ee o BE 79 diviuexact 214 diy ll da os Pad da dla 208 Givrem erica a ee ee 35 56 diVSUIM soe o Ah bi ht ES s 155 Giviumods Cars ae te be 208 AVERE ec OR 51 dymdit a a ii da 214 AVM TEA nia Sn ots ak ke ee ea 214 AVALS a ee Las Ae nas 213 AUMITEZ Ee soa ik Pd Ne eS 214 AMAS saaier at Me ad aa e o e 213 AVM STZ 4 a 214 dvmdss gt loe Ste a 3 213 GVMAUSSZ k a ee a ee Bk wd 214 E echo 4S beet i oho et 17 23 ECM cido tess Ae ec eed Aled oh RE da 76 81 editing characters 31 effective length 187 189 eygalid Gop aldeas ia Oe be a 212 Sigem LE AAA diz 144 o Lana ake Eek hk 72 element dit iio bo oan Ae 118 element_divmodpr 118 element mid coord ees 118 element_mulmodpr 118 element_pow o 118 236 element_powmodpr 118 element_reduce 119 element_val 0 0 119 SL tert wi Sh te E hk By 39 Clladd viir reii ad A 90 CITA sl Yt BM 90 A E ered 90 CUA a ia A Qt B 90 LLAD ir ID Pee Se Gate A 90 CLIDIL da a a 90 ellchangecurve 91 ellchangepoint 91 elleisnum mirrors A a eS 91 lleta kratego a t een ae eee AUS 91 ellglob
260. he calculator the general command line syntax is gp s stacksize p primelimit files where items within brackets are optional The files argument is a list of files written in the GP scripting language which will be loaded on startup The ones starting with a minus sign are flags setting some internal parameters of GP or defaults See Section 2 1 below for a list and explanation of all defaults there are many more than just those two These defaults can be changed by adding parameters to the input line as above or interactively during a GP session or in a preferences file also known as gprc Some features were developed on UNIX platforms and depend heavily on the operating system in use It is possible that some of these will be ported to other operating systems BeOS MacOS DOS OS 2 Windows etc in future versions most of them should be relatively easy tasks As for now most of them were not So whenever a specific feature of the UNIX version is discussed in a paragraph a UNIX sign sticks out in the left margin like here Just skip these if you re stranded on a different operating system the core GP functions i e at least everything which is even faintly mathematical in nature will still be available to you It should also be possible and then definitely advisable to install Linux or FreeBSD on your machine Note added in version 2 0 12 Most UNIX goodies are now available for DOS OS 2 and Windows thanks to the
261. he exponent is an integer then exact operations are performed using binary left shift powering techniques In particular in this case x cannot be a vector or matrix unless it is a square matrix and moreover invertible if the exponent is negative If x is a p adic number its precision will increase if v n gt 0 Powering a binary quadratic form types t_QFI and t_QFR returns a reduced representative of the class provided the input is reduced In particular 771 is identical to zx PARI is able to rewrite the multiplication x x of two identical objects as 1 or sqr x here identical means the operands are two different labels referencing the same chunk of memory no equality test is performed This is no longer true when more than two arguments are involved If the exponent is not of type integer this is treated as a transcendental function see Sec tion 3 3 and in particular has the effect of componentwise powering on vector or matrices As an exception if the exponent is a rational number p q and x an integer modulo a prime return a solution y of y x if it exists Currently q must not have large prime factors 56 Beware that Mod 7 19 7 1 2 1 Mod 11 19 is any square root sqrt Mod 7 19 42 Mod 8 19 is the smallest square root Mod 7 19 3 5 3 Mod 1 19 437 5 3 14 Mod 1 19 Mod 7 19 is just another cubic root The library syntax is gpow x n prec for xn
262. he length of the result is then equal to the length of b unless a partial remainder is encountered which is equal to zero in which case the expansion stops In the case of real numbers the stopping criterion is thus different from the one mentioned above since if b is too long some partial quotients may not be significant If b is an integer the command is understood as contfrac z nmaz The library syntax is contfracO z b nmazx Also available are gboundcf x nmax gcf x or gcf2 b x where nmaz is a C integer 3 4 10 contfracpnqn z when x is a vector or a one row matrix x is considered as the list of partial quotients a9 a1 of a rational number and the result is the 2 by 2 matrix Pn Pn 1 qn dn 1 in the standard notation of continued fractions so pn qn ao 1 a1 1 a If x is a matrix with two rows bp b1 b and fao a1 n this is then considered as a generalized continued fraction and we have similarly pn qn 1 bo ao bi ai bn an Note that in this case one usually has by 1 The library syntax is pnqn z 3 4 11 core n flag 0 if n is a non zero integer written as n df with d squarefree returns d If flag is non zero returns the two element row vector d f The library syntax is core0 n flag Also available are core n core n 0 and core2 n core n 1 3 4 12 coredisc n flag if n is a non zero integer written as n df with d fundamental di
263. he value of prettyprinter The default script tex2mail converts its input to readable two dimensional text Independently of the setting of this default an object can be printed in any of the three formats at any time using the commands Na m and Mb respectively see below 2 1 17 parisize default 1M bytes on the Mac 4M otherwise GP and in fact any program using the PARI library needs a stack in which to do its computations parisize is the stack size in bytes It is strongly recommended you increase this default using the s command line switch or a gprc if you can afford it Don t increase it beyond the actual amount of RAM installed on your computer or GP will spend most of its time paging In case of emergency you can use the allocatemem function to increase parisize once the session is started 2 1 18 path default gp on UNIX systems C C GP on DOS OS 2 and Windows and otherwise This is a list of directories separated by colons semicolons in the DOS world since colons are pre empted for drive names When asked to read a file whose name does not contain i e no explicit path was given GP will look for it in these directories in the order they were written in path Here as usual means the current directory and its immediate parent Environment expansion is performed 2 1 19 prettyprinter default tex2mail TeX noindent ragged by_par the name of an externa
264. her non string type two t_STR objects are compared using the standard lexicographic order The standard boolean functions inclusive or amp amp and and not are also available and the library syntax is gor x y gand x y and gnot respectively In library mode it is in fact usually preferable to use the two basic functions which are gcmp z y which gives the sign 1 0 or 1 of x y where x and y must be in R and gegal z y which can be applied to any two PARI objects x and y and gives 1 i e true if they are equal but not necessarily identical 0 i e false otherwise Particular cases of gegal which should be used are gemp0 z 1 0 gemp1 x x 1 and gemp_1 z x 1 Note that gemp0 x tests whether x is equal to zero even if x is not an exact object To test whether x is an exact object which is equal to zero one must use isexactzero Also note that the gemp and gegal functions return a C integer and not a GEN like gle etc GP accepts the following synonyms for some of the above functions since we thought it might easily lead to confusion we don t use the customary C operators for bitwise and or bitwise or use bitand or bitor hence and amp are accepted as synonyms of and amp amp respectively Also lt gt is accepted as a synonym for On the other hand is definitely not a synonym for since it is the assignment statement 57 3 1 13 lex z y gives the resul
265. hich can be recalled by typing t under GP The library syntax is compo x n where n is a long The two other methods are more natural but more restricted The function polcoeff z n gives the coefficient of degree n of the polynomial or power series x with respect to the main variable of x to check variable ordering or to change it use the function reorder see Section 3 11 2 22 In particular if n is less than the valuation of x or in the case of a polynomial greater than the degree the result is zero contrary to compo which would send an error message If x is a power series and n is greater than the largest significant degree then an error message is issued For greater flexibility vector or matrix types are also accepted for x and the meaning is then identical with that of compo Finally note that a scalar type is considered by polcoeff as a polynomial of degree zero The library syntax is truecoeff x n The third method is specific to vectors or matrices under GP If x is a row or column vector then x n represents the nt component of x i e compo x n It is more natural and shorter to 63 write If x is a matrix x m n represents the coefficient of row m and column n of the matrix x m represents the mt row of x and x n represents the n column of z Finally note that in library mode the macros coeff and mael are available to deal with the non recursivity of the GEN type from the compiler s point of view
266. ial as above and B is the integer basis as would be computed by nfbasis This can be useful if the integer basis is known in advance If flag 2 pol is changed into another polynomial P defining the same number field which is as simple as can easily be found using the polred algorithm and all the subsequent computations are done using this new polynomial In particular the first component of the result is the modified polynomial If flag 3 does a polred as in case 2 but outputs nf Mod a P where nf is as before and Mod a P Mod zx pol gives the change of variables This is implicit when pol is not monic first a linear change of variables is performed to get a monic polynomial then a polred reduction If flag 4 as 2 but uses a partial polred If flag 5 as 3 using a partial polred The library syntax is nfinitO z flag prec 3 6 94 nfisideal nf x returns 1 if x is an ideal in the number field nf 0 otherwise The library syntax is isideal z 122 3 6 95 nfisincl x y tests whether the number field K defined by the polynomial x is conjugate to a subfield of the field L defined by y where x and y must be in Q X If they are not the output is the number 0 If they are the output is a vector of polynomials each polynomial a representing an embedding of K into L i e being such that y zoa If y is a number field nf a much faster algorithm is used factoring x over y using nffactor Before version
267. ibrary syntax is gzeta s prec 3 4 Arithmetic functions These functions are by definition functions whose natural domain of definition is either Z or Zs or sometimes polynomials over a base ring Functions which concern polynomials exclusively will be explained in the next section The way these functions are used is completely different from transcendental functions in general only the types integer and polynomial are accepted as arguments If a vector or matrix type is given the function will be applied on each coefficient independently In the present version 2 2 7 all arithmetic functions in the narrow sense of the word Euler s totient function the Moebius function the sums over divisors or powers of divisors etc call after trial division by small primes the same versatile factoring machinery described under factorint It includes Shanks SQUFOF Pollard Rho ECM and MPQS stages and has an early exit option for the functions moebius and the integer function underlying issquarefree Note that it relies on a fairly strong probabilistic primality test see ispseudoprime 3 4 1 addprimes x adds the primes contained in the vector x or the single integer x to the table computed upon GP initialization by pari_init in library mode and returns a row vector entries contain all such user primes Whenever factor or smallfact is subsequently called first the primes in the table computed by pari_init will be check
268. icated functions in the next section let us see here how to input values of the different data types known to PARI Recall that blanks are ignored in any expression which is not a string see below 2 3 1 Integers type t_INT type the integer with an initial or if desired with no decimal point 2 3 2 Real numbers type t_REAL type the number with a decimal point The internal precision of the real number will be the supremum of the input precision and the default precision For example if the default precision is 28 digits typing 2 will give a number with internal precision 28 but typing a 45 significant digit real number will give a number with internal precision at least 45 although less may be printed You can also use scientific notation with the letter E or e in which case the non leading decimal point may be omitted like 6 02 E 23 or 1e 5 but not e10 By definition 0 E N or 0 E N returns a real 0 of decimal exponent N whereas 0 returns a real 0 of default precision of exponent defaultprecision see Section 1 2 6 3 2 3 3 Integermods type t_INTMOD to enter n mod m type Mod n m not n m see Section 3 2 3 2 3 4 Rational numbers types t_FRAC and t_FRACN under GP all fractions are automatically reduced to lowest terms so it is in principle impossible to work with reducible fractions of type t_FRACN although of course in library mode this is easy To enter n m just type it as written As e
269. ich you want to work This is done using the quadgen function in the following way Write something like w quadgen d where d is the discriminant of the quadratic order in which you want to work hence d is congruent to 0 or 1 modulo 4 The name w is of course just a suggestion but corresponds to traditional usage You can use any variable name that you like However quadratic numbers are always printed with a w regardless of the discriminant So beware two numbers can be printed in the same way and not be equal However GP will refuse to add or multiply them for example Now 1 w will be the canonical integral basis of the quadratic order i e w Vd 2 if d 0mod 4 and w 1 Vd 2 if d 1 mod 4 where d is the discriminant and to enter yw you just type x y w 2 3 8 Polmods type t_POLMOD exactly as for integermods to enter z mody where x and y are polynomials type Mod x y not x y see Section 3 2 3 Note that when y is an irreducible polynomial in one variable polmods whose modulus is y are simply algebraic numbers in the finite extension defined by the polynomial y This allows us to work easily in number fields finite extensions of the p adic field Qp or finite fields Important remark Since the variables occurring in a polmod are not free variables it is essential in order to avoid inconsistencies that polmods use the same variable in internal operations i e be tween polmods and variables of lower p
270. ick bidirectional exact division is not implemented void diviuexact GEN x ulong y returns the Euclidean quotient x y assuming y divides x and y is odd 214 5 2 9 Miscellaneous functions void addsii long s GEN x GEN z assigns the sum of the long s and the integer x into the integer z essentially identical to addsiz except that z is specifically an integer long divise GEN x GEN y if the integer y divides the integer x returns 1 true otherwise returns 0 false long divisii GEN x long s GEN z assigns the Euclidean quotient of the integer x and the long s into the integer z and returns the remainder as a long long mpdivis GEN x GEN y GEN z if the integer y divides the integer x assigns the quotient to the integer z and returns 1 true otherwise returns 0 false void mulsii long s GEN x GEN z assigns the product of the long s and the integer x into the integer z essentially identical to mulsiz except that z is specifically an integer void addumului ulong a ulong b GEN x return a b X 5 3 Level 2 kernel operations on general PARI objects The functions available to handle subunits are the following GEN compo GEN x long n creates a copy of the n th true component i e not counting the codewords of the object x GEN truecoeff GEN x long n creates a copy of the coefficient of degree n of x if x is a scalar polynomial or power series and otherwise of the n th component of x The re
271. if it exists BIGINT otherwise 3 3 Transcendental functions As a general rule which of course in some cases may have exceptions transcendental functions operate in the following way e If the argument is either an integer a real a rational a complex or a quadratic number it is if necessary first converted to a real or complex number using the current precision held in the default realprecision Note that only exact arguments are converted while inexact arguments such as reals are not Under GP this is transparent to the user but when programming in library mode care must be taken to supply a meaningful parameter prec as the last argument of the function if the first argument is an exact object This parameter is ignored if the argument is inexact Note that in library mode the precision argument prec is a word count including codewords i e represents the length in words of a real number while under GP the precision which is changed by the metacommand p or using default realprecision is the number of significant decimal digits Note that some accuracies attainable on 32 bit machines cannot be attained on 64 bit machines for parity reasons For example the default GP accuracy is 28 decimal digits on 32 bit machines corresponding to prec having the value 5 but this cannot be attained on 64 bit machines After possible conversion the function is computed Note that even if the argument is real the result may be compl
272. imes addprimes empties the extra prime table x can also be a single integer List the current extra primes if x is omitted The library syntax is removeprimes z 3 4 62 sigma z k 1 sum of the kt powers of the positive divisors of x x must be of type integer The library syntax is sumdiv x sigma x or gsumdivk z k sigma z k where k is a C long integer 3 4 63 sqrtint x integer square root of x which must be a non negative integer The result is non negative and rounded towards zero The library syntax is racine z 88 3 4 64 znlog x g g must be a primitive root mod a prime p and the result is the discrete log of x in the multiplicative group Z pZ This function uses a simple minded combination of Pohlig Hellman algorithm and Shanks baby step giant step which requires O q storage where q is the largest prime factor of p 1 Hence it cannot be used when the largest prime divisor of p 1 is greater than about 10 The library syntax is znlog z g 3 4 65 znorder x x must be an integer mod n and the result is the order of x in the multiplicative group Z nZ Returns an error if x is not invertible The library syntax is order z 3 4 66 znprimroot n returns a primitive root generator of Z nZ whenever this latter group is cyclic n 4 or n 2p or n p where p is an odd prime and k gt 0 The library syntax is gener z 3 4 67 znstar n gives the structure of the m
273. in gerepile But beware as well that the addresses of all the objects in the translated zone will have changed after a call to gerepile every pointer you may have kept around elsewhere outside the stack objects which previously pointed into the zone below 1top must be discarded If you need to recover more than one object use one of the gerepilemany functions below As a consequence of the preceding explanation we must now state the most important law about programming in PARI If a given PARI object is to be relocated by gerepile then apart from universal ob jects the chunks of memory used by its components should be in consecutive memory locations All GENs created by documented PARI function are guaranteed to satisfy this This is because the gerepile function knows only about two connected zones the garbage that will be erased between 1bot and 1top and the significant pointers that will be copied and updated If there is garbage interspersed with your objects disasters will occur when we try to update them and consider the corresponding pointers So be very wary when you allow objects to become disconnected Have a look at the examples it is not as complicated as it seems In practice this is achieved by the following programming idiom ltop avma garbage lbot avma gq anything return gerepile ltop lbot q returns the updated q Beware that ltop avma garbage return gerepile ltop avma anything
274. ing which of course accepts numeric arguments without adverse effects due to the expansion mechanism See Section 2 1 for a list of available defaults and Section 2 2 for some shortcut alternatives Typing default or Md yields the complete default list as well as their current values If val is omitted prints the current value of default key If flag is set returns the result instead of printing it 3 11 2 5 error str outputs its argument list each of them interpreted as a string then interrupts the running GP program returning to the input prompt Example error n n is not squarefree Note that due to the automatic concatenation of strings you could in fact use only one argument just by suppressing the commas 3 11 2 6 extern str the string str is the name of an external command i e one you would type from your UNIX shell prompt This command is immediately run and its input fed into GP just as if read from a file 3 11 2 7 getheap returns a two component row vector giving the number of objects on the heap and the amount of memory they occupy in long words Useful mainly for debugging purposes The library syntax is getheap 3 11 2 8 getrand returns the current value of the random number seed Useful mainly for debugging purposes The library syntax is getrand returns a C long 3 11 2 9 getstack returns the current value of top avma i e the number of bytes used up to now on
275. ing reduced forms an O D algorithm See also qfbhclassno 3 4 45 qfbcompraw z y composition of the binary quadratic forms x and y without reduction of the result This is useful e g to compute a generating element of an ideal The library syntax is compraw z y 85 3 4 46 qfbhclassno x Hurwitz class number of x where x is non negative and congruent to 0 or 3 modulo 4 See also qfbclassno The library syntax is hclassno z 3 4 47 qfbnucomp z y composition of the primitive positive definite binary quadratic forms x and y type t_QFI using the NUCOMP and NUDUPL algorithms of Shanks la Atkin I is any positive constant but for optimal speed one should take DY 4 where D is the common discriminant of x and y When z and y do not have the same discriminant the result is undefined The library syntax is nucomp z y The auxiliary function nudupl z 1 should be used instead for speed when x y 3 4 48 qfbnupow x n n th power of the primitive positive definite binary quadratic form x using Shanks s NUCOMP and NUDUPL algorithms see qfbnucomp The library syntax is nupow z n 3 4 49 qfbpowraw z 7 n th power of the binary quadratic form x computed without doing any reduction i e using qfbcompraw Here n must be non negative and n lt 2 31 The library syntax is powraw x n where n must be a long integer 3 4 50 qfbprimeform p prime binary quadratic form of discriminant x whose first coefficient is
276. ins the matrix B i e the matrix containing the expressions of the prime ideal factorbase in terms of the p It is an r x c matrix bnf 3 contains the complex logarithmic embeddings of the system of fundamental units which has been found It is an r 72 x r r2 1 matrix 100 bnf 4 contains the matrix M of Archimedean components of the relations of the matrix WIB bnf 5 contains the prime factor base i e the list of prime ideals used in finding the relations bnf 6 used to contain a permutation of the prime factor base but has been obsoleted It contains a dummy 0 bnf 9 is a 3 element row vector used in bnfisprincipal only and obtained as follows Let D UWV obtained by applying the Smith normal form algorithm to the matrix W bnf 1 and let U be the reduction of U modulo D The first elements of the factorbase are given in terms of bnf gen by the columns of U with Archimedean component ga let also GD be the Archimedean components of the generators of the principal ideals defined by the bnf gen i bnf cyc i Then bnf 9 U Ja Dal Finally bnf 10 is by default unused and set equal to 0 This field is used to store further information about the field as it becomes available which is rarely needed hence would be too expensive to compute during the initial bnfinit call For instance the generators of the principal ideals bnf gen i bnf cyc i during a call to bnrisprincipa
277. is centerliftO z v where v is a long and an omitted v is coded as 1 Also available is centerlift x centerlift0 x 1 3 2 24 changevar z y creates a copy of the object x where its variables are modified according to the permutation specified by the vector y For example assume that the variables have been introduced in the order x a b c Then if y is the vector x c a b the variable a will be replaced by c b by a and c by b x being unchanged Note that the permutation must be completely specified e g c a b would not work since this would replace x by c and leave a and b unchanged as well as c which is the fourth variable of the initial list In particular the new variable names must be distinct The library syntax is changevar z y 3 2 25 components of a PARI object There are essentially three ways to extract the components from a PARI object The first and most general is the function component x n which extracts the n component of x This is to be understood as follows every PARI type has one or two initial code words The components are counted starting at 1 after these code words In particular if x is a vector this is indeed the n component of x if x is a matrix the nt column if x is a polynomial the nt coefficient i e of degree n 1 and for power series the nt significant coefficient The use of the function component implies the knowledge of the structure of the different PARI types w
278. is type has a second codeword which is analogous to the one for integers It contains a sign 0 if the polynomial is equal to 0 and 1 if not see however the important remark below a variable number e g 0 for x 1 for y etc and an effective length These data can be handled with the following macros signe and setsigne as for reals and integers long lgef GEN z returns the effective length of z void setlgef GEN z long 1 sets the effective length of z to 1 long varn GEN z returns the variable number of the object z void setvarn GEN z long v sets the variable number of z to v The variable numbers encode the relative priorities of variables as discussed in Section 2 6 2 We will give more details in Section 4 6 Note also the function long gvar GEN z which tries to return a variable number for z even if z is not a polynomial or power series The variable number of a scalar type is set by definition equal to BIGINT The components z 2 z 3 z 1gef z 1 point to the coefficients of the polynomial in ascending order with z 2 being the constant term and so on Note that the degree of the polynomial is equal to its effective length minus three The function long degree GEN x returns the degree of x with respect to its main variable even when x is not a polynomial a rational function for instance By convention the degree of 0 is 1 Important remark A zero polynomial can be characterized by the fact that its si
279. ise Then the series will converge at least as fast as the usual one for et and the cutoff error will be easy to estimate In fact a larger value of n would be preferable but this is slightly machine dependent and more complicated and will be left to the reader Let us start writing our program So as to be able to use it in other contexts we will structure it in the following way a main program which will do the input and output and a function which we shall call matexp which does the real work The main program is easy to write It can be something like this include lt pari h gt GEN matexp GEN x long prec int main long d prec 3 GEN x take a stack of 10 bytes no prime table pari_init 1000000 2 printf precision of the computation in decimal digits n d itos lisGEN stdin 197 if d gt 0 prec long d pariK1 3 printf input your matrix in GP format n x matexp lisGEN stdin prec sort g d 0 exit 0 The variable prec represents the length in longwords of the real numbers used pariKl is a constant defined in paricom h equal to In 10 In 2 BITS_IN_LONG which allows us to convert from a number of decimal digits to a number of longwords independently of the actual bit size of your long integers The function lisGEN reads an expression here from standard input and converts it to a GEN like the GP parser itself would This means it takes care of whitespace
280. isticated functions are also implemented like solving Thue equations finding integral bases and discriminants of number fields computing class groups and fundamental units computing 10 in relative number field extensions including explicit class field theory and also many functions dealing with elliptic curves over Q or over local fields 1 3 6 Other functions Quite a number of other functions dealing with polynomials e g finding complex or p adic roots factoring etc power series e g substitution reversion linear algebra e g determinant charac teristic polynomial linear systems and different kinds of recursions are also included In addition standard numerical analysis routines like Romberg integration open or closed on a finite or infinite interval real root finding when the root is bracketed polynomial interpolation infinite series evaluation and plotting are included See the last sections of Chapter 3 for details 11 12 UNIX EMACS Chapter 2 Specific Use of the GP Calculator Originally GP was designed as a debugging tool for the PARI system library and hence not much thought had been given to making it user friendly The situation has now changed somewhat and GP is very useful as a stand alone tool The operations and functions available in PARI and GP will be described in the next chapter In the present one we describe the specific use of the GP programmable calculator For starting t
281. ists possible values Terminal options can be appended to the terminal name and space terminal size can be put immediately after the name as in gif 300 200 Positive return value means success 161 3 10 29 psdraw list same as plotdraw except that the output is a PostScript program appended to the psfile 3 10 30 psploth X a b expr same as ploth except that the output is a PostScript program appended to the psfile 3 10 31 psplothraw listz listy same as plothraw except that the output is a PostScript pro gram appended to the psfile 3 11 Programming under GP 3 11 1 Control statements A number of control statements are available under GP They are simpler and have a syntax slightly different from their C counterparts but are quite powerful enough to write any kind of program Some of them are specific to GP since they are made for number theorists As usual X will denote any simple variable name and seg will always denote a sequence of expressions including the empty sequence 3 11 1 1 break n 1 interrupts execution of current seg and immediately exits from the n innermost enclosing loops within the current function call or the top level loop n must be bigger than 1 If n is greater than the number of enclosing loops all enclosing loops are exited 3 11 1 2 for X a b seq the formal variable X going from a to b the seq is evaluated Nothing is done if a gt b a and b must be in R 3 11 1 3
282. it 3 5 1 elladd 21 22 sum of the points z1 and z2 on the elliptic curve corresponding to the vector E The library syntax is addell E z1 22 3 5 2 ellak E n computes the coefficient a of the L function of the elliptic curve E i e in principle coefficients of a newform of weight 2 assuming Taniyama Weil conjecture which is now known to hold in full generality thanks to the work of Breuil Conrad Diamond Taylor and Wiles E must be a medium or long vector of the type given by ellinit For this function to work for every n and not just those prime to the conductor E must be a minimal Weierstrass equation If this is not the case use the function ellminimalmodel first before using ellak The library syntax is akell E n 3 5 3 ellan E n computes the vector of the first n az corresponding to the elliptic curve E All comments in ellak description remain valid The library syntax is anell E n where n is a C integer 3 5 4 ellap E p flag 0 computes the a corresponding to the elliptic curve E and the prime number p These are defined by the equation E F p 1 ap where E F stands for the number of points of the curve E over the finite field F When flag is O this uses the baby step giant step method and a trick due to Mestre This runs in time O p and requires O p storage hence becomes unreasonable when p has about 30 digits If flag is 1 computes the a as a sum of Legendre symbols This i
283. it should satisfy gerepileupto assumptions see Section 4 4 193 4 7 2 Output For output there exist essentially three different functions with variants corresponding to the three main GP output formats as described in Section 2 1 16 plus three extra ones respectively devoted to T X output string output and advanced debugging e raw format obtained by using the function brute with the following syntax void brute GEN obj char x long n This prints the PARI object obj in format x0 n using the notations from Section 2 1 9 Recall that here x is either e f or g corresponding to the three numerical output formats and n is the number of printed significant digits and should be set to 1 if all of them are wanted these arguments only affect the printing of real numbers Usually you will not need that much flexibility so most of the time you will get by with the function void outbrute GEN obj which is equivalent to brute x g 1 or even better with void output GEN obj which is equivalent to outbrute obj followed by a newline and a buffer flush This is especially nice during debugging For instance using dbx or gdb if obj is a GEN typing print output obj will enable you to see the content of obj provided the optimizer has not put it into a register but it is rarely a good idea to debug optimized code e prettymatrix format this format is identical to the preceding one excep
284. its you input For instance to get 2 decimal digits you need one word of precision which on a 32 bit machine actually gives you 9 digits 9 lt log 232 lt 10 default realprecision 2 realprecision 9 significant digits 2 digits displayed 2 1 26 secure default 0 this is a toggle which can be either 1 on or 0 off If on the system and extern command are disabled These two commands are potentially dangerous when you execute foreign scripts since they let GP execute arbitrary UNIX commands GP will ask for confirmation before letting you or a script unset this toggle 2 1 27 seriesprecision default 16 number of significant terms when converting a polynomial or rational function to a power series see ps 20 2 1 28 simplify default 1 this is a toggle which can be either 1 on or 0 off When the PARI library computes something the type of the result is not always the simplest possible The only type conversions which the PARI library does automatically are rational numbers to integers when they are of type t_FRAC and equal to integers and similarly rational functions to polynomials when they are of type t_RFRAC and equal to polynomials This feature is useful in many cases and saves time but can be annoying at times Hence you can disable this and whenever you feel like it use the function simplify see Chapter 3 which allows you to simplify objects to the simplest possible types recursively see y 2
285. ity Use default defaultname to get the default help 7 pattern produces a list of sections in Chapter 3 of the manual related to your query As before if pattern ends by followed by a chapter number that chapter is searched instead you also have the option to append a simple without a chapter number to browse through the whole manual If your query contains dangerous characters e g or blanks it is advisable to enclose it within double quotes as for GP strings e g elliptic curve Note that extended help is much more powerful than the short help since it knows about operators as well you can type or amp amp whereas a single would just yield a not too helpful xxx unknown identifier message Also you can ask for extended help on section number n in Chapter 3 just by typing n where n would yield merely a list of functions Finally a few key concepts in GP are documented in this way metacommands e g defaults e g psfile and type names 22 e g t_INT or integer as well as various miscellaneous keywords such as edit short summary of line editor commands operator member user defined nf ell Last but not least without argument will open a dvi previewer xdvi by default GPXDVI if it is defined in your environment containing the full user s manual tutorial and refcard do the same with the tutorial and reference card respectively Technical note these fun
286. ject The library syntax is gtopoly z v where v is a variable number 3 2 5 Polrev z v x transform the object x into a polynomial with main variable v If x is a scalar this gives a constant polynomial If x is a power series the effect is identical to truncate see there i e it chops off the O X If x is a vector this function creates the polynomial whose coefficients are given in x with x 1 being the constant term Note that this is the reverse of Pol if x is a vector otherwise it is identical to Pol The library syntax is gtopolyrev z v where v is a variable number 3 2 6 Qfb a b c D 0 creates the binary quadratic form ax bry cy If b 4ac gt 0 initialize Shanks distance function to D The library syntax is QfbO a b c D prec Also available are qfi a b c when b 4ac lt 0 and qfr a b c d when b 4ac gt 0 3 2 7 Ser x v x transforms the object x into a power series with main variable v x by default If x is a scalar this gives a constant power series with precision given by the default serieslength corresponding to the C global variable precdl If x is a polynomial the precision is the greatest of precdl and the degree of the polynomial If x is a vector the precision is similarly given and the coefficients of the vector are understood to be the coefficients of the power series starting from the constant term i e the reverse of the function Pol The warni
287. l or those corresponding to the relations in W and B when the bnf internal precision needs to be increased e The less technical components are as follows bnf 7 or bnf nf is equal to the number field data nf as would be given by nfinit bnf 8 is a vector containing the last 6 components of bnfclassunit 1 i e the classgroup bnf clgp the regulator bnf reg the general check number which should be close to 1 the num ber of roots of unity and a generator bnf tu the fundamental units bnf fu and finally the check on their computation If the precision becomes insufficient GP outputs a warning fundamental units too large not given and does not strive to compute the units by default flag 0 When flag 1 GP insists on finding the fundamental units exactly the internal precision being doubled and the computation redone until the exact results are obtained The user should be warned that this can take a very long time when the coefficients of the fundamental units on the integral basis are very large When flag 2 on the contrary it is initially agreed that GP will not compute units Note that the resulting bnf will not be suitable for bnrinit and that this flag provides negligible time savings In short do not use it without a very good reason When flag 3 computes a very small version of bnfinit a small big number field or sbnf for short which contains enough information to recover the full bnf vector v
288. l A 82 Hilbert class field 87 Hilbert matrix 144 Hilbert symbol 82 121 Hilbert oos cor a Re he A 82 DTSUSIZE a a ANa 18 NE Beet ae sO ee Ee EE Ges 145 Hnfalt ir a Es od i 145 hnfmods a dd 145 240 hnfmodid Vii 145 HQTOVAL ec Hak AO a oe oe AS 136 Hurwitz class number 85 AYPOTUs Cuy a eben ts ok ob ee a 72 I Togs etn bod m tee hae HAS 25 31 69 THESSEC ira ra a 71 idealistas as hehe whedon ados 97 VCO Rata Ma a A a 96 Veal ad Gt ri sa A ee eS 112 idealaddtoone 112 idealaddtooneO 112 idealappr 112 idealappr0 112 idealchinese 112 idealcoprime 2 2 42 6 43 sea sos 112 Tdealdiy partorire Sata waa 112 113 idealdivO so 283 S68 Mak we 4 eee 112 AdeaLdIVEXKACE azi ei kh See el 113 idealfactor lt com Fo due eek wk 113 idealhermite 113 Tdealhnt s yt ak erate Se ir 113 129 id ealhntO sion aa Ve 113 idealintersect 113 146 idealinv 113 122 Idealista Std 113 ideallistO 2 25 8 f4 34 ee Aw a 113 id ealtistarch s cele a e 113 ideallistarchO 113 idealllltSd Laa he Paes et 115 ideallog rerai aane E dr a Dh 114 dsalmin a s aaea A A 114 de lm l iia hs a PG 114 idealmulred 114 ideal norm coca Ae eae AS Le 114 Tdealpow sI bos ea edt oe YY 114 idealpowred 114 idealpows 0 4
289. l directory if you have compiled the readline library somewhere else without installing it to one of its standard locations You can also invoke Configure with one of the following arguments with readline prefix to 1ib libreadline rr and include readline h with readline lib path to libreadline zz with readline include path to readline h 223 Linux Linux distributions have separate readline and readline devel packages You need both of them installed to compile gp with readline support If only readline is installed Configure will complain Configure may also complain about a missing libncurses so in which case you will have to install the ncurses devel package some distributions let you install readline devel without ncurses devel which is a bug in their package dependency handling Technical note Configure can build GP on different architectures simultaneously from the same toplevel sources Instead of the readline link alluded above you can create readline osname arch using the same naming conventions as for the Oxxzx directory e g readline linux i686 2 4 Debugging profiling If you also want to debug the PARI library Configure g will create a directory Orxx dbg containing a special Makefile ensuring that the GP and PARI library built there will be suitable for debugging if your compiler doesn t use standard flags e g g you may have to tweak that Makefile If you want to profile GP or the library using gpro
290. l number this type has a second codeword z 1 which also encodes its sign obtained or set using the same functions as for the integers and a binary exponent This exponent can be handled using the following macros long expo GEN z returns the exponent of z This is defined even when z is equal to zero see Section 1 2 6 3 void setexpo GEN z long e sets the exponent of z to e Note the functions long gexpo GEN z which tries to return an exponent for z even if z is not a real number long gsigne GEN z which returns a sign for z even when z is neither real nor integer a rational number for instance The real zero is characterized by having its sign equal to 0 If z is not equal to 0 then is is represented as 2 M where e is the exponent and M 1 2 is the mantissa of z whose digits are stored in z 2 z lg z 1 187 More precisely let m be the integer z 2 z 1g z 1 in base 2 BITS_IN_LONG here z 2 is the most significant longword and is normalized i e its most significant bit is 1 Then we have M m 9l bit accuracy lg z _ Thus the real number 3 5 to accuracy bit accuracy lg z is represented as z 0 encoding type t_REAL 1g z z 1 encoding sign 1 expo 1 z 2 0xe0000000 z 3 z 1g z 1 0x0 4 5 3 Type t_INTMOD integermod z 1 points to the modulus and z 2 at the number representing the class z Both are separate GEN objects and both must be of type inte
291. l or power series x to s where 0 lt s lt MAXVARN 5 2 2 Memory allocation on the PARI stack GEN cgetg long n long t allocates memory on the PARI stack for an object of length n and type t and initializes its first codeword GEN cgeti long n allocates memory on the PARI stack for an integer of length n and initializes its first codeword Identical to cgetg n t_INT GEN cgetr long n allocates memory on the PARI stack for a real of length n and initializes its first codeword Identical to cgetg n t_REAL void cgiv GEN x frees object x if it is the last created on the PARI stack otherwise nothing happens GEN gerepile pari_sp p pari sp q GEN x general garbage collector for the PARI stack See Section 4 4 for a detailed explanation and many examples 5 2 3 Assignments conversions and integer parts void mpaff GEN x GEN z assigns x into z where x and z are integers or reals void affsz long s GEN z assigns the long s into the integer or real z void affsi long s GEN z assigns the long s into the integer z void affsr long s GEN z assigns the long s into the real z void affii GEN x GEN z assigns the integer x into the integer z void affir GEN x GEN z assigns the integer x into the real z void affrs GEN x long s assigns the real x into the long s not This is a forbidden assign ment in PARI so an error message is issued void affri GEN x GEN z assigns the real x into the integer z no it doesn
292. l prettyprinter to use when output is 3 alternate prettyprinter This is experimental but the default tex2mail looks already much nicer than the built in beautified format output 2 2 1 20 primelimit default 200k on the Mac and 500k otherwise GP precomputes a list of all primes less than primelimit at initialization time These are used by many arithmetical functions If you don t plan to invoke any of them you can just set this to 1 19 EMACS 2 1 21 prompt default a string that will be printed as prompt Note that most usual escape sequences are available there Me for Esc n for Newline for Time expansion is performed This string is sent through the library function strftime on a Unix system you can try man strftime at your shell prompt This means that constructs have a special meaning usually related to the time and date For instance 4H hour 24 hour clock and M minute 00 59 use hh to get a real If you use readline escape sequences in your prompt will result in display bugs If you have a relatively recent readline see the comment at the end of Section 2 1 2 you can brace them with special sequences and and you will be safe If these just result in extra spaces in your prompt then you ll have to get a more recent readline See the file misc gprc dft for an example Caution Emacs needs to know about the prompt pattern to separate your input from previous GP resul
293. lar type is zero and the derivative of a vector or matrix is done componentwise One can use xz as a shortcut if the derivative is with respect to the main variable of x The library syntax is deriv x v where v is a long and an omitted v is coded as 1 When x is a t_POL derivpol z is a shortcut for deriv x 1 3 7 3 eval x replaces in x the formal variables by the values that have been assigned to them after the creation of x This is mainly useful in GP and not in library mode Do not confuse this with substitution see subst Applying this function to a character string yields the output from the corresponding GP command as if directly input from the keyboard see Section 2 6 6 The library syntax is geval x The more basic functions poleval q x qfeval q x and hqfeval q x evaluate q at x where q is respectively assumed to be a polynomial a quadratic form a symmetric matrix or an Hermitian form an Hermitian complex matrix 3 7 4 factorpadic pol p r flag 0 p adic factorization of the polynomial pol to precision r the result being a two column matrix as in factor The factors are normalized so that their leading coefficient is a power of p r must be strictly larger than the p adic valuation of the discriminant of pol for the result to make any sense The method used is a modified version of the round 4 algorithm of Zassenhaus If flag 1 use an algorithm due to Buchmann and Lenstra which is usually less
294. letter X will always denote any simple variable name and represents the formal parameter used in the function numerical integration A number of Romberg like integration methods are implemented see intnum as opposed to intformal which we already described The user should not require too much accuracy 18 or 28 decimal digits is OK but not much more In addition analytical cleanup of the integral must have been done there must be no singularities in the interval or at the boundaries In practice this can be accomplished with a simple change of variable Furthermore for improper integrals where one or both of the limits of integration are plus or minus infinity the function must decrease sufficiently rapidly at infinity This can often be accomplished through integration by parts Finally the function to be integrated should not be very small compared to the current precision on the entire interval This can of course be accomplished by just multiplying by an appropriate constant Note that infinity can be represented with essentially no loss of accuracy by 1e4000 However beware of real underflow when dealing with rapidly decreasing functions For example if one wants to compute the h e7 da to 28 decimal digits then one should set infinity equal to 10 for example and certainly not to 1e4000 153 The integrand may have values belonging to a vector space over the real numbers in particular it can be complex valued or vector valued
295. library function strftime This means that Achar combinations have a special meaning usually related to the time and date For instance H hour 24 hour clock and 4M minute 00 59 on a Unix system you can try man strftime at your shell prompt to get a complete list This is applied to prompt psfile and logfile For instance default prompt H M will prepend the time of day in the form hh mm to GP s usual prompt e environment expansion When the string contains a sequence of the form SOMEVAR e g HOME the environment is searched and if SOMEVAR is defined the sequence is replaced by the corresponding value Also the symbol has the same meaning as in the C and bash shells by itself stands for your home directory and user is expanded to user s home directory This is applied to all filenames 15 UNIX EMACS 2 1 1 buffersize default 30k GP input is buffered which means only so many bytes of data can be read at a time before a command is executed This used to be a very important variable to allow for very large input files to be read into GP for example large matrices without it complaining about unused characters Currently buffersize is automatically adjusted to the size of the data that are to be read It will never go down by itself though Thus this option may come in handy to decrease the buffer size after some unusually large read when you don t need to keep gigantic buffers aro
296. library syntax is simplify lt 3 2 45 sizebyte x outputs the total number of bytes occupied by the tree representing the PARI object x The library syntax is taille2 x which returns a long The function taille returns the number of words instead 3 2 46 sizedigit 1 outputs a quick bound for the number of decimal digits of the components of x off by at most 1 If you want the exact value you can use length Str x which is much slower The library syntax is sizedigit x which returns a long 3 2 47 truncate z amp e truncates x and sets e to the number of error bits When z is in R this means that the part after the decimal point is chopped away e is the binary exponent of the difference between the original and the truncated value the fractional part If the exponent of x is too large compared to its precision i e e gt 0 the result is undefined and an error occurs if e was not given The function applies componentwise on vector matrices e is then the maximal number of error bits If x is a rational function the result is the integer part Euclidean quotient of numerator by denominator and e is not set Note a very special use of truncate when applied to a power series it transforms it into a polynomial or a rational function with denominator a power of X by chopping away the O X Similarly when applied to a p adic number it transforms it into an integer or a rational number by chopping away th
297. lidean quotient of x and the integer or polynomial y GEN gdiventsg z long s GEN x GEN z yields the true Euclidean quotient of the long s by the integer x GEN gdiventgs z GEN x long s GEN z yields the true Euclidean quotient of the integer x by the long s GEN gdiventres GEN x GEN y creates a 2 component vertical vector whose components are the true Euclidean quotient and remainder of x and y GEN gdivmod GEN x GEN y GEN r If r is not equal to NULL or ONLY_REM creates the false Euclidean quotient of x and y and puts the address of the remainder into r If r is equal to NULL do not create the remainder and if r is equal to ONLY_REM create and output only the remainder The remainder is created after the quotient and can be disposed of individually with a cgiv r GEN poldivres GEN x GEN y GEN r same as gdivmod but specifically for polynomials x and y GEN gdeuc GEN x GEN y creates the Euclidean quotient of the polynomials x and y GEN gdivround GEN x GEN y if x and y are integers returns the quotient x y of x and y rounded to the nearest integer If x y falls exactly halfway between two consecutive integers then it is rounded towards 00 as for round If x and y are not both integers the result is the same as that of gdivent GEN gmodiz GEN x GEN y GEN z yields the true remainder of x modulo the integer or polynomial y 219 GEN gmodsg z long s GEN x GEN z yields the true remai
298. lift 3 0 379 92 3 lift Mod x x 2 1 43 x lift x Mod 1 3 Mod 2 3 4 x 2 1ift x Mod y y 2 1 Mod 2 3 15 y x Mod 2 3 do you understand this one lift x Mod y y 2 1 Mod 2 3 x 16 Mod y y 2 1 x Mod 2 y 2 1 The library syntax is liftO z v where v is a long and an omitted v is coded as 1 Also available is lift x 1i t0 x 1 3 2 34 norm x algebraic norm of zx i e the product of x with its conjugate no square roots are taken or conjugates for polmods For vectors and matrices the norm is taken componentwise and hence is not the L norm see norm12 Note that the norm of an element of R is its square so as to be compatible with the complex norm The library syntax is gnorm z 3 2 35 norml2 z square of the L norm of x More precisely if x is a scalar norm12 x is defined to be x conj x If x is a row or column vector or a matrix norm12 x is defined recursively as norml2 x where x run through the components of x In particular this yields the usual YN 2 1 resp Y 2 1 if x is a vector resp matrix with complex components norml2 1 2 3 vector 1 14 norml2 1 2 3 4 matrix 1 30 norml2 I x 43 x 2 4 1 norml2 1 2 8 4 5 6 recursively defined 74 91 The library syntax is gnorml2 7 3 2 36 numerator z numerator of x When x is a rational number or function the meaning
299. ling is simply the number of pixels the origin being at the upper left and the y coordinates going downwards Note that in the present version 2 2 7 all these plotting functions both low and high level have been written for the X11 window system hence also for GUI s based on X11 such as Open windows and Motif only though very little code remains which is actually platform dependent A Suntools Sunview Macintosh and an Atari Gem port were provided for previous versions These may be adapted in future releases Under X11 Suntools the physical window opened by plotdraw or any of the ploth func tions is completely separated from GP technically a fork is done and the non graphical memory is immediately freed in the child process which means you can go on working in the current GP session without having to kill the window first Under X11 this window can be closed enlarged or reduced using the standard window manager functions No zooming procedure is implemented though yet e Finally note that in the same way that printtex allows you to have a T X output corre sponding to printed results the functions starting with ps allow you to have PostScript output of the plots This will not be absolutely identical with the screen output but will be sufficiently close Note that you can use PostScript output even if you do not have the plotting routines enabled The PostScript output is written in a file whose name is derived from the ps
300. lip w clips the content of rectwindow w i e remove all parts of the drawing that would not be visible on the screen Together with plotcopy this function enables you to draw on a scratchpad before commiting the part you re interested in to the final picture 157 3 10 4 plotcolor w c set default color to c in rectwindow w In present version 2 2 7 this is only implemented for X11 window system and you only have the following palette to choose from 1 black 2 blue 3 sienna 4 red 5 cornsilk 6 grey 7 gainsborough Note that it should be fairly easy for you to hardwire some more colors by tweaking the files rect h and plotX c User defined colormaps would be nice and may be available in future versions 3 10 5 plotcopy wl w2 dx dy copy the contents of rectwindow w1 to rectwindow w2 with offset dx dy 3 10 6 plotcursor w give as a 2 component vector the current scaled position of the virtual cursor corresponding to the rectwindow w 3 10 7 plotdraw list physically draw the rectwindows given in list which must be a vector whose number of components is divisible by 3 If list wl x1 yl w2 12 y2 the windows wl w2 etc are physically placed with their upper left corner at physical position x1 y1 12 y2 respectively and are then drawn together Overlapping regions will thus be drawn twice and the windows are considered transparent Then display the whole drawing in a special window on your screen
301. listed above You will probably very rarely use the heap yourself and if so only as a collection of individual copies of objects taken from the stack called clones in the sequel Thus you need not bother with its internal structure which may change as PARI evolves Some complex PARI functions may create clones for special garbage collecting purposes usually destroying them when returning e a table of primes in fact of differences between consecutive primes called diffptr of type byteptr pointer to unsigned char Its use is described in appendix C e access to all the built in functions of the PARI library These are declared to the outside world when you include pari h but need the above things to function properly So if you forget the call to pari_init you will immediately get a fatal error when running your program 4 2 Important technical notes 4 2 1 Typecasts We have seen that due to the non recursiveness of the PARI types from the compiler s point of view many typecasts will be necessary when programming in PARI To take an example a vector V of dimension 2 two components will be represented by a chunk of memory pointed to by the GEN V V O contains coded information in particular about the type of the object its length etc V 1 and V 2 contain pointers to the two components of V Those coefficients V i themselves are in chunks of memory whose complexity depends on their own types and so on This is where typec
302. long Ig GEN x returns the length of x in BIL bit words long lgef GEN x returns the effective length of the polynomial x in BIL bit words long lgefint GEN x returns the effective length of the integer x in BIL bit words long signe GEN x returns the sign 1 0 or 1 of x Can be used for integers reals polynomials and power series for the last two types only 0 or 1 are possible long gsigne GEN x same as signe but also valid for rational numbers and marginally less efficient for the other types long expo GEN x returns the unbiased binary exponent of the real number x long gexpo GEN x same as expo but also valid when x is not a real number When x is an exact 0 this returns HIGHEXPOBIT long expi GEN x returns the binary exponent of the real number equal to the integer x This is a special case of gexpo above covering the case where x is of type t_INT long valp GEN x returns the unbiased 16 bit p adic valuation for a p adic or X adic valuation for a power series taken with respect to the main variable of x long precp GEN x returns the precision of the p adic x long varn GEN x returns the variable number of x between 0 and MAXVARN Should be used only for polynomials and power series long gvar GEN x returns the main variable number when any variable at all occurs in the composite object x the smallest variable number which occurs and BIGINT otherwise void settyp GEN x long s sets the type n
303. lt which is described in Sec tion 3 11 2 4 The basic syntax is default def value which sets the default def to value In interactive use most of these can be abbreviated using historic GP metacommands mostly starting with which we shall describe in the next section Here we will only describe the available defaults and how they are used Just be aware that typing default by itself will list all of them as well as their current values see d Just after the default name we give between parentheses the initial value when GP starts assuming you did not tamper with it using command line switches or a gprc Note the suffixes k M or G can be appended to a value which is a numeric argument with the effect of multiplying it by 10 10 and 10 respectively Case is not taken into account there so for instance 30k and 30K both stand for 30000 This is mostly useful to modify or set the defaults primelimit or stacksize which typically involve a lot of trailing zeroes somewhat technical Note As we will see in Section 2 6 6 the second argument to default will be subject to string context expansion which means you can use run time values In other words something like a 3 default logfile var some filename a log logs the output in some filename3 1log Some defaults will be expanded further when the values are used after the above expansion has been performed e time expansion the string is sent through the
304. lynomial of degree 0 and to rational functions provided the denominator is a monomial The library syntax is polcoeff0 z s v where v is a long and an omitted v is coded as 1 Also available is truecoeff z v 3 7 8 poldegree z v degree of the polynomial x in the main variable if v is omitted in the variable v otherwise This is to be understood as follows The degree of 0 is VERYBIGINT by convention VERYBIGINT is 2 1 for 32 bit machines or 263 1 for 64 bit machines When lt z is a non zero scalar its degree is 0 When x is non zero polynomial or rational function it is the ordinary degree of x Return an error otherwise The library syntax is poldegree x v where v and the result are longs and an omitted v is coded as 1 Also available is degree x which is equivalent to poldegree x 1 3 7 9 polcyclo n v x n th cyclotomic polynomial in variable v x by default The integer n must be positive The library syntax is cyclo n v where n and v are long integers v is a variable number usually obtained through varn 3 7 10 poldisc pol v discriminant of the polynomial pol in the main variable is v is omitted in v otherwise The algorithm used is the subresultant algorithm The library syntax is poldiscO z v Also available is discsr x equivalent to poldisc0 x 1 3 7 11 poldiscreduced f reduced discriminant vector of the integral monic polynomial f This is the vector of el
305. lynomial x in the field F defined by the irreducible poly nomial a over F The coefficients of x must be operation compatible with Z pZ The result is a two column matrix the first column being the irreducible polynomials dividing x and the second the exponents It is recommended to use for the variable of a which will be used as variable of a polmod a name distinct from the other variables used so that a 1ift of the result will be legible If all the coefficients of x are in F a much faster algorithm is applied using the computation of isomorphisms between finite fields The library syntax is factmod9 z p a 3 4 22 factorial x or x factorial of x The expression z gives a result which is an integer while factorial 1 gives a real number The library syntax is mpfact x for x and mpfactr z prec for factorial x x must be a long integer and not a PARI integer 3 4 23 factorint n flag 0 factors the integer n into a product of pseudoprimes see ispseu doprime using a combination of the Shanks SQUFOF and Pollard Rho method with modifications due to Brent Lenstra s ECM with modifications by Montgomery and MPQS the latter adapted from the LiDIA code with the kind permission of the LiDIA maintainers as well as a search for pure powers with exponents lt 10 The output is a two column matrix as for factor Use isprime on the result if you want to guarantee primality This gives direct access to the integer fact
306. m In x 7 7 The result is complex with imaginary part equal to 7 if x R and z lt 0 p adic arguments are also accepted for x with the convention that In p 0 Hence in particular exp In x x will not in general be equal to 1 but to a p 1 th root of unity or 1 if p 2 times a power of p If flag is equal to 1 AGM use an agm formula suggested by Mestre when z is real otherwise identical to log The library syntax is glog x prec or glogagm z prec 3 3 38 Ingamma x principal branch of the logarithm of the gamma function of x Can have much larger arguments than gamma itself In the present version 2 2 7 the p adic Ingamma function is not implemented The library syntax is gmgamma z prec 3 3 39 polylog m x flag 0 one of the different polylogarithms depending on flag If flag 0 or is omitted m polylogarithm of z i e analytic continuation of the power series Lin x gt 2 n The program uses the power series when x lt 1 2 and the power series expansion in log x otherwise It is valid in a large domain at least x lt 230 but should not be used too far away from the unit circle since it is then better to use the functional equation linking the value at x to the value at 1 x which takes a trivial form for the variant below Power series polynomial rational and vector matrix arguments are allowed For the variants to follow we need a notation let Rm denotes R
307. maining two are macros NOT functions see Section 4 2 1 for a detailed explanation long coeff GEN x long i long j applied to a matrix x type t_MAT this gives the address of the coefficient at row i and column j of x long mael n GEN x long a1 long an stands for x a az an where 2 lt n lt 5 with all the necessary typecasts 5 3 1 Copying and conversion GEN cgetp GEN x creates space sufficient to hold the p adic x and sets the prime p and the p adic precision to those of x but does not copy the p adic unit or zero representative and the modulus of x GEN gcopy GEN x creates a new copy of the object x on the PARI stack For permanent subob jects only the pointer is copied GEN forcecopy GEN x same as copy except that even permanent subobjects are copied onto the stack long taille GEN x returns the total number of BIL bit words occupied by the tree representing x GEN gclone GEN x creates a new permanent copy of the object x on the heap GEN greffe GEN x long 1 int use_stack applied to a polynomial x type t_POL creates a power series type t_SER of length 1 starting with x but without actually copying the coefficients just the pointers If use_stack is zero this is created through malloc and must be freed after use Intended for internal use only 215 double rtodbl GEN x applied to a real x type t_REAL converts x into a C double if possible GEN dbltor double x converts the C dou
308. mal vectors As a side note to old timers this used to fail bluntly when x had more than 5000 minimal vectors Beware that the computations can now be very lengthy when x has many minimal vectors The library syntax is perf x 3 8 48 qfrep q B flag 0 q being a square and symmetric matrix with integer entries repre senting a positive definite quadratic form outputs the vector whose i th entry 1 lt i lt B is half the number of vectors v such that q v i This routine uses a naive algorithm based on qfminin and will fail if any entry becomes larger than 231 The binary digits of flag mean e 1 count vectors of even norm from 1 to 2B e 2 return a t_VECSMALL instead of a t_GEN The library syntax is qfrep0 q B flag 150 3 8 49 qfsign x signature of the quadratic form represented by the symmetric matrix x The result is a two component vector The library syntax is signat x 3 8 50 setintersect x y intersection of the two sets x and y The library syntax is setintersect z y 3 8 51 setisset 1 returns true 1 if x is a set false 0 if not In PARI a set is simply a row vector whose entries are strictly increasing To convert any vector and other objects into a set use the function Set The library syntax is setisset x and this returns a long 3 8 52 setminus z y difference of the two sets x and y i e set of elements of x which do not belong to y The library syntax is setminus z y 3
309. marks once again First note the use of the function gscalmat with the following syntax GEN gscalmat GEN x long m The effect of this function is to create the m x m scalar matrix whose diagonal entries are x Hence the length of the matrix including the codeword will in fact be m 1 There is a corresponding function gscalsmat which takes a long as a first argument If we refer to what has been said above the main loop should be self evident When we do the final squarings according to the fundamental dogma on the use of gerepile we keep the value of avma in 1bot just before the squaring so that if it is the last one 1bot will indeed be the bottom address of the garbage pile and gerepile will work Note that it takes a completely negligible time to do this in each loop compared to a matrix squaring However when 200 n is initially equal to 0 no squaring has to be done and we have our final result ready but we lost the address of the bottom of the garbage pile Hence we use the trick of copying y again to the top of the stack This is inefficient but does the trick If we wanted to avoid this using only gerepile the best thing to do would be to put the instruction 1bot avma just before both occurrences of the instruction y gadd p2 y Of course we could also rewrite the last block as follows now square back n times for i 0 i lt n i y gsqr y return gerepileupto ltop y because it does not matter to gerepileupto t
310. me of the GP executable the stack size and the prime limit before the execution of GP begins If for any of these you simply type return the default value will be used On UNIX machines it will be the place you told Configure usually usr local bin gp for the executable 10M for the stack and 500k for the prime limit You can then work as usual under GP but with two notable advantages which don t really matter if readline is available to you see below First and foremost you have at your disposal all the facilities of a text editor like Emacs in particular for correcting or copying blocks Second you can have an on line help which is much more complete than what you obtain by typing name This is done by typing M In the minibuffer Emacs asks what function you want to describe and after your reply you obtain the description which is in the users manual including the description of functions such as which use special symbols This help system can also be menu driven by using the command M c which opens a help menu window which enables you to choose the category of commands for which you want an explanation Nevertheless if extended help is available on your system see Section 2 2 1 you should use it instead of the above since it s nicer it ran through TEX and understands many more keywords Finally you can use command completion in the following way After the prompt type the first few letters of the command then
311. med to have integral entries but needs not be of maximal rank The result is a two component vector of matrices the columns of the first matrix represent a basis of the integer kernel of x not necessarily LLL reduced and the second matrix is the transformation matrix T such that x T is an LLL reduced Z basis of the image of the matrix zx If flag 5 case as case 4 but x may have polynomial coefficients If flag 8 same as case 0 where x may have polynomial coefficients The library syntax is qflll0 z flag prec Also available are U x prec flag 0 Mlint x flag 1 and Ilkerim z flag 4 149 3 8 45 qflllgram z flag 0 same as qf111 except that the matrix x is the Gram matrix of the lattice vectors and not the coordinates of the vectors themselves In particular x must now be a square symmetric real matrix corresponding to a positive definite quadratic form The result is again the transformation matrix T which gives as columns the coefficients with respect to the initial basis vectors The flags have more or less the same meaning but some are missing In brief flag 0 numerically unstable in the present version 2 2 7 flag 1 x has integer entries the computations are all done in integers flag 4 x has integer entries gives the kernel and reduced image flag 5 same as 4 for generic x The library syntax is qflllgram0 z flag prec Also available are Wlgram z prec flag 0 lllgramint x
312. might be enough On the other hand beware that this is not permanent independant storage but part of the PARI stack Note that objects allocated through these two functions cannot be gerepile d They are not valid GENs since they have no PARI type Examples 1 z cgeti DEFAULTPREC and cgetg DEFAULTPREC t_INT create an integer ob ject whose precision is bit_accuracy DEFAULTPREC 64 This means z can hold rational integers of absolute value less than 264 Note that in both cases the second codeword will not be filled Of course we could use numerical values e g cgeti 4 but this would have different mean ings on different machines as bit_accuracy 4 equals 64 on 32 bit machines but 128 on 64 bit machines 2 The following creates a complex number whose real and imaginary parts can hold real numbers of precision bit_accuracy MEDDEFAULTPREC 96 bits z cgetg 3 t_COMPLEX z 1 lgetr MEDDEFAULTPREC z 2 lgetr MEDDEFAULTPREC 3 To create a matrix object for 4 x 3 matrices z cgetg 4 t_MAT for i 1 i lt 4 i z i lgetg 5 t_COL If one wishes to create space for the matrix elements themselves one has to follow this with a double loop to fill each column vector These last two examples illustrate the fact that since PARI types are recursive all the branches of the tree must be created The function cgetg creates only the root and other calls to cgetg must be made to produce the whole tre
313. might work but should be frowned upon We cannot predict whether avma is going to be evaluated after or before the call to anything it depends on the compiler If we are out of luck it will be after the call so the result will belong to the garbage zone and the gerepile statement becomes equivalent to avma ltop Thus we would return a pointer to random garbage e A simple variant is GEN gerepileupto pari_sp ltop GEN q which cleans the stack between ltop and the connected object q and returns q updated For this to work q must have been created before all its components otherwise they would belong to the 181 garbage zone Documented PARI functions guarantee this If you stumble upon one that does not consider it a bug worth reporting e Another variant a special case of gerepilemany below where n 1 is GEN gerepilecopy pari_sp ltop GEN x which is functionnally equivalent to gerepileupto ltop gcopy x but more efficient In this case the GEN parameter x need not satisfy any property before the garbage collection it may be disconnected components created before the root and so on Of course this is less efficient than either gerepileupto or gerepile because x has to be copied to a clean stack zone first e To cope with complicated cases where many objects have to be preserved you can use void gerepileall pari_sp ltop int n where the routine expects n further arguments which are the addresses of the GENs yo
314. modular parametrization of the elliptic curve E where E is given in the long or medium format output by ellinit in the form of a two component vector u v of power series given to the current default series precision This vector is characterized by the following two properties First the point x y u v satisfies the equation of the elliptic curve Second the differential du 24 a u gt a3 is equal to f z dz a differential form on H To N where N is the conductor of the curve The variable used in the power series for u and v is 2 which is implicitly understood to be equal to exp 2imz It is assumed that the curve is a strong Weil curve and the Manin constant is equal to 1 The equation of the curve E must be minimal use ellminimalmodel to get a minimal equation The library syntax is taniyama and the precision of the result is determined by the global variable precdl 3 5 27 elltors E flag 0 if E is an elliptic curve defined over Q outputs the torsion subgroup of E as a 3 component vector t v1 v2 where t is the order of the torsion group v1 gives the structure of the torsion group as a product of cyclic groups sorted by decreasing order and v2 gives generators for these cyclic groups amp must be a long vector as output by ellinit E ellinit 0 0 0 1 0 elltors E 41 4 2 2 Lo 0 1 0 Here the torsion subgroup is isomorphic to Z 2Z x Z 2Z with generators 0 0 and 1 0 If flag
315. mp e About library programming the library function foo as defined at the beginning of this section is seen to have two mandatory arguments x and flag no PARI mathematical function has been implemented so as to accept a variable number of arguments so all arguments are mandatory when programming with the library often variants are provided corresponding to the various flag values When not mentioned otherwise the result and arguments of a function are assumed implicitly to be of type GEN Most other functions return an object of type long integer in C see Chapter 4 The variable or parameter names prec and flag always denote long integers The entree type is used by the library to implement iterators loops sums integrals etc when a formal variable has to successively assume a number of values in a given set When programming with the library it is easier and much more efficient to code loops and the like directly Hence this type is not documented although it does appear in a few library function prototypes below See Section 3 9 for more details 54 3 1 Standard monadic or dyadic operators 3 1 1 The expressions x and x refer to monadic operators the first does nothing the second negates 2 The library syntax is gneg x for x 3 1 2 The expression x y is the sum and x y is the difference of x and y Among the prominent impossibilities are addition subtraction between a scalar type and a vector or
316. n V 1 r for i 2 n Vli Vli 1 xz V addhelp sqrtnall sqrtnall x n compute the vector of nth roots of x The library syntax is gsqrtn x n amp z prec 3 3 46 tan x tangent of x The library syntax is gtan z prec 3 3 47 tanh hyperbolic tangent of x The library syntax is gth z prec 3 3 48 teichmuller x Teichm ller character of the p adic number z The library syntax is teich x 3 3 49 theta q z Jacobi sine theta function The library syntax is theta q z prec 75 3 3 50 thetanullk q k k th derivative at z 0 of theta q z The library syntax is thetanullk q k prec where k is a long 3 3 51 weber z flag 0 one of Weber s three f functions If flag 0 returns f a exp in 24 n w 1 2 m x such that j f24 16 3 f 4 where j is the elliptic j invariant see the function e11j If flag 1 returns fx n a 2 n x such that j f 16 f7 Finally if flag 2 returns fol V2n 2x n x such that j fz 16 f2 Note the identities f8 fS f and f fifo V2 The library syntax is weber0 zx flag prec or wf x prec wf1 x prec or wf2 z prec 3 3 52 zeta s Riemann s zeta function s gt gt n computed using the Euler Maclaurin summation formula except when s is of type integer in which case it is computed using Bernoulli numbers for s lt 0 or s gt 0 and even and using modular forms for s gt 0 and odd The l
317. n from your preferences file for instance see Section 2 9 and that s it you can use the new function gpname under GP and we would very much like to hear about it 4 9 5 Integration the hard way If install is not available for your Operating System things are more complicated you have to hardcode your function in the GP binary or install Linux Here is what needs to be done You need to choose a section and add a file functions section gpname containing the follow ing keeping the notation above Function gpname Section section C Name libname Prototype code Help some help text At this point you can rebuild the database by running make Def in the directory desc Then you can recompile GP 204 4 9 6 Example A complete description could look like this install bnfinitO GDO L DGp ClassGroupInit libpari so addhelp ClassGroupInit ClassGroupInit P flag 0 data compute the necessary data for which means we have a function ClassGroupInit under GP which calls the library function bn finitO The function has one mandatory argument and possibly two more two D in the code plus the current real precision More precisely the first argument is a GEN the second one is con verted to a long using itos 0 is passed if it is omitted and the third one is also a GEN but we pass NULL if no argument was supplied by the user This matches the C prototype from paridecl h GEN bnfinit
318. n 4 6 2 2 to create new ones The other objects are not initialized by default bern i This is the 2i th Bernoulli number Bp 1 By 1 6 B4 1 30 etc To initialize them use the function void mpbern long n long prec This creates the even numbered Bernoulli numbers up to Ba 2 as real numbers of precision prec They can then be used with the macro bern i Note that this is not a function but simply an abbreviation hence care must be taken that i is inside the right bounds i e 0 lt i lt n 1 before using it since no checking is done by PARI itself geuler This is Euler s constant It is initialized by the first call to mpeuler see Section 3 3 2 gpi This is the number 7 It is initialized by the first call to mppi see Section 3 3 4 The use of both geuler and gpi is deprecated since it s always possible that some library function increases the precision of the constant after you ve computed it hence modifying the computation accuracy without your asking for it and increasing your running times for no good reason You should always use mpeuler and mppi note that only the first call will actually compute the constant unless a higher precision is required In addition some single or double precision real numbers like PI are predefined and their list is in the file paricom h Finally one has access to a table of differences of primes through the pointer diffptr This is used as follows when
319. n 8 component vector gal gal 1 contains the polynomial pol gal po1 gal 2 is a three components vector p e q where p is a prime number gal p such that pol totally split modulo p e is an integer and q p gal mod is the modulus of the roots in gal roots gal 3 is a vector L containing the p adic roots of pol as integers implicitly modulo gal mod gal roots gal 4 is the inverse of the Van der Monde matrix of the p adic roots of pol multiplied by gal 5 gal 5 is a multiple of the least common denominator of the automorphisms expressed as polynomial in a root of pol gal 6 is the Galois group G expressed as a vector of permutations of L gal group gal 7 is a generating subset S s1 5 of G expressed as a vector of permutations of L gal gen gal 8 contains the relative orders o1 0y of the generators of S gal orders Let H be the maximal normal supersolvable subgroup of G we have the following properties o if G H As then o1 0g ends by 2 2 3 o if G H S4 then 01 0 ends by 2 2 3 2 e else G is super solvable e for 1 lt i lt g the subgroup of G generated by s1 Sg is normal with the exception of i g 2 in the second case and of i g 3 in the third e the relative order o of s is its order in the quotient group G s1 5 1 with the same exceptions e for any x G there exists a unique family e1 ey such that no exceptions
320. n dealing with polynomials with inexact entries since polynomial Euclidean division doesn t make much sense in this case The library syntax is sylvestermatrix z y 3 7 25 polsym x n creates the vector of the symmetric powers of the roots of the polynomial x up to power n using Newton s formula The library syntax is polsym z 3 7 26 poltchebi n v x creates the nt Chebyshev polynomial in variable v The library syntax is tchebi n v where n and v are long integers v is a variable number 139 3 7 27 polzagier n m creates Zagier s polynomial PL used in the functions sumalt and sumpos with flag 1 One must have m lt n The exact definition can be found in Convergence acceleration of alternating series Cohen et al Experiment Math vol 9 2000 pp 3 12 The library syntax is polzagreel n m prec if the result is only wanted as a polynomial with real coefficients to the precision prec or polzag n m if the result is wanted exactly where n and m are longs 3 7 28 serconvol x y convolution or Hadamard product of the two power series x and y in other words if x Y az X and y bp X then serconvol z y Y az by XP The library syntax is convol z y 3 7 29 serlaplace x x must be a power series with only non negative exponents If YN ax k X then the result is Daz X The library syntax is laplace z 3 7 30 serreverse 1 reverse power series i e 271 not 1 x
321. n manually gzip some of the documentation files if you wish usersch tex and all dvi files assuming your xdvi knows how to deal with compressed files the online help system will handle it By default if a dynamic library libpari so could be built the static library libpari a will not be created If you want it as well you can use the target make install lib sta You can 225 install a statically linked gp with the target make install bin sta As a rule programs linked statically with libpari a may be slightly faster about 5 gain but use much more disk space and take more time to compile They are also harder to upgrade you will have to recompile them all instead of just installing the new dynamic library On the other hand there s no risk of breaking them by installing a new pari library 3 1 Extra packages The following optional packages endow PARI with some extra capabilities a single package for now e galdata The default polgalois function can only compute Galois groups of polynomials of degree less or equal to 7 Install this package if you want to handle polynomials of degree bigger than 7 and less than 11 To install package pack you need to fetch the separate archive pack tgz which you can download from the pari server Copy the archive in the PARI toplevel directory then extract its contents these will go to data pack Typing make install data will then install all such packages 3 2 The GPRC file Copy
322. n on integers The function gshift is the PARI analogue of the C or GP operators lt lt and gt gt We now come to the heart of the function We have a GEN p1 which points to a certain matrix of which we want to take the exponential We will want to transform this matrix into a matrix with real or complex of real entries before starting the computation To do this we simply multiply by the real number 1 in precision prec 1 to be on the side of safety To sum the series we will use three variables a variable p2 which at stage k will contain p1 k a variable y which will contain par p1 i and a variable r which will contain the size estimate s k Note that we do not use Horner s rule This is simply because we are lazy and do not want to compute in advance the number of terms that we need We leave this modification and many other improvements to the reader The program continues as follows initializations before the loop r cgetr precti gaffsg 1 r p1 gmul r p1 y gscalmat r 1x 1 creates scalar matrix with r on diagonal p2 pil r 8 k 1 y gadd y p2 now the main loop while expo r gt BITS_IN_LONG prec 1 k p2 gdivgs gmul p2 p1 k r gdivgs gmul s r k y gadd y p2 now square back n times if necessary if In lbot avma y gcopy y else for i 0 i lt n i lbot avma y gsqr y return gerepile ltop lbot y A few re
323. n significant terms Prints its current value if n is omitted 2 2 18 q quits the GP session and returns to the system Shortcut for the function quit see Section 3 11 2 20 2 2 19 r filename reads into GP all the commands contained in the named file as if they had been typed from the keyboard one line after the other Can be used in combination with the w command see below Related but not equivalent to the function read see Section 3 11 2 21 in particular if the file contains more than one line of input there will be one history entry for each of them whereas read would only record the last one If filename is omitted re read the previously used input file fails if no file has ever been successfully read in the current session If a GP binary file see Section 3 11 2 30 is read using this command it is silently loaded without cluttering the history This command accepts compressed files in compressed Z or gzipped gz or z format They will be uncompressed on the fly as GP reads them without changing the files themselves 2 2 20 s prints the state of the PARI stack and heap This is used primarily as a debugging device for PARI and is not intended for the casual user 2 2 21 t prints the internal longword format of all the PARI types The detailed bit or byte format of the initial codeword s is explained in Chapter 4 but its knowledge is not necessary for a GP user 2 2 22 u prints the definitions of all
324. nal type number of the PARI object x Otherwise makes a copy of x and sets its type equal to type t which can be either a number or preferably since internal codes may eventually change a symbolic name such as t_FRACN you can skip the t_ part here so that FRACN by itself would also be all right Check out existing type names with the metacommand t GP will not let you create meaningless objects in this way where the internal structure does not match the type This function can be useful to create reducible rationals type t_FRACN or rational functions type t_RFRACN In fact it s the only way to do so in GP In this case the created object as well as the objects created from it will not be reduced automatically making some operations a bit faster There is no equivalent library syntax since the internal functions typ and settyp are available Note that settyp does not create a copy of x contrary to most PARI functions It also doesn t check for consistency settyp just changes the type in place and returns nothing typ returns a C long integer Note also the different spellings of the internal functions set typ and of the GP function type which is due to the fact that type is a reserved identifier for some C compilers 3 11 2 27 whatnow key if keyword key is the name of a function that was present in GP version 1 39 15 or lower outputs the new function name and syntax if it changed at all 387 out of 560 did 3 11 2 28 writ
325. nce more generally the condition on a is tested after execution of the seq not before as in while 3 11 1 12 while a seq while a is non zero evaluate the expression sequence seg The test is made before evaluating the seq hence in particular if a is initially equal to zero the seq will not be evaluated at all 3 11 2 Specific functions used in GP programming In addition to the general PARI functions it is necessary to have some functions which will be of use specifically for GP though a few of these can be accessed under library mode Before we start describing these we recall the difference between strings and keywords see Section 2 6 6 the latter don t get expanded at all and you can type them without any enclosing quotes The former are dynamic objects where everything outside quotes gets immediately expanded 3 11 2 1 addhelp S str changes the help message for the symbol S The string str is expanded on the spot and stored as the online help for S If S is a function you have defined its definition will still be printed before the message str It is recommended that you document global variables and user functions in this way Of course GP won t protest if you don t do it There s nothing to prevent you from modifying the help of built in PARI functions but if you do we d like to hear why you needed to do it 3 11 2 2 alias newkey key defines the keyword newkey as an alias for keyword key key must correspo
326. nd x being either a Z basis of an ideal in the number field not necessarily in HNF or a prime ideal in the format output by the function idealprimedec this function tests whether the ideal is principal or not The result is more complete than a simple true false answer it gives a row vector v1 va where v is the vector of components c of the class of the ideal x in the class group expressed on the generators g given by bnfinit specifically bnf gen The c are chosen so that 0 lt ci lt ni where n is the order of g the vector of n being bnf cyc v2 gives on the integral basis the components of a such that x a 9 In particular x is principal if and only if v is equal to the zero vector In the latter case x aZxK where a is given by vg Note that if a is too large to be given a warning message will be printed and va will be set equal to the empty vector If flag 0 outputs only v which is much easier to compute If flag 2 does as if flag were 0 but doubles the precision until a result is obtained If flag 3 as in the default behaviour flag 1 but doubles the precision until a result is obtained 102 The user is warned that these two last setting may induce very lengthy computations The library syntax is isprincipalall bnf x flag 3 6 10 bnfisunit bnf x bnf being the number field data output by bnfinit and x being an algebraic number type integer rational or polmod this outputs the d
327. nd to an existing function name This is different from the general user macros in that alias expansion takes place immediately upon execution without having to look up any function code and is thus much faster A sample alias file misc gpalias is provided with the standard distribution Alias commands are meant to be read upon startup from the gprc file to cope with function names you are dissatisfied with and should be useless in interactive usage 164 UNIX 3 11 2 3 allocatemem z 0 this is a very special operation which allows the user to change the stack size after initialization x must be a non negative integer If x 0 a new stack of size 16 x x 16 bytes will be allocated all the PARI data on the old stack will be moved to the new one and the old stack will be discarded If x 0 the size of the new stack will be twice the size of the old one Although it is a function this must be the last instruction in any GP sequence The technical reason is that this routine usually moves the stack so objects from the current sequence might not be correct anymore Hence to prevent such problems this routine terminates by a longjmp just as an error would and not by a return The library syntax is allocatemoremem z where x is an unsigned long and the return type is void GP uses a variant which ends by a longjmp 3 11 2 4 default key val flag sets the default corresponding to keyword key to value val val is a str
328. nd y returns the lower BIL bits and put the carry borrow bit into overflow ulong subllx ulong x ulong y subtracts overflow from the difference of the ulongs x and y returns the lower BIL bits and puts the carry borrow bit into overflow ulong shiftl ulong x ulong y shifts the ulong x left by y bits returns the lower BIL bits and stores the high order BIL bits into hiremainder We must have 1 lt y lt BIL In particular y must be non zero the caller is responsible for testing this ulong shiftlr ulong x ulong y shifts the ulong x lt lt BIL right by y bits returns the higher BIL bits and stores the low order BIL bits into hiremainder We must have 1 lt y lt BIL In particular y must be non zero int bfffo ulong x returns the number of leading zero bits in the ulong x i e the number of bit positions by which it would have to be shifted left until its leftmost bit first becomes equal to 1 which can be between 0 and BIL 1 for nonzero x When x is 0 BIL is returned 207 ulong mulll ulong x ulong y multiplies the ulong x by the ulong y returns the lower BIL bits and stores the high order BIL bits into hiremainder ulong addmul ulong x ulong y adds hiremainder to the product of the ulongs x and y returns the lower BIL bits and stores the high order BIL bits into hiremainder ulong divll ulong x ulong y returns the Euclidean quotient of hiremainder lt lt BIL x and the ulong divisor y and stores the
329. nder of the long s modulo the integer x GEN gmodgs z GEN x long s GEN z yields the true remainder of the integer x modulo the long s GEN gres GEN x GEN y creates the Euclidean remainder of the polynomial x divided by the polynomial y GEN ginvmod GEN x GEN y creates the inverse of x modulo y when it exists y must be of type t_INT in which case x is of type t_INT or t_POL in which case x is either a scalar type or of type t_POL GEN gpow GEN x GEN y long 1 creates xY The precision 1 is taken into account only if y is not an integer and x is an exact object If y is an integer binary powering is done Otherwise the result is exp y log x computed to precision 1 GEN ggcd GEN x GEN y creates the GCD of x and y GEN glem GEN x GEN y creates the LCM of x and y GEN subres GEN x GEN y creates the resultant of the polynomials x and y computed using the subresultant algorithm GEN gpowgs GEN x long n creates x using binary powering GEN gsubst GEN x long v GEN y substitutes the object y into x for the variable number v int gdivise GEN x GEN y returns 1 true if y divides x 0 otherwise GEN gbezout GEN x GEN y GEN u GEN v creates the GCD of x and y and puts the ad dresses of objects u and v such that ux vy ged x y into u and v 220 Appendix A Installation Guide for the UNIX Versions 1 Required tools We assume that you have either an ANSI C or a C compiler availa
330. nding operation is put into z The size of the PARI stack does not change void gaddz GEN x GEN y GEN z void gaddgsz GEN x long y GEN z void gaddsgz GEN x GEN y GEN z There are also low level functions which are special cases of the above GEN addii GEN x GEN y here x and y are GENs of type t_INT this is not checked GEN addrr GEN x GEN y here x and y are GEN reals type t_REAL There also exist functions addir addri mpadd whose two arguments can be of type integer or real addis to add a t_INT and a long and so on All these functions can of course be called by the user but we feel that the few microseconds lost in calling more general functions in this case gadd are compensated by the fact that one needs to remember a much smaller number of functions and also because there is a hidden danger here the types of the objects that you use if they are themselves results of a previous computation are not 173 completely predetermined For instance the multiplication of a type real t_REAL by a type integer t_INT usually gives a result of type real except when the integer is 0 in which case according to the PARI philosophy the result is the exact integer 0 Hence if afterwards you call a function which specifically needs a real type argument you are going to be in trouble If you really want to use these functions their names are self explanatory once you know that i stands for a PARI integer r for a PARI real mp
331. ned via library mode would produce an illegal object and eventually a disaster In any case if you are working with expressions involving several variables and want to have them ordered in a specific manner in the internal representation just described the simplest is just to write down the variables one after the other under GP before starting any real computations You could also define variables from your GPRC to have a consistant ordering of common variable names in all your GP sessions e g read in a file variables gp containing x y z t a b c d If you already have started working and want to change the names of the variables in an object use the function changevar If you only want to have them ordered when the result is printed you can also use the function reorder but this won t change anything to the internal representation and is not recommended This is not strictly true if an identifier is interpreted as a user function no variable is registered Also the variable x is predefined and always has the highest possible priority 34 Important note PARI allows Euclidean division of multivariate polynomials but assumes that the computation takes place in the fraction field of the coefficient ring if it is not an integral domain the result will a priori not make sense This can be very tricky for instance assume x has highest priority which is always the case then y 7xhy 11 0 Pyhx 12 y these two take
332. nflllgram which deserves to be improved The library syntax is rnfpolredabs nf pol flag prec 3 6 143 rnfpseudobasis nf pol given a number field nf as output by nfinit and a polynomial pol with coefficients in nf defining a relative extension L of nf computes a pseudo basis A J for the maximal order Zr viewed as a Zkx module and the relative discriminant of L This is output as a four element row vector 4 1 D d where D is the relative ideal discriminant and d is the relative discriminant considered as an element of nf nf uz The library syntax is rnfpseudobasis nf pol 3 6 144 rnfsteinitz nf x given a number field nf as output by nfinit and either a polynomial x with coefficients in nf defining a relative extension L of nf or a pseudo basis x of such an extension as output for example by rnfpseudobasis computes another pseudo basis A T not in HNF in general such that all the ideals of J except perhaps the last one are equal to the ring of integers of nf and outputs the four component row vector 4 I D d as in rnfpseudobasis The name of this function comes from the fact that the ideal class of the last ideal of J which is well defined is the Steinitz class of the Zx module Zz its image in SKo Zx The library syntax is rnfsteinitz nf x 3 6 145 subgrouplist bnr bound flag 0 bnr being as output by bnrinit or a list of cyclic components of a finite Abelian group G outputs the list of subgroups of G
333. ng given for Pol applies here this is not a substitution function The library syntax is gtoser z v where v is a variable number i e a C integer 3 2 8 Set x converts x into a set i e into a row vector with strictly increasing entries x can be of any type but is most useful when x is already a vector The components of x are put in canonical form type t_STR so as to be easily sorted To recover an ordinary GEN from such an element you can apply eval to it The library syntax is gtoset x 3 2 9 Str 1 x converts its argument list into a single character string type t_STR the empty string if z is omitted To recover an ordinary GEN from a string apply eval to it The arguments of Str are evaluated in string context see Section 2 6 6 x2 0 i 2 Str x i 41 x2 eval 42 0 This function is mostly useless in library mode Use the pair strtoGEN GENtostr to convert between GEN and char The latter returns a malloced string which should be freed after usage 60 3 2 10 Strchr x converts x to a string translating each integer into a character Strchr 97 1 a Vecsmall hello world A2 Vecsmall 104 101 108 108 111 32 119 111 114 108 100 Strchr 43 hello world 3 2 11 Strexpand x converts its argument list into a single character string type t_STR the empty string if x is omitted Then performe environment expansion see Section 2 1 This featu
334. ng information in the first 13 components a1 42 43 04 06 b2 ba be bg Ca C6 A J In particular the discriminant is E 12 or E disc and the j invariant is E 13 or E j The other six components are only present if flag is 0 or omitted Their content depends on whether the curve is defined over R or not e When E is defined over R E 14 E roots is a vector whose three components contain the roots of the right hand side of the associated Weierstrass equation y 12 2 a3 2 g x If the roots are all real then they are ordered by decreasing value If only one is real it is the first component E 15 E omega 1 is the real period of E integral of dx 2y aix a3 over the connected component of the identity element of the real points of the curve and E 16 E omega 2 is a complex period In other words w E 15 and w2 E 16 form a basis of the complex lattice defining E E omega with T a having positive imaginary part E 17 and E 18 are the corresponding values 7 and nz such that 7 w3 nowi im and both can be retrieved by typing E eta as a row vector whose components are the 7 Finally E 19 E area is the volume of the complex lattice defining E e When E is defined over Qp the p adic valuation of j must be negative Then E 14 E roots is the vector with a single component equal to the p adic root of the associated Weierstrass equation corresponding to 1 under the Tate p
335. ng the not enough primes error You can use the function initprimes from the file arith2 c to compute a new table on the fly and assign it to diffptr or to a similar variable of your own Beware that before changing diffptr you should really free the malloced precomputed table first and then all pointers into the old table will become invalid PARI currently guarantees that the first 6547 primes up to and including 65557 will be present in the table even if you set maxnum to zero 232 Some Word refers to PARI GP concepts SomeWord is a PARI GP keyword SomeWord is a generic index entry A Abelian extension 127 133 ODS DA eee eed 69 abgi emp o soe ee bl wie Oe bois 212 absite qual ol A GS 212 ADSH CMP Cerda boo a os eee ee 212 ACCULACY i p A ados 9 ACOS el a ae ee ee a 70 ACOSH i oh gat GU BAe a Bee he 70 addeld 266 se see An iE a GE tea 90 addhelp o o o oo o 204 AAGHE MP co ita ad Mees 42 164 add 2 Da Sosy tit Pen eee th ee ot 173 BODE a ee ee ea eee ee An 173 Bd das js oly bi at ot ee Bd aie Ag ea 173 add clo nse te beet eke ke GE oid eS 207 ACCU ua a abled Sgt 207 addmul carre ia ek a 207 addprimes 43 76 126 add a SONS ete Pe tere ee ees 173 add ta saree SEs Be ha he eae eee Pere 173 AOS ok enh We eth ee e Se a 214 addumudlui s s i ele u a ae ee eee 8 215 adduumod 0 208 AY sts eves e Shee mead ai 144 adjoint matrix 143 APHID ii A
336. ng the user files mentioned above we look for etc gpre etc gprc for a system wide gpre file you will need root privileges to set up such a file yourself Note that on Unix systems the gprc s default name starts with a and thus is hidden to regular 1s commands you need to type 1s a to list it 47 2 9 2 Syntax The syntax in the gprc file and valid in this file only is simple minded but should be sufficient for most purposes The file is read line by line as usual white space is ignored unless surrounded by quotes and the standard multiline constructions using braces or are available multiline comments between are also recognized 2 9 2 1 Preprocessor Two types of lines are first dealt with by a preprocessor e comments are removed This applies to all text surrounded by as well as to everything following on a given line e lines starting with if boolean are treated as comments if boolean evaluates to false and read normally otherwise The condition can be negated using either if not or if If the rest of the current line is empty the test applies to the next line same behaviour as under GP Only three tests can be performed EMACS true if GP is running in an Emacs or TeXmacs shell see Section 2 10 READL true if GP is compiled with readline support see Section 2 11 1 VERSION op number where op is in the set gt lt lt gt and number is a PARI version n
337. nnounce to announce major version changes You can t write to this one but you should probably subscribe e pari dev for everything related to the development of PARI including suggestions tech nical questions bug reports or patch submissions e pari users for everything else To subscribe send empty messages respectively to pari announce subscribe list cr yp to pari users subscribe list cr yp to pari dev subscribe list cr yp to The PARI home page maintained by Gerhard Niklasch at the address http pari math u bordeaux fr maintains an archive of all discussions as well as a download area If don t want to subscribe to those lists you can write to us at the address pari math u bordeaux fr At the very least we will forward you mail to the lists above and correct faulty behaviour if necessary But we cannot promise you will get an individual answer If you have used PARI in the preparation of a paper please cite it in the following form BibTeX format manual PARI2 organization The PARI Group title PARI GP Version 2 2 7 year 2002 address Bordeaux note available from tt http pari math u bordeaux fr In any case if you like this software we would be indebted if you could send us an email message giving us some information about yourself and what you use PARI for Good luck and enjoy 227 228 Appendix B A Sample program and Makefile We assume that you have i
338. nomials Sometimes faster Divergent alternating series can sometimes be summed by this method as well as series which are not exactly alternating see for example Section 2 6 4 The library syntax is sumalt entree ep GEN a char expr long flag long prec 3 9 8 sumdiv n X expr sum of expression expr over the positive divisors of n Arithmetic functions like sigma use the multiplicativity of the underlying expression to speed up the computation In the present version 2 2 7 there is no way to indicate that expr is multi plicative in n hence specialized functions should be preferred whenever possible The library syntax is divsum entree ep GEN num char expr 155 3 9 9 suminf X a expr infinite sum of expression expr the formal parameter X starting at a The evaluation stops when the relative error of the expression is less than the default precision The expressions must always evaluate to a complex number The library syntax is suminf entree ep GEN a char expr long prec 3 9 10 sumpos X a expr flag 0 numerical summation of the series expr which must be a series of terms having the same sign the formal variable X starting at a The algorithm used is Van Wijngaarden s trick for converting such a series into an alternating one and is quite slow If flag 1 use slightly different polynomials Sometimes faster The library syntax is sumpos entree ep GEN a char expr long flag long prec
339. ns types t_RFRAC and t_RFRACN as for fractions all rational func tions are automatically reduced to lowest terms under GP All that was said about fractions in Section 2 3 4 remains valid here 2 3 12 Binary quadratic forms of positive or negative discriminant type t_QFR and t_QFI these are input using the function Qfb see Chapter 3 For example Qfb 1 2 3 will create the binary form g 2ry 3y It will be imaginary of internal type t_QFI since 2 4x3 8 is negative In the case of forms with positive discriminant type t_QFR you may add an optional fourth component related to the regulator more precisely to Shanks and Lenstra s distance which must be a real number See also the function qfbprimeform which directly creates a prime form of given discriminant see Chapter 3 2 3 13 Row and column vectors types t_VEC and t_COL to enter a row vector type the components separated by commas and enclosed between brackets and e g 1 2 3 To enter a column vector type the vector horizontally and add a tilde to transpose yields the empty row vector The function Vec can be used to transform any object into a vector see Chapter 3 27 2 3 14 Matrices type t_MAT to enter a matrix type the components line by line the components being separated by commas the lines by semicolons and everything enclosed in brackets and eg x y z t
340. ns the i th component of vector x x i j x 5 and x i respectively return the entry of coordinates i 7 the j th column and the th row of matrix x If the assignment operator immediately follows a sequence of selections it assigns its right hand side to the selected component E g x 1 1 0 is valid but beware that x 1 1 0 is not because the parentheses force the complete evaluation of x 1 and the result is not modifiable e Priority 6 unary prefix quote its argument a variable name without evaluating it Ta x 1 x 1 subst a x 1 Hook variable name expected subst a x 1 a subst a x 1 Yi 2 powering unary postfix derivative with respect to the main variable If f is a GP or user function f x is allowed If x is a scalar the operator performs numerical derivation defined as f a e f a e 2e for a suitably small epsilon depending on current precision It behaves as f x otherwise unary postfix vector matrix transpose unary postfix factorial x x x 1 1 member unary postfix x member extracts member from structure x see Section 2 6 5 29 e Priority 5 unary prefix logical not lx return 1 if x is equal to 0 specifically if gemp0 x 1 and 0 otherwise unary prefix cardinality 2 returns length zx e Priority 4 unary prefix toggles the sign of its argument has no effect whatsoever
341. nside an open brace close brace pair all your input lines will be concatenated suppressing any newlines Thus all newlines should occur after a semicolon a comma or an operator for clarity s sake we don t recommend splitting an identifier over two lines in this way For instance the following program a b b c would silently produce garbage since what GP will really see is a bb c which will assign the value of c to both bb and a if this really is what you intended you re a hopeless case 32 2 6 The GP PARI programming language The GP calculator uses a purely interpreted language The structure of this language is reminiscent of LISP with a functional notation f x y rather than f x y all programming constructs such as if while etc are functions see Section 3 11 for a complete list and the main loop does not really execute but rather evaluates sequences of expressions Of course it is by no means a true LISP 2 6 1 Variables and symbolic expressions In GP you can use up to 16383 variable names up to 65535 on 64 bit machines These names can be any standard identifier names i e they must start with a letter and contain only valid keyword characters _ or alphanumeric characters _A Za z0 9 To avoid confusion with other symbols you must not use other non alphanumeric symbols like or In addition to the function names which you must not use see the list with c there are exac
342. nsion is transparent to the user under GP but must be given as a second argument in library mode The library syntax is tayl x y n where the long integer n is the desired number of terms in the expansion 3 7 33 thue tnf a sol solves the equation P x y a in integers x and y where tnf was created with thueinit P sol if present contains the solutions of Norm x a modulo units of positive norm in the number field defined by P as computed by bnfisintnorm If tnf was computed without assuming GRH flag 1 in thueinit the result is unconditional For instance here s how to solve the Thue equation 21 5y 3 4 tnf thueinit x 13 5 thue tnf 4 41 1 1 Hence assuming GRH the only solution is x 1 y 1 The library syntax is thue tnf a sol where an omitted sol is coded as NULL 3 7 34 thueinit P flag 0 initializes the tnf corresponding to P It is meant to be used in conjunction with thue to solve Thue equations P x y a where a is an integer If flag is non zero certify the result unconditionnally Otherwise assume GRH this being much faster of course The library syntax is thueinit P flag prec 3 8 Vectors matrices linear algebra and sets Note that most linear algebra functions operating on subspaces defined by generating sets such as mathnf qf111 etc take matrices as arguments As usual the generating vectors are taken to be the columns of the given matrix
343. nstalled the PARI library and include files as explained in Appendix A or in the installation guide If you chose differently any of the directory names change them accordingly in the Makefiles If the program example that we have given is in the file matexp c say as the first of several matrix transcendental functions then a sample Makefile might look as follows Note that the actual file examples Makefile is much more elaborate and you should have a look at it if you intend to use install on custom made functions see Section 3 11 2 13 CC cc INCDIR usr local include pari LIBDIR usr local lib CFLAGS 0 I INCDIR L LIBDIR all matexp matexp matexp c CC CFLAGS o matexp matexp c lpari lm We then give the listing of the program examples matexp c seen in detail in Section 4 8 with the slight modifications explained at the end of that section Id matexp c v 1 4 2002 10 02 15 28 56 karim Exp include pari h GEN matexp GEN x long prec pari_sp ltop avma long 1x lg x i k n GEN y r s p1 p2 check that x is a square matrix if typ x t_MAT err typeer matexp if 1x 1 return cgetg 1 t_MAT if 1x lg x 1 err talker not a square matrix convert x to real or complex of real and compute its Lz norm s gzero r cgetr prec 1 affsr 1 r x gmul r x for i 1 i lt lx i s gadd s gnorml2 GEN x i if typ s t_REAL setlg s 3 s gs
344. nteger this is the smallest non negative integer congruent to x modulo y If y is a polynomial this is the polynomial of smallest degree congruent to z modulo y When y is a non integral real number x y is defined as x 2 y y This coincides with the definition for y integer if and only if x is an integer but still belongs to 0 y For instance 1 2 3 11 2 0 5 3 Hook forbidden division t_REAL t_INT 1 2 3 0 12 1 2 Note that when y is an integer and x a polynomial y is first promoted to a polynomial of degree 0 When zx is a vector or matrix the operator is applied componentwise The library syntax is gmod z y for x y 3 1 8 divrem x y v creates a column vector with two components the first being the Eu clidean quotient x y the second the Euclidean remainder x a y y of the division of x by y This avoids the need to do two divisions if one needs both the quotient and the remainder If v is present and x y are multivariate polynomials divide with respect to the variable v Beware that divrem x y 2 is in general not the same as x y there is no operator to obtain it in GP divrem 1 2 3 2 41 1 2 1 2 3 12 2 divrem Mod 2 9 3 2 forbidden division t_INTMOD t_INT Mod 2 9 6 13 Mod 2 3 The library syntax is divrem z y v where v is a long Also available as gdiventres z y when v is not needed 3 1 9 The expression xn is powering If t
345. ntent which should be easier to grasp than a parser s logic For instance 1 0 division by zero in gdiv gdivgs or ginv The first half of the sentence is crystal clear and the second one only gives more context as to where exactly the problem occured in the library Unfortunately library errors do not give context so it can be hard to track down exactly where in your program the error occured 2 7 2 Error recovery It is quite annoying to wait for some program to finish and find out the hard way that there was a mistake in it like the division by 0 above sending you back to the prompt First you may lose some valuable intermediate data Also correcting the error may not be ovious you might have to change your program adding a number of extra statements and tests to try and narrow down the problem A slightly different situation still related to error recovery is when you you actually foresee that some error may occur are unable to prevent it but quite capable of recovering from it given the chance Examples include lazy factorization cf addprimes where you knowingly use a pseudo prime N as if it were prime you may then encounter an impossible situation but this would 43 usually exhibit a factor of N enabling you to refine the factorization and go on Or you might run an expensive computation at low precision to guess the size of the output hence the right precision to use You can then encounter errors like
346. nusoidal curves ploth X 0 2 Pi X X sin X cos X 1 will draw a circle and the line y x 158 e 2 Recursive recursive plot If this flag is set only one curve can be drawn at time i e expr must be either a two component vector for a single parametric curve and the parametric flag has to be set or a scalar function The idea is to choose pairs of successive reference points and if their middle point is not too far away from the segment joining them draw this as a local approximation to the curve Otherwise add the middle point to the reference points This is very fast and usually more precise than usual plot Compare the results of ploth X 1 1 sin 1 X 2 and ploth X 1 1 sin 1 X for instance But beware that if you are extremely unlucky or choose too few reference points you may draw some nice polygon bearing little resemblance to the original curve For instance you should never plot recursively an odd function in a symmetric interval around 0 Try ploth x 20 20 sin x 2 to see why Hence it s usually a good idea to try and plot the same curve with slightly different parameters The other values toggle various display options e 4 no_Rescale do not rescale plot according to the computed extrema This is meant to be used when graphing multiple functions on a rectwindow as a plotrecth call in conjunction with plotscale e 8 no_X_axis do not print the z axis e 16 no_Y axis do not prin
347. nvalid object you couldn t lift it 2 3 9 Polynomials type t_POL type the polynomial in a natural way not forgetting to put a x between a coefficient and a formal variable this x does not appear in beautified output Any variable name can be used except for the reserved names I used exclusively for the square root of 1 Pi 3 14 Euler Euler s constant and all the function names predefined functions as described in Chapter 3 use c to get the complete list of them and user defined functions which you ought to know about use u if you are subject to memory lapses The total number of different variable names is limited to 16384 and 65536 on 32 bit and 64 bit machines respectively which should be enough If you ever need hundreds of variables you should probably be using vectors instead See Section 2 6 2 for a discussion of multivariate polynomial rings 2 3 10 Power series type t_SER type a rational function or polynomial expression and add to it OCexpr k where expr is an expression which has non zero valuation it can be a polynomial power series or a rational function the most common case being simply a variable name This indicates to GP that it is dealing with a power series and the desired precision is k times the valuation of expr with respect to the main variable of expr to check the ordering of the variables or to modify it use the function reorder see Section 3 11 2 22 2 3 11 Rational functio
348. o a real zero using for instance the instructions s cgetr prec 1 gaffsg 0 s This raises the question which real zero does this produce have a look at Section 1 2 6 3 In fact the following choice has been made it will give you the zero with exponent equal to BITS_IN_LONG times the number of longwords in the mantissa i e bit accuracy 1g s Instead of the above idiom you can also use the function GEN realzero long prec which simply returns a real zero to accuracy bit_accuracy prec The sixth remark here is about how to determine the approximate size of a real number The fastest way to do this is to look at its binary exponent Hence we need to have s actually represented as a real number and not as an integer or a rational number The result of transcendental functions is guaranteed to be of type t_REAL or complex with t_REAL components thus this is indeed the case after the call to gsqrt since its argument is a nonnegative real number Finally note the use of the function gmul2n It has the following syntax GEN gmul2n GEN x long n 199 and the effect is simply to multiply x by 2 where n can be positive or negative This is much faster than gmul or gmulgs There is another function gshift with exactly the same syntax When n is non negative the effects of these two functions are the same However when n is negative gshift acts like a right shift of n hence does not normally perform an exact divisio
349. o ish ge EO E ge el E E EE 75 Ganlyama sos ai seed a ea A 95 Taniyama Weil conjecture 90 ALG 2 28 ick IT Bede eee Ses 89 tate ee A ence eA enn die oe Mt ag A 89 Gayl sd col koe Eke ee aed ahs 141 Taylor series o o 55 Val coed ae aie oe amp A at a 8 90 TAYLO ach ite tee eal tata ales 140 TEME Avil t Se Gh begs whe AE eee 139 COUCH E en Sy A ee 75 teichmuller wie ce ea ee ek BA 75 tex2mail cc SR Se PG eS 19 Pexprint ss acs a eed ea a as 194 TeXstyle 18 21 theta se 4 60 ee ie ee Pe OM 75 thetanulik ooo See a 75 DOS far a ta te Be ae ee Be de 141 thieiniti 0505 a Se 141 time expansion 15 TIMET 4008 2 eh och ke 4 abn 2 aaran amp sk ge ha Seen 21 EFEMER dc ec echt oo a e ak radat 196 TIMEN a AE a 196 TIMETZ ac a A 196 ELACC E alado Sets 151 Trager hs bent ee ith es Rote 108 CAP bee gs hoe Bs AS hes MK ASE A 42 44 168 truecoeff lt fe hee eee 63 137 215 truedvmdii 0 4 214 CrUNnCAatS Met Le ey bale he es 64 67 tschirnhaus ae ira end ei oo 127 TUS AA a Micke AS 98 CUBU ia a ee as 98 CUTAN a A BA eee AO 22 YDI oe a 186 209 type number 186 ey Peed ay bec ca Ae ode Abd kD 42 169 ty PECASt 4 or ts Po a aoe eb 172 CY POS a Aa ek a eels eee Hie eh ee ee a 6 ECO 4 cee ety wh hal gee a Ean 7 27 190 t_COMPLEX 7 25 188 EZERAC aster be gu ao word 7 25 188 t FRACN ode hk hee a a 7 25 188 ti do ae Aas
350. o wait for the next line or lines of input before doing anything There are three ways of doing this The first one is simply to use the backslash character at the end of the line that you are typing just before hitting lt Return gt This tells GP that what you will write on the next line is the physical continuation of what you have just written In other words it makes GP forget your newline character For example if you use this while defining a function and if you ask for the definition of the function using name you will see that your backslash has disappeared and that everything is on the same line You can type a anywhere It will be interpreted as above only if apart from ignored whitespace characters it is immediately followed by a newline For example you can type 73 4 instead of typing 3 4 The second one is a slight variation on the first and is mostly useful when defining a user function see Section 2 6 4 since an equal sign can never end a valid expression GP will disregard a newline immediately following an a 123 1 123 The third one cannot be used everywhere but is in general much more useful It is the use of braces and When GP sees an opening brace at the beginning of a line modulo spaces as usual it understands that you are typing a multi line command and newlines will be ignored until you type a closing brace However there is an important but easily obeyed restriction i
351. ompare with the C prototypes as they stand in the code Remark If you need to implement complicated control statements probably for some improved summation functions you will need to know about the entree type which is not documented Check the comment before the function list at the end of language init c and the source code in language sumiter c You should be able to make something of it 4 9 3 Coding guidelines Code your function in a file of its own using as a guide other functions in the PARI sources One important thing to remember is to clean the stack before exiting your main function usually using gerepile since otherwise successive calls to the function will clutter the stack with unnecessary garbage and stack overflow will occur sooner Also if it returns a GEN and you want it to be accessible to GP you have to make sure this GEN is suitable for gerepileupto see Section 4 4 If error messages are to be generated in your function use the general error handling routine err see Section 4 7 3 Recall that apart from the warn variants this function does not return but ends with a longjmp statement As well instead of explicit printf fprintf statements use the following encapsulated variants void pariputs char s write s to the GP output stream void fprintferr char s write s to the GP error stream this function is in fact much more versatile see Section 4 7 4 Declare all public functions in an appropriate
352. on seq2 is evaluated Of course seq or seq2 may be empty so if a seq evaluates seq if a is not equal to zero you don t have to write the second comma and does nothing otherwise whereas if a seq evaluates seg if a is equal to zero and does nothing otherwise You could get the same result using the not operator if a seq Note that the boolean operators amp amp and are evaluated according to operator precedence as explained in Section 2 4 but that contrary to other operators the evaluation of the arguments is stopped as soon as the final truth value has been determined For instance if reallydoit amp amp longcomplicatedfunction is a perfectly safe statement Recall that functions such as break and next operate on loops such as forxxx while until The if statement is not a loop obviously 3 11 1 9 next n 1 interrupts execution of current seq resume the next iteration of the innermost enclosing loop within the current function call or top level loop If n is specified resume at the n th enclosing loop If n is bigger than the number of enclosing loops all enclosing loops are exited 3 11 1 10 return z 0 returns from current subroutine with result x If x is omitted return the void value return no result like print 3 11 1 11 until a seq evaluates expression sequence seg until a is not equal to 0 i e until a is true If a is initially not equal to 0 seq is evaluated o
353. only advised when using the calculator GP As an alternative one can replace a numeric flag by a character string containing symbolic identifiers For a generic flag the mnemonic corresponding to the numeric identifier is given after it as in taken from description of log z flag 0 If flag is equal to 1 AGM use an agm formula which means that one can use indifferently log x 1 or log x AGM For a binary flag mnemonics corresponding to the various toggles are given after each of them They can be negated by prepending no_ to the mnemonic or by removing such a prefix These toggles are grouped together using any punctuation character such as or For instance taken from description of ploth X a b expr flag 0 n 0 Binary digits of flags mean 1 Parametric 2 Recursive so that instead of 1 one could use the mnemonic Parametric no_Recursive or simply Para metric since Recursive is unset by default default value of flag is 0 i e everything unset Pointers If a parameter in the function prototype is prefixed with a amp sign as in foo z amp e it means that besides the normal return value the function may assign a value to e as a side effect When passing the argument the amp sign has to be typed in explicitly As of version 2 2 7 this pointer argument is optional for all documented functions hence the amp will always appear between brackets as in issquare z a
354. only its square is in which case the output is t 1 t E must be a long vector output by ellinit The library syntax is zell E z prec 3 5 22 ellpow E z n computes n times the point z for the group law on the elliptic curve E Here n can be in Z or n can be a complex quadratic integer if the curve E has complex multiplication by n if not an error message is issued The library syntax is powell E z n 3 5 23 ellrootno E p 1 E being a medium or long vector given by ellinit this computes the local if p 4 1 or global if p 1 root number of the L series of the elliptic curve E Note that the global root number is the sign of the functional equation and conjecturally is the parity of the rank of the Mordell Weil group The equation for E must have coefficients in Q but need not be minimal The library syntax is ellrootno E p and the result equal to 1 is a long 3 5 24 ellsigma F z flag 0 value of the Weierstrass o function of the lattice associated to E as given by ellinit alternatively E can be given as a lattice w1 w2 If flag 1 computes an arbitrary determination of log a z If flag 2 3 same using the product expansion instead of theta series The library syntax is ellsigma E z flag 94 3 5 25 ellsub 21 22 difference of the points z1 and 22 on the elliptic curve corresponding to the vector E The library syntax is subell E z1 22 3 5 26 elltaniyama E computes the
355. ontrary that you are giving a Z generating set If this is not the case you must change it into a Z generating set using idealhnf nf x Concerning relative extensions some additional definitions are necessary When defining a relative extension the base field nf must be defined by a variable having a lower priority see Section 2 6 2 than the variable defining the extension For example under GP you can use the variable name y or t to define the base field and the variable name x to define the relative extension e A relative matrix is a matrix whose entries are elements of a fixed number field nf always expressed as column vectors on the integral basis nf zk Hence it is a matrix of vectors e An ideal list is a row vector of fractional ideals of the number field nf e A pseudo matrix is a pair A I where A is a relative matrix and J an ideal list whose length is the same as the number of columns of A This pair is represented by a 2 component row vector e The module generated by a pseudo matrix A T is the sum gt ajA where the a are the ideals of J and A is the j th column of A e A pseudo matrix A 1 is a pseudo basis of the module it generates if A is a square matrix with non zero determinant and all the ideals of J are non zero We say that it is in Hermite Normal Form HNF if it is upper triangular and all the elements of the diagonal are equal to 1 e The determinant of a pseudo basis A T is the ideal
356. oo 44 168 Drake RA 44 162 O 2025 6 9 84 seve a eed AAN o 90 A 6 0 os E aod wc ee ee Bete 193 195 DUGCHEU ye ee et Yee a 104 DUCHiMAL aot GP eas ea ey ae rd idee Be 87 Buchmann 98 99 115 136 Buchmann McCurley 87 buchharrow o tdo paurs we Aw d eb 103 buchreal Xs cot gees Fair ee 87 butfersiZe via rada id A ee 15 C Cantor Zassenhaus 81 Carat dates Pie Seale Qube does 142 CAVA sie och ty te he ones Fe ele eh ied 142 arhesg CLA A See ano be ee 142 case distinction 31 COLT a rete RAE MR eS 62 centerlift 62 63 centert ifto cscs a He a a 63 certifybuchall 99 CESTE a ne kee 175 176 185 210 oe ahaa a aTa A A a AT 175 210 CLOUD 0 ahd alae awe A a oe a Eea 215 ERELT io EE E AE ees 175 210 CEL E aaa E E E 180 210 changevar 34 63 character string 28 190 character 104 106 107 characteristic polynomial 141 Char poly aior ee ge Sl ae A 141 charpolyO a ie Mad AG eed aye 142 Chebyshev 0 00005 139 CHINES cr BLE eed Bede ed BS 77 CHIDOS eee BS ek aoe eed aed 77 ClaSsno 2 oak A ee ae a ee g 85 Classno2 edn ea eh ek ee EAs 85 CID is ts So A ok ai didas 98 CEISP ei a aoe a ale we A 46 A he gece Gates Beds Got rar 172 178 Cmdtooh foot a eh a Taa 20 CMP h vig ce Ad Pe ome te 212 CMP IT oie a ee et te tel 212 EMPIRE 212 EMPED diia a ro A 212 CMPFES oo eta Ve 212 C
357. or S depending whether m is odd or even If flag 1 modified mt polylogarithm of x called Dm x in Zagier defined for x lt 1 by m 1 k ed m 1 t les EDT g a CRRA hog 4 k 73 If flag 2 modified mt polylogarithm of x called D x in Zagier defined for x lt 1 by de amp Ca par k 2 m k 0 If flag 3 another modified mt polylogarithm of x called P w in Zagier defined for z lt 1 by m l ok m 1 2 B ot Be m Rm X E log x Lim 2 log x m k k 0 These three functions satisfy the functional equation fm 1 2 1 1 f x The library syntax is polylog0 m x flag prec 3 3 40 psi x the function of x i e the logarithmic derivative I x T The library syntax is gpsi z prec 3 3 41 sin x sine of x The library syntax is gsin prec 3 3 42 sinh x hyperbolic sine of x The library syntax is gsh z prec 3 3 43 sqr x square of x This operation is not completely straightforward i e identical to xxx since it can usually be computed more efficiently roughly one half of the elementary multiplications can be saved Also squaring a 2 adic number increases its precision For example 1 0 274 72 1 1 0 275 1 0 274 1 0 274 42 1 0 274 Note that this function is also called whenever one multiplies two objects which are known to be identical e g they a
358. or gerepileal1 to fix up the stack for you If you followed us this far congratulations and rejoice the rest is much easier 4 5 Implementation of the PARI types Although it is a little tedious we now go through each type and explain its implementation Let z be a GEN pointing at a PARI object In the following paragraphs we will constantly mix two points of view on the one hand z will be treated as the C pointer it is in the context of program fragments like z 1 on the other as PARI s handle on the internal representation of some mathematical entity so we will shamelessly write z 0 to indicate that the value thus represented is nonzero in which case the pointer z certainly will be non NULL We offer no apologies for this style In fact you had better feel comfortable juggling both views simultaneously in your mind if you want to write correct PARI programs Common to all the types is the first codeword z 0 which we do not have to worry about since this is taken care of by cgetg Its precise structure will depend on the machine you are using but it always contain the following data the internal type number associated to the symbolic type name the length of the root in longwords and a technical bit which indicates whether the object is a clone see below or not This last one is used by GP for internal garbage collecting you will not have to worry about it These data can be handled through the following macros lon
359. oring engine called by most arithmetical functions flag is optional its binary digits mean 1 avoid MPQS 2 skip first stage ECM we may still fall back to it later 4 avoid Rho and SQUFOF 8 don t run final ECM as a result a huge composite may be declared to be prime Note that a strong probabilistic primality test is used thus composites might very rarely not be detected The machinery underlying this function is still in a somewhat experimental state but should be much faster on average than pure ECM as used by all PARI versions up to 2 0 8 at the expense of heavier memory use You are invited to play with the flag settings and watch the internals at work by using GP s debuglevel default parameter level 3 shows just the outline 4 turns on time keeping 5 and above show an increasing amount of internal details If you see anything funny happening please let us know The library syntax is factorint n flag 81 3 4 24 factormod z p flag 0 factors the polynomial x modulo the prime integer p using Berlekamp The coefficients of x must be operation compatible with Z pZ The result is a two column matrix the first column being the irreducible polynomials dividing x and the second the exponents If flag is non zero outputs only the degrees of the irreducible polynomials for example for computing an L function A different algorithm for computing the mod p factorization is factorcantor which is sometimes faster
360. ormation about the field as it becomes available which is rarely needed hence would be too expensive to compute during the initial rnfinit call The library syntax is rnfinitalg nf pol prec 3 6 135 rnfisfree bnf x given a big number field bnf as output by bnfinit and either a polynomial x with coefficients in bnf defining a relative extension L of bnf or a pseudo basis x of such an extension returns true 1 if L bnf is free false 0 if not The library syntax is rnfisfree bnf x and the result is a long 3 6 136 rnfisnorm T a flag 0 similar to bnfisnorm but in the relative case T is as output by rnfisnorminit applied to the extension L K This tries to decide whether the element a in K is the norm of some z in the extension L K The output is a vector x q where a Norm x q The algorithm looks for a solution x which is an S integer with S a list of places of K containing at least the ramified primes the generators of the class group of L as well as those primes dividing a If L K is Galois then this is enough otherwise flag is used to add more primes to S all the places above the primes p lt flag resp p flag if flag gt 0 resp flag lt 0 The answer is guaranteed i e a is a norm iff q 1 if the field is Galois or under GRH if S contains all primes less than 12 log disc M where M is the normal closure of L K If rnfisnorminit has determined or was told that L K is Galois and flag
361. orrectly detected although it is expected that infinitely many such numbers exist If flag gt O checks whether x is a strong Miller Rabin pseudo prime for flag randomly chosen bases with end matching to catch square roots of 1 The library syntax is gispseudoprime z flag but the simpler function ispseudoprime z which returns a 1ong should be used if x is known to be of type integer 3 4 32 issquare x amp n true 1 if x is square false 0 if not x can be of any type If n is given and an exact square root had to be computed in the checking process puts that square root in n This is in particular the case when zx is an integer or a polynomial This is not the case for intmods use quadratic reciprocity or series only check the leading coefficient The library syntax is gcarrecomplet x amp n Also available is gcarreparfait x 3 4 33 issquarefree x true 1 if x is squarefree false 0 if not Here x can be an integer or a polynomial The library syntax is gissquarefree x but the simpler function issquarefree x which returns a long should be used if x is known to be of type integer This issquarefree is just the square of the Moebius function and is computed as a multiplicative arithmetic function much like the latter 83 3 4 34 kronecker z y Kronecker symbol ly where x and y must be of type integer By definition this is the extension of Legendre symbol to Z x Z by total multiplicativity in both argum
362. out that the garbage does not overflow the currently available stack If it ever did we would have several options allocate a larger stack in the main program for instance change 1000000 into 2000000 do some gerepileing along the way or if you know what you are doing use allocatemoremen 198 Secondly the err function is the general error handler for the PARI library This will abort the program after printing the required message Thirdly notice how we handle the special case 1x 1 empty matrix before accessing 1x x 1 Doing it the other way round could produce a fatal error Indeed if x is of length 1 then x 1 is not a component of x It is just the contents of the memory cell which happens to follow the one pointed to by x and thus has no reason to be a valid GEN Now recall that none of the codeword handling macros do any kind of type checking see Section 4 5 thus lg would consider x 1 as a valid address and try to access GEN x 1 the first codeword which is unlikely to be a legal memory address In the fourth place to compute the square of the L norm of x we just add the squares of the L norms of the column vectors which we obtain using the library function gnorml2 Had this function not existed the norm computation would of course have been just as easy to write but we would have needed a double loop We then take the square root of s in precision 3 the smallest possible The prec argument of transcend
363. outputs U V D where U and V are two unimodular matrices such that UXV is the diagonal matrix D Otherwise output only the diagonal of D 2 generic input if set allows polynomial entries in which case the input matrix must be square Otherwise assume that X has integer coefficients with arbitrary shape 4 cleanup if set cleans up the output This means that elementary divisors equal to 1 will be deleted i e outputs a shortened vector D instead of D If complete output was required returns U V D so that U XV D holds If this flag is set X is allowed to be of the form D or U V D as would normally be output with the cleanup flag unset The library syntax is matsnf0 X flag Also available is smith X flag 0 3 8 38 matsolve x y x being an invertible matrix and y a column vector finds the solution u of xr u y using Gaussian elimination This has the same effect as but is a bit faster than z7 x y The library syntax is gauss z y 3 8 39 matsolvemod m d y flag 0 m being any integral matrix d a vector of positive integer moduli and y an integral column vector gives a small integer solution to the system of congruences gt mu yi mod d if one exists otherwise returns zero Shorthand notation y resp d can be given as a single integer in which case all the y resp d above are taken to be equal to y resp d m 1 2 3 4 matsolvemod m 3 4 1 2 42
364. p adic unit except when the number is zero in which case u is zero the significand having a certain precision k i e being defined modulo p Then this p adic zero is understood to be equal to O p i e there are infinitely many distinct p adic zeros The number k is thus irrelevant For power series the situation is similar with p replaced by X i e a power series zero will be O X the number k here the length of the power series being also irrelevant For real numbers the precision k is also irrelevant and a real zero will in fact be O 2 where e is now usually a negative binary exponent This of course will be printed as usual for a real number 0 0000 in f format or 0 Exx in e format and not with a O symbol as with p adics or power series With respect to the natural ordering on the reals we make the following convention whatever its exponent a real zero is smaller than any positive number and any two real zeroes are equal 1 3 Operations and functions 1 3 1 The PARI philosophy The basic philosophy which governs PARI is that operations and functions should firstly give as exact a result as possible and secondly be permitted if they make any kind of sense More specifically if you do an operation not a transcendental one between exact objects you will get an exact object For example dividing 1 by 3 does not give 0 33333 as you might expect but simply the rational number 1 3 If you really want th
365. p for quicker computations next time the function is called 69 3 3 5 abs 1 absolute value of x modulus if x is complex Power series and rational functions are not allowed Contrary to most transcendental functions an exact argument is not converted to a real number before applying abs and an exact result is returned if possible abs 1 Zi 1 abs 3 7 4 7x1 42 5 7 abs 1 I 3 1 414213562373095048801688724 If x is a polynomial returns x if the leading coefficient is real and negative else returns x For a power series the constant coefficient is considered instead The library syntax is gabs z prec 3 3 6 acos x principal branch of cos z i e such that Re acos x 0 7 If x R and x gt 1 then acos x is complex The library syntax is gacos z prec 3 3 7 acosh x principal branch of cosh x i e such that Im acosh z 0 7 If x R and x lt 1 then acosh x is complex The library syntax is gach z prec 3 3 8 agm z y arithmetic geometric mean of x and y In the case of complex or negative numbers the principal square root is always chosen p adic or power series arguments are also allowed Note that a p adic agm exists only if x y is congruent to 1 modulo p modulo 16 for p 2 x and y cannot both be vectors or matrices The library syntax is agm z y prec 3 3 9 arg x argument of the complex number z such that 7 lt arg x lt r The library
366. peri ment with this mechanism as often as possible you ll probably find it very convenient For instance you can obtain default seriesprecision 10 just by hitting def lt TAB gt se lt TAB gt 10 which saves 18 keystrokes out of 27 Hitting M h will give you the usual short online help concerning the word directly beneath the cursor M H will yield the extended help corresponding to the help default program usually opens a dvi previewer or runs a primitive tex to ASCII program None of these disturb the line you were editing recall that you can always undo the effect of the preceding keys by hitting C _ ol 52 Chapter 3 Functions and Operations Available in PARI and GP The functions and operators available in PARI and in the GP PARI calculator are numerous and everexpanding Here is a description of the ones available in version 2 2 7 It should be noted that many of these functions accept quite different types as arguments but others are more restricted The list of acceptable types will be given for each function or class of functions Except when stated otherwise it is understood that a function or operation which should make natural sense is legal In this chapter we will describe the functions according to a rough classification The general entry looks something like foo x flag 0 short description The library syntax is foo z flag This means that the GP function foo has one mandatory argument x and an
367. place in Q y x x Mod 1 y 13 Mod 1 y x A n Q y yQ y z Qlz Mod x y 74 0 In the last exemple the division by y takes place in Q y r hence the Mod object is a coset in Q y z yQ y z which is the null ring since y is invertible So be very wary of variable ordering when your computations involve implicit divisions and many variables This also affects functions like numerator denominator or content denominator x y hi 1 denominator y x 12 x content x y 3 1 y content y x 14 1 content 2 x 45 2 Can you see why Hint 2 y 1 y x is in Q y x and denominator is taken with respect to Q y x y x y x z is in Q y x so y is invertible in the coefficient ring On the other hand 2 x involves a single variable and the coefficient ring is simply Z These problems arise because the variable ordering defines an implicit variable with respect to which division takes place This is the price to pay to allow and operators on polynomials instead of requiring a more cumbersome divrem z y var which also exists Unfortunately in some functions like content and denominator there is no way to set explicitly a main variable like in divrem and remove the dependance on implicit orderings This will hopefully be corrected in future versions 2 6 3 Expressions and expression sequences An expression is formed by combining the GP operators functions including
368. ppr nf x flag 0 if x is a fractional ideal given in any form gives an element a in nf such that for all prime ideals gp such that the valuation of x at gp is non zero we have vola v x and vela gt 0 for all other go If flag is non zero x must be given as a prime ideal factorization as output by idealfactor but possibly with zero or negative exponents This yields an element a such that for all prime ideals p occurring in x vg a is equal to the exponent of p in x and for all other prime ideals vola gt 0 This generalizes idealappr nf x 0 since zero exponents are allowed Note that the algorithm used is slightly different so that idealappr nf idealfactor nf x may not be the same as idealappr nf x 1 The library syntax is idealapprO n x flag 3 6 44 idealchinese nf x y x being a prime ideal factorization i e a 2 by 2 matrix whose first column contain prime ideals and the second column integral exponents y a vector of elements in nf indexed by the ideals in x computes an element b such that Ve b Yo gt ve x for all prime ideals in x and v b gt 0 for all other The library syntax is idealchinese nf x y 3 6 45 idealcoprime nf x y given two integral ideals x and y in the number field nf finds a 8 in the field expressed on the integral basis nf 7 such that 8 is an integral ideal coprime to y The library syntax is idealcoprime nf x y 112 3 6 46 idealdiv nf x y flag
369. qrt s 3 we do not need much precision on s ifs lt 1 we are happy k expo s if k lt 0 n 0 pl x else n k 1 pl gmul2n x n setexpo s 1 initializations before the loop 229 y gscalmat r 1x 1 creates scalar matrix with r on diagonal p2 pi r s k 1 y gadd y p2 the main loop while expo r gt BITS_IN_LONG prec 1 k p2 gdivgs gmul p2 p1 k r gdivgs gmul s r k y gadd y p2 square back n times if necessary for i 0 i lt n i y gsqr y return gerepileupto ltop y int main long d prec 3 GEN x take a stack of 10 bytes no prime table pari_init 1000000 2 printf precision of the computation in decimal digits n d itos lisGEN stdin if d gt 0 prec long d parik1 3 printf input your matrix in GP format n x matexp lisGEN stdin prec sor x g d 0 exit 0 230 Appendix C Summary of Available Constants In this appendix we give the list of predefined constants available in the PARI library All of them are in the heap and not on the PARI stack We start by recalling the universal objects introduced in Section 4 1 t_INT gzero zero gun un gdeux deux t_FRAC ghalf lhalf t_COMPLEX gi t_POL polun lpolun polx lpolx Only polynomials in the variables 0 and MAXVARN are defined initially Use fetch_var see Sectio
370. qual to pZ g aZx where Z x is the ring of integers of the field and a gt ajw where the w form the integral basis nf zk e is the ramification index f is the residual index and b is an n component column vector representing a 3 Zg such that vp Zg B pZk which will be useful for computing valuations but which the user can ignore The number a is guaranteed to have a valuation equal to 1 at the prime ideal this is automatic if e gt 1 The library syntax is primedec nf p 3 6 59 idealprincipal nf x creates the principal ideal generated by the algebraic number x which must be of type integer rational or polmod in the number field nf The result is a one column matrix The library syntax is principalideal nf x 3 6 60 idealred nf J vdir 0 LLL reduction of the ideal J in the number field nf along the direction vdir If vdir is present it must be an rl r2 component vector r1 and r2 number of real and complex places of nf as usual This function finds a small a in T it is an LLL pseudo minimum along direction vdir The result is the Hermite normal form of the LLL reduced ideal rI a where r is a rational number such that the resulting ideal is integral and primitive This is often but not always a reduced ideal in the sense of Buchmann If J is an idele the logarithmic embeddings of a are subtracted to the Archimedean part More often than not a principal ideal will yield the identity matrix
371. qual to the power series 1 O X for a certain k they will be done using power series of precision at most k These are the three most common initializations As an extreme example compare prod i 1 100 1 X7i MAN this has degree 5050 time 3 335 ms prod i 1 100 1 X7i 1 0 X7101 time 43 ms 42 1 X X72 X75 X 7 X712 X715 X 22 X 26 X 35 X 40 X 51 X 57 X 70 X 77 X 92 X7100 O X7101 The library syntax is produit entree ep GEN a GEN b char expr GEN x 3 9 3 prodeuler X a b expr product of expression expr initialized at 1 i e to a real number equal to 1 to the current realprecision the formal parameter X ranging over the prime numbers between a and b The library syntax is prodeuler entree ep GEN a GEN b char expr long prec 3 9 4 prodinf X a expr flag 0 infinite product of expression expr the formal parameter X starting at a The evaluation stops when the relative error of the expression minus 1 is less than the default precision The expressions must always evaluate to an element of C If flag 1 do the product of the 1 expr instead The library syntax is prodinf entree ep GEN a char expr long prec flag 0 or prodinf1 with the same arguments flag 1 154 3 9 5 solve X a b expr find a real root of expression expr between a and b under the condition expr X a x expr X b lt 0 This routine uses Brent s m
372. quivalent it is in Landscape orientation and assumes A4 paper size If the pdftex package is part of your T X setup you can produce these documents in PDF format which may be more convenient for online browsing the manual is complete with hyperlinks type make docpdf All these documents are available online from PARI home page and on ftp pari math u bordeaux fr pub pari in any case 226 4 2 C programming Once all libraries and include files are installed you can link your C programs to the PARI library A sample makefile examples Makefile is provided to illustrate the use of the various libraries Type make all in the examples directory to see how they perform on the mattrans c program which is commented in the manual 4 3 GP scripts Several complete sample GP programs are also given in the examples directory for example Shanks s SQUFOF factoring method the Pollard rho factoring method the Lucas Lehmer primality test for Mersenne numbers and a simple general class group and fundamental unit algorithm much worse than the built in bnfinit See the file examples EXPLAIN for some explanations 4 4 EMACS If you want to use gp under GNU Emacs read the file emacs pariemacs txt If you are familiar with Emacs we suggest that you do so 4 5 The PARI Community There are three mailing lists devoted to the PARI GP package run courtesy of Dan Bernstein and most feedback should be directed to those They are e pari a
373. r 3 10 25 plotrpoint w dx dy draw the point x1 dx y1 dy on the rectwindow w if it is inside w where xl yl is the current position of the cursor and in any case move the virtual cursor to position x1 dx yl dy 3 10 26 plotscale w x1 1x2 yl y2 scale the local coordinates of the rectwindow w so that x goes from xl to x2 and y goes from yl to y2 12 lt x1 and y2 lt yl being allowed Initially after the initialization of the rectwindow w using the function plotinit the default scaling is the graphic pixel count and in particular the y axis is oriented downwards since the origin is at the upper left The function plotscale allows to change all these defaults and should be used whenever functions are graphed 3 10 27 plotstring w x flag 0 draw on the rectwindow w the String x see Section 2 6 6 at the current position of the cursor flag is used for justification bits 1 and 2 regulate horizontal alignment left if 0 right if 2 center if 1 Bits 4 and 8 regulate vertical alignment bottom if 0 top if 8 v center if 4 Can insert additional small gap between point and string horizontal if bit 16 is set vertical if bit 32 is set see the tutorial for an example 3 10 28 plotterm term sets terminal where high resolution plots go this is currently only taken into account by the gnuplot graphical driver Using the gnuplot driver possible terminals are the same as in gnuplot If term is l
374. r GEN coeff hence cannot be put on the left side of an assignment To retrieve the values of elements of lists of of lists of vectors without getting infuriated by gigantic lists of typecasts we have the mael macros for multidimensional array element The syntax is maeln x a1 an where x is a GEN the a are indexes and n is an integer between 2 and 5 with a standalone mael as a synonym for mae12 This stands for x a a2 an with all the necessary typecasts and returns a long i e they are valid lvalues The gmaeln macros are synonyms for GEN maeln Note that due to the implementation of matrix types in PARI i e as horizontal lists of vertical vectors coeff x y is actually completely equivalent to mael y x It is suggested that you use coeff in matrix context and mael otherwise 4 2 2 Variations on basic functions In the library syntax descriptions in Chapter 3 we have only given the basic names of the functions For example gadd x y assumes that x and y are PARI objects of type GEN and creates the result x y on the PARI stack For most of the basic operators and functions many other variants are available We give some examples for gadd but the same is true for all the basic operators as well as for some simple common functions a more complete list is given in Chapter 5 GEN gaddgs GEN x long y GEN gaddsg long x GEN y In the following three z is a preexisting GEN and the result of the correspo
375. r z p2 p1 Recall that since the stack grows downward from the top the most recent object comes first We need a way to get rid of this garbage in this case it causes no harm except that it occupies memory space but in other cases it could disconnect other PARI objects and this is dangerous It would not have been possible to get rid of p1 p2 before z is computed since they are used in the final operation We cannot record avma before p1 is computed and restore it later since this would destroy z as well It is not possible either to use the function cgiv since p1 and p2 are not at the bottom of the stack and we do not want to give back z 182 But using gerepile we can give back the memory locations corresponding to p1 p2 and move the object z upwards so that no space is lost Specifically ltop avma remember the current address of the top of the stack pi gsqr x p2 gsqr y lbot avma keep the address of the bottom of the garbage pile z gadd p1 p2 z is now the last object on the stack Z gerepile ltop lbot z garbage collecting Of course the last two instructions could also have been written more simply z gerepile ltop lbot gadd p1 p2 In fact gerepileupto is even simpler to use because the result of gadd will be the last object on the stack and gadd is guaranteed to return an object suitable for gerepileupto ltop avma z gerepileupto ltop gadd gsqr x gsqr y
376. r of distinct prime divisors of x x must be of type integer The library syntax is omega x the result is a long 3 4 41 precprime x finds the largest pseudoprime see ispseudoprime less than or equal to x x can be of any real type Returns 0 if x lt 1 Note that if x is a prime this function returns x and not the largest prime strictly smaller than x To rigorously prove that the result is prime use isprime The library syntax is precprime z 3 4 42 prime x the x prime number which must be among the precalculated primes The library syntax is prime x x must be a long 84 3 4 43 primes x creates a row vector whose components are the first x prime numbers which must be among the precalculated primes The library syntax is primes x x must be a long 3 4 44 qfbclassno D flag 0 ordinary class number of the quadratic order of discriminant D In the present version 2 2 7 a O D algorithm is used for D gt 0 using Euler product and the functional equation so D should not be too large say D lt 108 for the time to be reasonable On the other hand for D lt 0 one can reasonably compute qfbclassno D for D lt 107 since the routine uses Shanks s method which is in O D 4 For larger values of D see quadclassunit If flag 1 compute the class number using Euler products and the functional equation However it is in O D 2 Important warning For D lt 0 this function may give incorrec
377. r read commands For example it is turned on at the beginning of the test files used to check whether GP has been built correctly see e 17 UNIX 2 1 9 format default g0 28 and g0 38 on 32 bit and 64 bit machines respectively of the form xm n where x is a letter in e f g and n m are integers If x is f real numbers will be printed in fixed floating point format with no explicit exponent e g 0 000033 unless their integer part is not defined not enough significant digits if the letter is e they will be printed in scientific format always with an explicit exponent e g 3 3e 5 If the letter is g real numbers will be printed in f format except when their absolute value is less than 2 or they are real zeroes of arbitrary exponent in which case they are printed in e format The number n is the number of significant digits printed for real numbers except if n lt 0 where all the significant digits will be printed initial default 28 or 38 for 64 bit machines and the number m is the number of characters to be used for printing integers but is ignored if equal to 0 which is the default This is a feeble attempt at formatting 2 1 10 help default the location of the gphelp script the name of the external help program which will be used from within GP when extended help is invoked usually through a or request see Section 2 2 1 or M H under readline see Section 2 11 1 2 1 11 histsize default 50
378. r s totient function of x in other words Z 1Z x must be of type integer The library syntax is phi z 3 4 18 factor x lim 1 general factorization function If x is of type integer rational polynomial or rational function the result is a two column matrix the first column being the irreducibles dividing x prime numbers or polynomials and the second the exponents If x is a vector or a matrix the factoring is done componentwise hence the result is a vector or matrix of two column matrices By definition 0 is factored as 0 If x is of type integer or rational the factors are pseudoprimes see ispseudoprime and in general not rigorously proven primes In fact any factor which is lt 10 is a genuine prime number Use isprime to prove primality of other factors as in fa factor 2 2 7 1 isprime fa 1 An argument lim can be added meaning that we look only for factors up to lim or to primelimit whichever is lowest except when lim 0 where the effect is identical to setting lim primelimit In this case the remaining part may actually be a proven composite See factorint for more information about the algorithms used The polynomials or rational functions to be factored must have scalar coefficients In particular PARI does not know how to factor multivariate polynomials See factormod and factorff for the algorithms used over finite fields factornf for the algorithms over number fields Over
379. ral sense such that x u y v gcd z y The arguments must be both integers or both polynomials and the result is a row vector with three components u v and gcd z y The library syntax is vecbezout z y to get the vector or gbezout z y amp u amp v which gives as result the address of the created gcd and puts the addresses of the corresponding created objects into u and v 3 4 4 bezoutres z y as bezout with the resultant of x and y replacing the gcd The library syntax is vecbezoutres z y to get the vector or subresext z y amp u amp v which gives as result the address of the created gcd and puts the addresses of the corresponding created objects into u and v 3 4 5 bigomega x number of prime divisors of x counted with multiplicity x must be an integer The library syntax is bigomega z the result is a long 3 4 6 binomial z y binomial coefficient o Here y must be an integer but x can be any PARI object The library syntax is binome z y where y must be a long 3 4 7 chinese x y if x and y are both integermods or both polmods creates with the same type a z in the same residue class as x and in the same residue class as y if it is possible This function also allows vector and matrix arguments in which case the operation is recur sively applied to each component of the vector or matrix For polynomial arguments it is applied to each coefficient If y is omitted and x is a vector
380. rary syntax is idmat n where n is a long Related functions are gscalmat x n which creates x times the identity matrix x being a GEN and n a long and gscalsmat x n which is the same when z is a long 3 8 22 matimage z flag 0 gives a basis for the image of the matrix x as columns of a matrix A priori the matrix can have entries of any type If flag 0 use standard Gauss pivot If flag 1 use matsupplement The library syntax is matimage0 xz flag Also available is image x flag 0 145 3 8 23 matimagecompl x gives the vector of the column indices which are not extracted by the function matimage Hence the number of components of matimagecomp1 x plus the number of columns of matimage x is equal to the number of columns of the matrix z The library syntax is imagecompl z 3 8 24 matindexrank z x being a matrix of rank r gives two vectors y and z of length r giving a list of rows and columns respectively starting from 1 such that the extracted matrix obtained from these two vectors using vecextract z y z is invertible The library syntax is indexrank z 3 8 25 matintersect z y x and y being two matrices with the same number of rows each of whose columns are independent finds a basis of the Q vector space equal to the intersection of the spaces spanned by the columns of x and y respectively See also the function idealintersect which does the same for free Z modules The library syntax is intersect z
381. re 1 yl is not necessarily the position of the virtual cursor use plotmove w x1 y1 before using this function 3 10 15 plotlinetype w type change the type of lines subsequently plotted in rectwindow w type 2 corresponds to frames 1 to axes larger values may correspond to something else w 1 changes highlevel plotting This is only taken into account by the gnuplot interface 3 10 16 plotmove w x y move the virtual cursor of the rectwindow w to position x y 3 10 17 plotpoints w X Y draw on the rectwindow w the points whose x y coordinates are in the vectors of equal length X and Y and which are inside w The virtual cursor does not move This is basically the same function as plothraw but either with no scaling factor or with a scale chosen using the function plotscale As was the case with the plotlines function X and Y are allowed to be simultaneously scalar In this case draw the single point X Y on the rectwindow w if it is actually inside w and in any case move the virtual cursor to position x y 3 10 18 plotpointsize w size changes the size of following points in rectwindow w If w 1 change it in all rectwindows This only works in the gnuplot interface 3 10 19 plotpointtype w type change the type of points subsequently plotted in rectwindow w type 1 corresponds to a dot larger values may correspond to something else w 1 changes highlevel plotting This is only t
382. re available to deal with the output of ellinit al a6 b2 b8 c4 c6 coefficients of the elliptic curve area volume of the complex lattice defining E disc discriminant of the curve j j invariant of the curve omega w1 w3 periods forming a basis of the complex lattice defining E w1 is the real period and w2 w belongs to Poincar s half plane eta quasi periods 71 2 such that m1w2 Now ir 89 roots roots of the associated Weierstrass equation tate u u v in the notation of Tate W Mestre s w this is technical Their use is best described by an example assume that E was output by ellinit then typing FE disc will retrieve the curve s discriminant The member functions area eta and omega are only available for curves over Q Conversely tate and w are only available for curves defined over Qp Some functions in particular those relative to height computations see ellheight require also that the curve be in minimal Weierstrass form This is achieved by the function ellminimalmodel All functions related to elliptic curves share the prefix ell and the precise curve we are interested in is always the first argument in either one of the three formats discussed above unless otherwise specified For instance in functions which do not use the extra information given by long vectors the curve can be given either as a five component vector or by one of the longer vectors computed by ellin
383. re can be used to read environment variable values Strexpand HOME doc 41 home pari doc The individual arguments are read in string context see Section 2 6 6 3 2 12 Strtex x x translates its arguments to TeX format and concatenates the results into a single character string type t_STR the empty string if x is omitted The individual arguments are read in string context see Section 2 6 6 3 2 13 Vec x transforms the object x into a row vector The vector will be with one component only except when x is a vector matrix or a quadratic form in which case the resulting vector is simply the initial object considered as a row vector a character string a vector of individual characters is returned but more importantly when x is a polynomial or a power series In the case of a polynomial the coefficients of the vector start with the leading coefficient of the polynomial while for power series only the significant coefficients are taken into account but this time by increasing order of degree The library syntax is gtovec z 3 2 14 Vecsmall x transforms the object x into a row vector of type t_VECSMALL This acts as Vec but only on a limited set of objects the result must be representable as a vector of small integers In particular polynomials and power series are forbidden If x is a character string a vector of individual characters in ASCII encoding is returned Strchr yields back the character strin
384. re the value of the same variable or we are computing a power x 1 0 274 x x 13 1 0 275 1 0 274 74 74 1 0 276 note the difference between 2 and 3 above The library syntax is gsqr x 74 3 3 44 sqrt x principal branch of the square root of x i e such that Arg sqrt 1 7 2 7 2 or in other words such that R sqrt 1 gt 0 or R sqrt x 0 and S sqrt x gt 0 If x R and x lt 0 then the result is complex with positive imaginary part Integermod a prime and p adics are allowed as arguments In that case the square root if it exists which is returned is the one whose first p adic digit or its unique p adic digit in the case of integermods is in the interval 0 p 2 When the argument is an integermod a non prime or a non prime adic the result is undefined The library syntax is gsqrt x prec 3 3 45 sqrtn z n amp z principal branch of the nth root of x i e such that Arg sqrt x a n x n Integermod a prime and p adics are allowed as arguments If z is present it is set to a suitable root of unity allowing to recover all the other roots If it was not possible z is set to zero The following script computes all roots in all possible cases sqrtnall x n local V r z r2 r sqrtn x n amp z if z error Impossible case in sqrtn if type x t_INTMOD type x t_PADIC r2 rxz n 1 while r2 r r2 z nt V vector
385. red x and factoredpolred z fa both corresponding to flag 0 3 6 108 polredabs z flag 0 finds one of the polynomial defining the same number field as the one defined by x and such that the sum of the squares of the modulus of the roots i e the T gt norm is minimal All x accepted by nfinit are also allowed here e g non monic polynomials nf bnf x Z_K_basis 126 Warning this routine uses an exponential time algorithm to enumerate all potential generators and may be exceedingly slow when the number field has many subfields hence a lot of elements of small T gt norm E g do not try it on the compositum of many quadratic fields use polred instead The binary digits of flag mean 1 outputs a two component row vector P a where P is the default output and a is an element expressed on a root of the polynomial P whose minimal polynomial is equal to zx 4 gives all polynomials of minimal T norm of the two polynomials P x and P x only one is given 16 possibly use a suborder of the maximal order The primes dividing the index of the order chosen are larger than primelimit or divide integers stored in the addprimes table In that case it may happen that the output polynomial does not have minimal T norm The library syntax is polredabs0 z flag 3 6 109 polredord z finds polynomials with reasonably small coefficients and of the same degree as that of x defining suborders of the order defined by x One
386. resent version 2 2 7 x must be irreducible and the degree of x must be less than or equal to 7 On certain versions for which the data file of Galois resolvents has been installed available in the Unix distribution as a separate package degrees 8 9 10 and 11 are also implemented The output is a 3 component vector n s k with the following meaning n is the cardinality of the group s is its signature s 1 if the group is a subgroup of the alternating group An s 1 otherwise k is more arbitrary and the choice made up to version 2 2 3 of PARI is rather unfortunate for n gt 7 k is the numbering of the group among all transitive subgroups of Sn as given in The transitive groups of degree up to eleven G Butler and J McKay Communications in Algebra vol 11 1983 pp 863 911 group k is denoted Ty there And for n lt 7 it was ad hoc so as to ensure that a given triple would design a unique group Specifically for polynomials of degree lt 7 the groups are coded as follows using standard notations In degree 1 S 1 1 1 In degree 2 Sa 2 1 1 In degree 3 43 C3 3 1 1 S3 6 1 1 In degree 4 C4 4 1 1 Va 4 1 1 Da 8 1 1 A4 12 1 1 S4 24 1 1 In degree 5 Cs 5 1 1 Ds 10 1 1 Mao 20 1 1 As 60 1 1 S5 120 1 1 In degree 6 Cg 6 1 1 S3 6 1 2 De 12 1 1 44 12 1 1 Gig 18 1 1
387. ress bot and ends just before top This means that the quantity top bot sizeof long is equal to the size argument of pari_init The PARI stack also has a current stack pointer called avma which stands for available memory address These three variables are global declared by pari h They are of type pari_sp which means pari stack pointer The stack is oriented upside down the more recent an object the closer to bot Accordingly initially avma top and avma gets decremented as new objects are created As its name indicates avma always points just after the first free address on the stack and GEN avma is always a pointer to the latest created object When avma reaches bot the stack overflows aborting all computations and an error message is issued To avoid this you will need to clean up the stack 179 from time to time when some bunch of intermediate objects will not be needed anymore This is called garbage collecting We are now going to describe briefly how this is done We will see many concrete examples in the next subsection e First PARI routines will do their own garbage collecting which means that whenever a doc umented function from the library returns only its result s will have been added to the stack non documented ones may not do this In particular a PARI function that does not return a GEN does not clutter the stack Thus if your computation is small enough i e you call few PARI ro
388. riority which have been introduced later in the GP session for external operations typically between a polynomial and a polmod For example PARI will not recognize that Mod y y 2 1 is the same as Mod x x72 1 Hopefully this problem will pass away when type element of a number field is eventually introduced See Section 2 6 2 for a definition of priority and a discussion of PARI s idea of multivariate polynomial arithmetic On the other hand Mod x x72 1 Mod x x72 1 which gives Mod 2 x x72 1 and x Mod y y 2 1 which gives a result mathematically equivalent to x i with i 1 are completely correct while y Mod x x72 1 gives Mod x y x 2 1 which may not be what you want y is treated here as a numerical parameter not as a polynomial variable On the other hand one can argue that there is no reason to consider these quantities equal E g one can be the opposite of another Compare with numerous discussions on whether the algebraic closure of Q is canonically defined or one needs to consider a groupoid of algebraic closures 26 Note added in version 2 0 16 As long as the main variables are the same it is allowed to mix t_POL and t_POLMODs The result will be the expected t_POLMOD For instance x Mod x x72 1 is equal to Mod 2 x x72 1 This wasn t the case prior to version 2 0 16 it returned a polynomial in x equivalent to x 7 which was in fact an i
389. rm type t_FRACN Q Rational numbers not necessarily in irreducible form type t_COMPLEX T i Complex numbers type t_PADIC Qp p adic numbers type t_QUAD Qfw Quadratic Numbers where Z w Z 2 type t_POLMOD T X P X T X Polmods polynomials modulo P type t_POL T X Polynomials type t_SER T X Power series finite Laurent series type t_RFRAC T X Rational functions in irreducible form type t_RFRACN T X Rational functions not necessarily in irreducible form type t_VEC q Row i e horizontal vectors type t_COL T Column i e vertical vectors type t_MAT Mm n T Matrices type t_LIST I Lists type t_STR Character strings and where the types T in recursive types can be different in each component In addition there exist types t_QFR and t_QFI for binary quadratic forms of respectively positive and negative discriminants which can be used in specific operations but which may disappear in future versions Every PARI object called GEN in the sequel belongs to one of these basic types Let us have a closer look 1 2 1 Integers and reals they are of arbitrary and varying length each number carrying in its internal representation its own length or precision with the following mild restrictions given for 32 bit machines the restrictions for 64 bit machines being so weak as to be considered inexistent integers must be in absolute value less than 2288435454 ie roughly 80807123 digits The precision of real numb
390. roots of the polynomial generating the field sign bnr bnf nf r r2 the signature of the field This means that the field has r real embeddings 2r9 complex ones t2 bnr bnf nf the T2 matrix see nfinit tu bnr bnf a generator for the torsion units tufu bnr bnf as futu but outputs w u zk bnr bnf nf integral basis i e a Z basis of the maximal order zkst bnr structure of Zx m can be extracted also from an idealstar For instance assume that bnf bnfinit pol for some polynomial Then bnf clgp retrieves the class group and bnf clgp no the class number If we had set bnf nfinit pol both would have output an error message All these functions are completely recursive thus for instance bnr bnf nf zk will yield the maximal order of bnr which you could get directly with a simple bnr zk Some of the functions starting with bnf are implementations of the sub exponential algorithms for finding class and unit groups under GRH due to Hafner McCurley Buchmann and Cohen Diaz Olivier The general call to the functions concerning class groups of general number fields i e excluding quadclassunit involves a polynomial P and a technical vector tech c c2 nrpid where the parameters are to be understood as follows 98 P is the defining polynomial for the number field which must be in Z X irreducible and preferably monic In fact if you supply a non monic polynomial at this point G
391. rrect i e whether it is possible to remove the assumption of the Generalized Riemann Hypothesis If it is correct the answer is 1 If not the program may output some error message but more probably will loop indefinitely In no occasion can the program give a wrong answer barring bugs of course if the program answers 1 the answer is certified The library syntax is certifybuchall bnf and the result is a C long 3 6 2 bnfclassunit P flag 0 tech Buchmann s sub exponential algorithm for com puting the class group the regulator and a system of fundamental units of the general algebraic number field K defined by the irreducible polynomial P with integer coefficients The result of this function is a vector v with many components it is not a bnf you need bnfinit for that which for ease of presentation is in fact output as a one column matrix First we describe the default behaviour flag 0 vfl is equal to the polynomial P Note that for optimum performance P should have gone through polred or nfinit z 2 v 2 is the 2 component vector r1 r2 where rl and r2 are as usual the number of real and half the number of complex embeddings of the number field K v 3 is the 2 component vector containing the field discriminant and the index v 4 is an integral basis in Hermite normal form v 5 v clgp is a 3 component vector containing the class number v clgp no the structure of the class group as a prod
392. s nf nfinit P L nffactor nf Q 1 vector L i rnfequation nf L i to obtain the same result If you are only interested in the degrees of the simple factors the rnfequation instruction can be replaced by a trivial poldegree P poldegree L i If flag 1 outputs a vector of 4 component vectors R a b k where R ranges through the list of all possible compositums as above and a resp b expresses the root of P resp Q as an element of Q X R Finally k is a small integer such that b ka X modulo R A compositum is quite often defined by a complicated polynomial which it is advisable to reduce before further work Here is a simple example involving the field Q 5 51 5 z polcompositum x 5 5 polcyclo 5 1 1 pol z 1 pol defines the compositum 12 x720 5 x719 15 x718 35x x 17 70 x716 141 x 15 260 x714 355 x713 95 x712 1460 x711 3279 x710 3660 x79 2005 x78 705 x77 9210 x 6 13506 x75 7145 x74 2740 x73 1040 x72 124 320 x 256 a z 2 a5 5 a is a fifth root of 5 43 0 z polredabs pol 1 look for a simpler polynomial pol z 1 5 x 20 25 x710 5 a subst a pol x z 2 a in the new coordinates 46 Mod 5 22 x719 1 22 x714 123 22x x 9 9 11 x 4 x 20 25 x710 5 The library syntax is poleompositum0 P Q flag 3 6 106 polgalois x Galois group of the non constant polynomial x Q X In the p
393. s each being given by g h where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nf This routine uses J Kltiners s algorithm in the general case and B Allombert s galoissubfields when nf is Galois with weakly supersolvable Galois group The library syntax is subfields nf d 3 6 101 nfroots nf x roots of the polynomial x in the number field nf given by nfinit without multiplicity in Q if nf is omitted x has coefficients in the number field scalar polmod polynomial column vector The main variable of nf must be of lower priority than that of x see Section 2 6 2 However if the coefficients of the number field occur explicitly as polmods as coefficients of x the variable of these polmods must be the same as the main variable of t see nffactor The library syntax is nfroots nf x 123 3 6 102 nfrootsof1 nf computes the number of roots of unity w and a primitive w th root of unity expressed on the integral basis belonging to the number field nf The result is a two component vector w z where z is a column vector expressing a primitive w th root of unity on the integral basis nf zk The library syntax is rootsof1 nf 3 6 103 nfsnf nf x given a torsion module x as a 3 component row vector A J J where A is a square invertible n x n matrix J and J are two ideal lists outputs an ideal list d d which is the Smith normal form of x
394. s slower than the previous method as soon as p is greater than 100 say No checking is done that p is indeed prime E must be a medium or long vector of the type given by ellinit defined over Q F or Q E must be given by a Weierstrass equation minimal at p The library syntax is ellap0 E p flag Also available are apell E p corresponding to flag 0 and apell2 E p flag 1 90 3 5 5 ellbil E z1 22 if 21 and 22 are points on the elliptic curve E this function computes the value of the canonical bilinear form on z1 22 ellheight E 21 22 ellheight E 21 ellheight F 22 where denotes of course addition on E In addition z1 or 22 but not both can be vectors or matrices Note that this is equal to twice some normalizations E is assumed to be integral given by a minimal model The library syntax is bilhell E z1 22 prec 3 5 6 ellchangecurve v changes the data for the elliptic curve E by changing the coordinates using the vector v u r s t i e if x and y are the new coordinates then z u22 r y uy su2x t The vector E must be a medium or long vector of the type given by ellinit The library syntax is coordch F v 3 5 7 ellchangepoint z v changes the coordinates of the point or vector of points x using the vector v u r s tJ i e if x and y are the new coordinates then x u z r y uy su x t see also ellchangecurve The library syntax is pointch
395. scriminant including 1 returns d If flag is non zero returns the two element row vector d f Note that if n is not congruent to 0 or 1 modulo 4 f will be a half integer and not an integer The library syntax is corediscO n flag Also available are coredisc n coredisc n 0 and coredisc2 n coredisc n 1 3 4 13 dirdiv x y x and y being vectors of perhaps different lengths but with y 1 4 0 considered as Dirichlet series computes the quotient of x by y again as a vector The library syntax is dirdiv z y 78 3 4 14 direuler p a b expr c computes the Dirichlet series to b terms of the Euler product of expression expr as p ranges through the primes from a to b expr must be a polynomial or rational function in another variable than p say X and expr X is understood as the Dirichlet series or more precisely the local factor expr p If c is present output only the first c coefficients in the series The library syntax is direuler entree ep GEN a GEN b char expr 3 4 15 dirmul x y x and y being vectors of perhaps different lengths considered as Dirichlet series computes the product of x by y again as a vector The library syntax is dirmul z y 3 4 16 divisors x creates a row vector whose components are the positive divisors of the integer x in increasing order The factorization of x as output by factor can be used instead The library syntax is divisors z 3 4 17 eulerphi x Eule
396. scriminant of the polynomial x Note that it does not have to be a complete factorization This is especially useful if only a local integral basis for some small set of places is desired only factors with exponents greater or equal to 2 will be considered The library syntax is nfbasisO z flag fa An extended version is nfbasis x amp d flag fa where d will receive the discriminant of the number field not of the polynomial x and an omitted fa should be input as NULL Also available are base x amp d flag 0 base2 x amp d flag 2 and factoredbase z fa amp d 3 6 71 nfbasistoalg nf this is the inverse function of nfalgtobasis Given an object x whose entries are expressed on the integral basis nf zk transforms it into an object whose entries are algebraic numbers i e polmods The library syntax is basistoalg nf x 3 6 72 nfdetint nf x given a pseudo matrix x computes a non zero ideal contained in i e mul tiple of the determinant of x This is particularly useful in conjunction with nfhnfmod The library syntax is nfdetint nf x 117 3 6 73 nfdisc x flag 0 fa field discriminant of the number field defined by the integral preferably monic irreducible polynomial x flag and fa are exactly as in nfbasis That is fa provides the matrix of a partial factorization of the discriminant of x and binary digits of flag are as follows 1 assume that no square of a prime greater than primelimit
397. sed with such arguments and only here you will get an error message if you try these outside of string context e Writing two strings alongside one another will just concatenate them producing a longer string Thus it is equivalent to type in a b or a b A little tricky point in the first expression the first whitespace is enclosed between quotes and so is part of a string while the second before the b is completely optional and GP actually suppresses it as it would with any number of whitespace characters at this point i e outside of any string e If you insert any expression when GP expects a string it gets expanded it is evaluated as a standard GP expression and the final result as would have been printed if you had typed it by itself is then converted to a string as if you had typed it directly For instance a 1 1 b is equivalent to a2b three strings get created the middle one being the expansion of 1 1 and these are then concatenated according to the rule described above Another tricky point here assume you did not assign a value to aaa in a GP expression before Then typing aaa by itself in a string context will actually produce the correct output i e the string whose content is aaa but in a fortuitous way This aaa gets expanded to the monomial of degree one in the variable aaa which is of course printed as aaa and thus will expand to the three letters you were expecting Warning expression invol
398. ses GP opens the gprc file and processes the commands in there before doing anything else e g creating the PARI stack If the file does not exist or cannot be read GP will proceed to the initialization phase at once eventually emitting a prompt If any explicit command line switches are given they override the values read from the preferences file 2 9 1 Where is it When GP is started it looks for a customization file or gprc in the following places in this order only the first one found will be loaded e On the Macintosh only GP looks in the directory which contains the GP executable itself for a file called gprc No other places are examined e If the operating system supports environment variables essentially anything but MacOS GP checks whether the environment variable GPRC is set Under DOS you can set it in AUTOEXEC BAT On Unix this can be done with something like GPRC my dir anyname export GPRC in sh syntax for instance in your profile setenv GPRC my dir anyname in csh syntax in your login or cshrc file If so the file named by GPRC is the gprc e If GPRC is not set and if the environment variable HOME is defined GP then tries HOME gprc on a Unix system HOME _gpre on a DOS OS 2 or Windows system e If HOME also leaves us clueless we try gprc on a Unix system where as usual stands for your home directory or gprc on a DOS OS 2 or Windows system e Finally if no gprc was found amo
399. seudobasis 127 134 tnfsteinitz a A E 134 RO DIGG y ais Eos olas 117 YootMod s ea a ls a na 138 LOO0EMO Z cua ds a 138 FOOtPadIG cy a dF es os 139 FOCUS iia BY eA ion 89 98 138 TOOUCSOLA ci He bes 123 Yootsold Li Ae ee eek Se AS 138 247 rond 2 gt e he BS PR 117 YOUNG 4 0 5 3 kets Ghee Ae ew 117 136 YOUNG 2 4 8 es ae ee eR Re as 66 row vector 7 27 190 RO e tp od seme ds E Soe pte 13 o dAran ele ee BO 179 215 S scalar product 55 scalar type e o ae ae ae be x 8 SchoOnare Lia meir ME sah porte et a SS 138 scientific format 17 BeCure A hon Weer ees eA ee 20 SOL 2 te Ea RA SE SU eet 60 SOECON VOL ai ee ee BA ee es 140 seriesprecision 20 24 Serlaplace 4 63 2 a hae a ays a 140 Se TTreverse x 6 2 Salas oy ele da boas 140 Sete sat a ha ee Bel eae SoS 60 SOCOXPO e sald ce Goel ot 187 189 210 Setintersect 2 poems Fane fos ai 151 SCLISSEt owe ct See Gee ee ee 151 SOULS a ar lA 186 199 209 A o sarani zn Wea tense a 189 190 209 setlgefint 187 209 SeCtMAnus ao e a ta 151 setprecp o ooo ooo o 188 210 Setrand 2 404 be eae ee ee 168 setsearch mercat ae aes 151 setsigne 187 189 210 SCUCYD 64 rp a Ge a Thra 4 186 209 SCtUNLON pedir o Bone gon Reus cd 151 S6tvalpisk doe a 188 189 210 getvari tee a Fee bee PGS 177 189 210 Shanks SQUFOF 76 81 Shanks ssy 405 ze sey bye ates 60 84 85 86
400. sions use GEN flisseq char s In fact these two functions start by filtering out all spaces and comments in the input string that is what the initial stands for They then call the underlying basic functions the GP parser proper GEN lisexpr char s and GEN lisseq char s which are slightly faster but which you probably do not need To read a GEN from a file you can use the simpler interface GEN lisGEN FILE file which reads a character string of arbitrary length from the stream file up to the first newline character applies flisexpr to it and returns the resulting GEN This way you will not have to worry about allocating buffers to hold the string To interactively input an expression use lisGEN stdin This function returns NULL if EOF is encountered before a complete expression could be read Once in a while it may be necessary to evaluate a GP expression sequence involving a call to a function you have defined in C This is easy using install which allows you to manipulate quite an arbitrary function GP knows about pointers The syntax is void install void f char name char code where f is the address of the function cast to the C type void name is the name by which you want to access your function from within your GP expressions and code is a character string describing the function call prototype see Section 4 9 2 for the precise description of prototype strings In case the function returns a GEN
401. ssing Baillie PSW test guarantees its primality currently x lt 1013 2 if x is a large prime whose primality could only sensibly be proven given the algorithms implemented in PARI using the APRCL test Otherwise x is large and x 1 is smooth output a three column matrix as a primality certificate The first column contains the prime factors p of x 1 the second the corresponding elements a as in Proposition 8 3 1 in GTM 138 and the third the output of isprime p 1 The algorithm fails if one of the pseudo prime factors is not prime which is exceedingly unlikely and well worth a bug report If flag 2 use APRCL The library syntax is gisprime z flag but the simpler function isprime which returns a long should be used if x is known to be of type integer 3 4 31 ispseudoprime z flag true 1 if x is a strong pseudo prime see below false 0 otherwise If this function returns false x is not prime if on the other hand it returns true it is only highly likely that x is a prime number Use isprime which is of course much slower to prove that x is indeed prime If flag 0 checks whether x is a Baillie Pomerance Selfridge Wagstaff pseudo prime strong Rabin Miller pseudo prime for base 2 followed by strong Lucas test for the sequence P 1 P smallest positive integer such that P 4 is not a square mod zx There are no known composite numbers passing this test in particular all composites lt 101 are c
402. ssion sequences In the keyword case only a very small set of words will actually be meaningful the default function is a prominent example Here is a useful example used to create generic matrices genmat u v s x matrix u v i j eval Str s i j genmat 2 3 genmat 2 3 m 1 x11 m11 x12 m12 x13 m13 x21 m21 x22 m22 x23 m23 41 Note that the argument of Str is evaluated in string context and really consists of 5 pieces Exercise why are the empty strings necessary This part could also have been written as concat concat Str s i j but not as concat Str s concat i j More simply we could have written concat Str s i j or even concat s i j silently assuming that s will indeed be a string In version 2 2 5 the prototype of Str was extended to allow more than one argument which are concatenated into a single string So finally Str s i j should now be preferred to the above solution since it is less cryptic and more efficient A last example the function hist returns all history entries from a to 4b neatly packed into a single vector hist a b vector b a 1 i eval Str a 1 i Reference The arguments of the following functions are processed in string context Str addhelp second argument default second argument error extern plotstring second argument plotterm first argument read system all the printrxxw functions all the writexrzrx functions
403. structure for instance as output by idealhnf nf use rnfidealabstorel rnf nf zk x to convert it to a relative ideal The library syntax is rnfidealabstorel rnf x 3 6 126 rnfidealdown rnf x let rnf be a relative number field extension L K as output by rnfinit and x an ideal of L given either in relative form or by a Z basis of elements of L see Section 3 6 125 returns the ideal of K below zx i e the intersection of x with K The library syntax is rnfidealdown rnf x 129 3 6 127 rnfidealhnf rnf x rnf being a relative number field extension L K as output by rn finit and zx being a relative ideal which can be as in the absolute case of many different types including of course elements computes the HNF pseudo matrix associated to x viewed as a Zg module The library syntax is rnfidealhermite rnf x 3 6 128 rnfidealmul rnf z y rnf being a relative number field extension L K as output by rnfinit and z and y being ideals of the relative extension L K given by pseudo matrices outputs the ideal product again as a relative ideal The library syntax is rnfidealmul rnf x y 3 6 129 rnfidealnormabs rnf x rnf being a relative number field extension L K as output by rnfinit and z being a relative ideal which can be as in the absolute case of many different types including of course elements computes the norm of the ideal x considered as an ideal of the absolute extension L Q This is identical to ide
404. syntax is denom z 3 2 29 floor x floor of x When z is in R the result is the largest integer smaller than or equal to x Applied to a rational function floor returns the euclidian quotient of the numerator by the denominator The library syntax is gfloor x 3 2 30 frac x fractional part of x Identical to x floor x If x is real the result is in 0 1 The library syntax is gfrac z 3 2 31 imag x imaginary part of x When x is a quadratic number this is the coefficient of w in the canonical integral basis 1 w The library syntax is gimag x This returns a copy of the imaginary part The internal routine imag_i is faster since it returns the pointer and skips the copy 3 2 32 length x number of non code words in x really used i e the effective length minus 2 for integers and polynomials In particular the degree of a polynomial is equal to its length minus 1 If x has type t_STR output number of letters The library syntax is glength x and the result is a C long 64 3 2 33 lift x v lifts an element z a mod n of Z nZ to a in Z and similarly lifts a polmod to a polynomial if v is omitted Otherwise lifts only polmods whose modulus has main variable v if v does not occur in zx lifts only intmods If x is of recursive non modular type the lift is done coefficientwise For p adics this routine acts as truncate It is not allowed to have x of type t_REAL lift Mod 5 3 1 2
405. t oc ade Ra Se eB gow ea OR a G 231 DEINTLAC fos bce cel Ree da 70 Bernoulli numbers 70 71 76 bernreal licita os hE oe 70 71 DEYNVEC mimado e eee Da e ed 71 besselht 71 besselh2 ii ad ll besse lingus ts eas Sie RA eS 71 bessely yest ace vies Goby Hee gate tos 71 besseljh o o 71 b ss lk ries sor a ees 71 besseln oaa OR Ba a Ba es 71 Destapp s en surani aai wee 76 77 bestapprO co ooo ooo oo o 77 BEZOUY ut he Gh Pe OAD eit res 77 bezoutres 0 0000 0 77 Biff ia a OE E Sek Ew A 207 BIGDEFAULTPREC 174 BEGINT pon A Ba 189 190 bigomega o o 77 bibhedl ok oe oe A E 91 binaire eso se ha SE OG ae i 61 binary files 20 64 Dae d arid ee aoe a a 170 binary file ans a ota 24 168 binary flag oa ee ea eka ca 53 binary quadratic form 7 27 60 Danary in ee eth Ae Me kG 61 binme scs bose Ss PM AE eS 77 binomial coefficient 77 Binomial A seren OS bee ae e ok ae Ges 77 Birch and Swinnerton Dyer conjecture 91 Ditanda Paste Yee S 57 61 DLENCS 2 seh tr a ht ie Set GER eek Ss 61 bitnegimply ocaso 62 o 6 6 Selks a Mee eS eon J 57 62 BITS IN SLONG z bsan g bale tee oe bts 174 bittest oda SER Pane eA ee es 62 bitwise and 57 61 bitwise exclusive or 62 bitwise inclusive or 62 bitwise negation 0 61 itwises OR Lem ae a Goh ee we OS 57 DITXOL ic
406. t for matrices The relevant functions are void matbrute GEN obj char x long n void outmat GEN obj which is followed by a newline and a buffer flush e prettyprint format the basic function has an additional parameter m corresponding to the minimum field width used for printing integers void sor GEN obj char x long n long m The simplified version is void outbeaut GEN obj which is equivalent to sor obj g 1 0 followed by a newline and a buffer flush e The first extra format corresponds to the texprint function of GP and gives a TFX output of the result It is obtained by using void exe GEN obj char x long n e The second one is the function GENtostr which converts a PARI GEN to an ASCII string The syntax is charx GENtostr GEN obj wich returns a malloc ed character string which you should free after use e The third and final one outputs the hexadecimal tree corresponding to the GP command x using the function void voir GEN obj long nb which will only output the first nb words corresponding to leaves very handy when you have a look at big recursive structures If you set this parameter to 1 all significant words will be printed Usually this last type of output would only be used for debugging purposes 194 Remark Apart from GENtostr all PARI output is done on the stream outfile which by default is initialized to stdout If you want that your output be directed to another file
407. t of a GP expression sequence Upon exit the stack decreases You can kill a variable decreasing the stack yourself However the stack has a bottom the value of a variable is the monomial of degree 1 in this variable as is natural for a mathematician 2 6 2 Variable priorities PARI has no intelligent sparse representation of polynomials So a multivariate polynomial in PARI is just a polynomial in one variable whose coefficients are themselves polynomials arbitrary but for the fact that they do not involve the main variable All computations are then just done formally on the coefficients as if the polynomial was univariate This is not symmetrical So if I enter x y in a clean session what happens This is understood as a y x Z y x but how can GP decide that x is more important than y Why not y x x y which is the same mathematical entity after all The answer is that variables are ordered implicitly by the GP interpreter when a new identifier e g x or y as above is input the corresponding variable is registered as having a strictly lower priority than any variable in use at this point To see the ordering used by GP at any given time type reorder Given such an ordering multivariate polynomials are stored so that the variable with the highest priority is the main variable And so on recursively until all variables are exhausted A different storage pattern which could only be obtai
408. t of a lexicographic comparison between x and y as 1 0 or 1 This is to be interpreted in quite a wide sense It is admissible to compare objects of different types scalars vectors matrices provided the scalars can be compared as well as vectors matrices of different lengths The comparison is recursive In case all components are equal up to the smallest length of the operands the more complex is considered to be larger More precisely the longest is the largest when lengths are equal we have matrix gt vector gt scalar For example lex 1 3 1 2 5 Vil lex 1 3 1 3 1 42 1 lex 1 1 13 1 lex 1 1 714 0 The library syntax is lexcmp z y 3 1 14 sign x sign 0 1 or 1 of x which must be of type integer real or fraction The library syntax is gsigne x The result is a long 3 1 15 max z y and min z y creates the maximum and minimum of x and y when they can be compared The library syntax is gmax x y and gmin z y 3 1 16 vecmax x if x is a vector or a matrix returns the maximum of the elements of z otherwise returns a copy of x Returns oo in the form of 2 1 or 2 1 for 64 bit machines if x is empty The library syntax is vecmax z 3 1 17 vecmin x if x is a vector or a matrix returns the minimum of the elements of x otherwise returns a copy of x Returns 00 in the form of 2 1 or 263 1 for 64 bit machines if x is
409. t results when the class group has a low exponent has many cyclic factors because implementing Shanks s method in full generality slows it down immensely It is therefore strongly recommended to double check results using either the version with flag 1 the function qfbhclassno D or the function quadclassunit Warning contrary to what its name implies this routine does not compute the number of classes of binary primitive forms of discriminant D which is equal to the narrow class number The two notions are the same when D lt 0 or the fundamental unit e has negative norm when D gt 0 and Ne gt 0 the number of classes of forms is twice the ordinary class number This is a problem which we cannot fix for backward compatibility reasons Use the following routine if you are only interested in the number of classes of forms QFBclassno D qfbclassno D if D lt O norm quadunit D lt 0 1 2 Here are a few examples qfbclassno 400000028 time 3 140 ms hi 1 quadclassunit 400000028 no time 20 ms much faster 12 1 gqfbclassno 400000028 time O ms 13 7253 correct and fast enough quadclassunit 400000028 no time O ms 14 7253 The library syntax is gfbclassno0 D flag Also available classno D qfbclassno D classno2 D qfbclassno D 1 and finally there exists the function hclassno D which com putes the class number of an imaginary quadratic field by count
410. t the y axis e 32 no_Frame do not print frame e 64 no Lines only plot reference points do not join them e 128 Points oo plot both lines and points e 256 Splines use splines to interpolate the points e 512 no_X_ticks plot no x ticks e 1024 no Y ticks plot no y ticks e 2048 Same_ticks plot all ticks with the same length 3 10 10 plothraw listz listy flag 0 given listr and listy two vectors of equal length plots in high precision the points whose x y coordinates are given in liste and listy Automatic positioning and scaling is done but with the same scaling factor on x and y If flag is 1 join points other non 0 flags toggle display options and should be combinations of bits 2 k gt 3 as in ploth 3 10 11 plothsizes return data corresponding to the output window in the form of a 6 component vector window width and height sizes for ticks in horizontal and vertical directions this is intended for the gnuplot interface and is currently not significant width and height of characters 159 3 10 12 plotinit w x y flag initialize the rectwindow w destroying any rect objects you may have already drawn in w The virtual cursor is set to 0 0 The rectwindow size is set to width x and height y If flag 0 x and y represent pixel units Otherwise x and y are understood as fractions of the size of the current output device hence must be between 0 and 1 and internally converted to pi
411. t to do with the remaining arguments and what message to print The list of valid keywords is in language errmessages c together with the basic corresponding message For instance err typeer matexp will print the message xxx incorrect type in matexp Among the codewords are warning keywords all those which start with the prefix warn In that case err does not abort the computation just print the requested message and go on The basic example is err warner Strategy 1 failed Trying strategy 2 which is the exact equivalent of err talker except that you certainly do not want to stop the program at this point just inform the user that something important has occurred in particular this output would be suitably highlighted under GP whereas a simple printf would not 195 4 7 4 Debugging output The global variables DEBUGLEVEL and DEBUGMEM corresponding to the default debug and debugmem see Section 2 1 are used throughout the PARI code to govern the amount of diagnostic and debugging output depending on their values You can use them to debug your own functions especially after having made them accessible under GP through the command install see Section 3 11 2 13 For debugging output you can use printf and the standard output functions brute or output mainly but also some special purpose functions which embody both concepts the main one being void fprintferr char pariformat Now let us define what
412. tants Index oO Dm a al 13 14 21 25 28 31 32 42 46 47 49 50 53 54 58 68 76 89 96 135 141 153 156 162 171 171 172 175 179 186 190 192 197 201 207 207 208 215 221 229 231 233 Chapter 1 Overview of the PARI system 1 1 Introduction The PARI system is a package which is capable of doing formal computations on recursive types at high speed it is primarily aimed at number theorists but can be used by anybody whose primary need is speed Although quite an amount of symbolic manipulation is possible in PARI this system does very badly compared to much more sophisticated systems like Axiom Macsyma Maple Mathematica or Reduce on such manipulations e g multivariate polynomials formal integration etc On the other hand the three main advantages of the system are its speed which can be between 5 and 100 times better on many computations the possibility of using directly data types which are familiar to mathematicians and its extensive algebraic number theory module which has no equivalent in the above mentioned systems It is possible to use PARI in three different ways 1 as a library which can be called from an upper level language application for instance written in C or C 2 as a sophisticated programmable calculator named GP which contains most of the control instructions of a standard language like C 3 the compiler GP2C
413. th many components and one wants to get the exponential of each one This could easily be done either under GP or in library mode but in fact PARI assumes that this is exactly what you want to do when you take the exponential of a vector so no work is necessary Most transcendental functions work in the same way see Chapter 3 for details An ambiguity would arise with square matrices PARI always considers that you want to do componentwise function evaluation hence to get for example the exponential of a square matrix you would need to use a function with a different name matexp for instance In the present version 2 2 7 this is not yet implemented See however the program in Appendix C which is a first attempt for this particular function The available operations and functions in PARI are described in detail in Chapter 3 Here is a brief summary 1 3 2 Standard operations Of course the four standard operators exist It should once more be emphasized that division is as far as possible an exact operation 4 divided by 3 gives 4 3 In addition to this operations on integers or polynomials like Euclidean division Euclidean remainder exist and for integers computes the quotient such that the remainder has smallest possible absolute value There is also the exponentiation operator when the exponent is of type integer Otherwise it is considered as a transcendental function Finally the logical operators
414. the number field occur explicitly as polmods as coefficients of x the variable of these polmods must be the same as the main variable of t see nffactor The library syntax is nffactormod nf x pr 119 3 6 88 nfgaloisapply nf aut x nf being a number field as output by nfinit and aut being a Galois automorphism of nf expressed either as a polynomial or a polmod such automorphisms being found using for example one of the variants of nfgaloisconj computes the action of the automorphism aut on the object x in the number field x can be an element scalar polmod polynomial or column vector of the number field an ideal either given by Zx generators or by a Z basis a prime ideal given as a 5 element row vector or an idele given as a 2 element row vector Because of possible confusion with elements and ideals other vector or matrix arguments are forbidden The library syntax is galoisapply nf aut x 3 6 89 nfgaloisconj nf flag 0 d nf being a number field as output by nfinit computes the conjugates of a root r of the non constant polynomial x nf 1 expressed as polynomials in r This can be used even if the number field nf is not Galois since some conjugates may lie in the field nf can simply be a polynomial if flag 1 If no flags or flag 0 if nf is a number field use a combination of flag 4 and 1 and the result is always complete else use a combination of flag 4 and 2 and the result is subject to the r
415. the prime number p By abuse of notation p 1 is a valid special case which returns the unit form Returns an error if x is not a quadratic residue mod p In the case where x gt 0 the distance component of the form is set equal to zero according to the current precision The library syntax is primeform z p prec where the third variable prec is a long but is only taken into account when x gt 0 3 4 51 qfbred z flag 0 D isqrtD sqrtD reduces the binary quadratic form x up dating Shanks s distance function if x is indefinite The binary digits of flag are toggles meaning 1 perform a single reduction step 2 don t update Shanks s distance D isqrtD sqrtD if present supply the values of the discriminant VD and VD respectively no checking is done of these facts If D lt 0 these values are useless and all references to Shanks s distance are irrelevant The library syntax is qfbredO z flag D isqrtD sqrtD Use NULL to omit any of D isqrtD sqrtD Also available are redimag zx qfbred x where xv is definite and for indefinite forms redreal xz qfbred z rhoreal x qfbred z 1 redrealnod z sq qfbred z 2 isqrtD rhorealnod z sq qfbred z 3 isqrtD 86 3 4 52 qfbsolve Q p Solve the equation Q x y p over the integers where Q is a imaginary binary quadratic form and p a prime number Return z y as a two components vector or zero if
416. the stack Should be equal to 0 in between commands Useful mainly for debugging purposes The library syntax is getstack returns a C long 3 11 2 10 gettime returns the time in milliseconds elapsed since either the last call to get time or to the beginning of the containing GP instruction if inside GP whichever came last The library syntax is gettime returns a C long 165 UNIX 3 11 2 11 global list of variables declares the corresponding variables to be global From now on you will be forbidden to use them as formal parameters for function definitions or as loop indexes This is especially useful when patching together various scripts possibly written with different naming conventions For instance the following situation is dangerous p 3 fiz characteristic forprime p 2 N f p since within the loop or within the function s body even worse in the subroutines called in that scope the true global value of p will be hidden If the statement global p 3 appears at the beginning of the script then both expressions will trigger syntax errors Calling global without arguments prints the list of global variables in use In particular eval global will output the values of all global variables 3 11 2 12 input reads a string interpreted as a GP expression from the input file usually standard input i e the keyboard If a sequence of expressions is given the result is the result of the
417. the window and its current value can be obtained using the function plotcursor A number of primitive graphic objects called rect objects can then be drawn in these windows using a default color associated to that window which can be changed under X11 using the plotcolor function black otherwise and only the part of the object which is inside the window will be drawn with the exception of polygons and strings which are drawn entirely but the virtual cursor can move outside of the window The ones sharing the prefix plotr draw relatively to the current position of the virtual cursor the others use absolute coordinates Those having the prefix plotrecth put in the rectwindow a large batch of rect objects corresponding to the output of the related ploth function Finally the actual physical drawing is done using the function plotdraw Note that the windows are preserved so that further drawings using the same windows at different positions or different windows can be done without extra work If you want to erase a window and free the corresponding memory use the function plotkill It is not possible to partially erase a window Erase it completely initialize it again and then fill it with the graphic objects that you want to keep 156 In addition to initializing the window you may want to have a scaled window to avoid un necessary conversions For this use the function plotscale below As long as this function is not called the sca
418. thms using scripts before actually coding them in C for the sake of speed Note that the install command enables you to concentrate on critical parts of your programs only which can of course be written with the help of other mathematical libraries than PARI and to easily and efficiently import foreign functions for use under GP see Section 3 11 2 13 We are aware of three PARI related public domain libraries We neither endorse nor support any of them You might want to give them a try if you are familiar with the languages they are based on First there are PariPer1 written by Ilya Zakharevich ilya math ohio state edu and PariPython by St fane Fermigier fermigie math jussieu fr Finaly Michael Stoll see http nswt tuwien ac at 8000 htdocs internet unix perl math pari html see http www math jussieu fr fermigie PariPython readme html 46 Michael_Stoll math uni bonn de has integrated PARI into CLISP which is a Common Lisp implementation by Bruno Haible Marcus Daniels and others These provide interfaces to GP functions for use in perl python or Lisp programs To our knowledge only the python and perl interfaces have been upgraded to version 2 0 of PARI the CLISP one being still based on version 1 39 xx 2 9 The preferences file This file called gprc in the sequel is used to modify or extend GP default behaviour in all GP sessions e g customize default values or load common user functions and alia
419. tion e provided no other trap as above intercepts it first The error message is printed as well as the result of the evaluation of rec and control is given back to the GP prompt In particular current computation is then lost The following error handler prints the list of all user variables then stores in a file their name and their values trap print reorder writebin crash If no recovery code is given rec is omitted a break loop will be started see Section 2 7 3 In particular trap by itself installs a default error handler that will start a break loop whenever an exception is raised If rec is the empty string the default handler for that error if e is present is disabled 168 Note The interface is currently not adequate for trapping individual exceptions In the current version 2 2 7 the following keywords are recognized but the name list will be expanded and changed in the future all library mode errors can be trapped it s a matter of defining the keywords to GP and there are currently far too many useless ones accurer accuracy problem gdiver division by 0 invmoder impossible inverse modulo archer not available on this architecture or operating system siginter SIGINT received usually from Control C talker miscellaneous error typeer wrong type errpile the PARI stack overflows 3 11 2 26 type z t this is useful only under GP If t is not present returns the inter
420. tional function in general To obtain the approximate real value of the quotient of two integers add 0 to the result to obtain the approximate p adic value of the quotient of two integers add O p k to the result finally to obtain the Taylor series expansion of the quotient of two polynomials add 0 X k to the result or use the taylor function see Section 3 7 32 The library syntax is gdiv x y for x y 3 1 5 The expression x y is the Euclidean quotient of x and y If y is a real scalar this is defined as floor x y if y gt 0 and ceil x y if y lt 0 and the division is not exact Hence the remainder x x y y is in 0 y Note that when y is an integer and x a polynomial y is first promoted to a polynomial of degree 0 When x is a vector or matrix the operator is applied componentwise The library syntax is gdivent x y for x y 3 1 6 The expression x y evaluates to the rounded Euclidean quotient of x and y This is the same as x y except for scalar division the quotient is such that the corresponding remainder is smallest in absolute value and in case of a tie the quotient closest to 00 is chosen hence the remainder would belong to y 2 y 2 When z is a vector or matrix the operator is applied componentwise The library syntax is gdivround z y for x y 55 3 1 7 The expression x y evaluates to the modular Euclidean remainder of x and y which we now define If y is an i
421. tions modulo units of positive norm of the absolute norm equation Norm a x where a is an integer in bnf If bnf has not been certified the correctness of the result depends on the validity of GRH See also bnfisnorm The library syntax is bnfisintnorm bnf x 3 6 7 bnfisnorm bnf x flag 1 tries to tell whether the rational number x is the norm of some element y in bnf Returns a vector a b where x Norm a xb Looks for a solution which is an S unit with S a certain set of prime ideals containing among others all primes dividing z If bnf is known to be Galois set flag 0 in this case x is a norm iff b 1 If flag is non zero the program adds to S the following prime ideals depending on the sign of flag If flag gt 0 the ideals of norm less than flag And if flag lt 0 the ideals dividing flag Assuming GRH the answer is guaranteed i e x is a norm iff b 1 if S contains all primes less than 12 log disc Bnf where Bnf is the Galois closure of bnf See also bnfisintnorm The library syntax is bnfisnorm bnf x flag prec where flag and prec are longs 3 6 8 bnfissunit bnf sfu x bnf being output by bnfinit sfu by bnfsunit gives the column vector of exponents of x on the fundamental S units and the roots of unity If x is not a unit outputs an empty vector The library syntax is bnfissunit bnf sfu x 3 6 9 bnfisprincipal bnf x flag 1 bnf being the number field data output by bnfinit a
422. tive one for small operands and much faster for huge ones say about twice faster around 10000 decimal digits A third version is written entirely in ANSI C with a C compatible syntax and should be portable without much trouble to any 32 or 64 bit computer having no real memory constraints It is about 2 times slower than versions including an assembly kernel This version has been tested for example on MIPS based DECstations 3100 and 5000 and SGI computers on various Macs based on 680x0 chips or more recent PowerPCs like the G3 or G4 new Intel chips like ARM handheld devices or ia64 Itanium In addition PARI has been ported to a considerable number of smaller and larger machines for example the VAX 68000 based machines like the Atari Mac Classic or Amiga 500 68020 machines such as the Amiga 2500 or 3000 or mainframes like the IBM S390 An historical version of the PARI GP kernel written in 1985 was specific to 680x0 based computers and was entirely writen in MC68020 assembly language around 6000 lines It ran on SUN 3 xx Sony News NeXT cubes and on 680x0 based Macs Since 1997 this version was unmaintained it has been removed from the PARI distribution in version 2 2 5 To run PARI with a 68k assembler micro kernel one now uses the GMP version 1 2 The PARI types The crucial word in PARI is recursiveness most of the types it knows about are recursive For example the basic type Complex exists actually called t_COM
423. tly three special variable names which you are not allowed to use Pi and Euler which represent well known constants and T 1 Note that GP names are case sensitive since version 1 900 This means for instance that the symbol i is perfectly safe to use and will not be mistaken for 1 and that o is not synonymous anymore to 0 If you grew addicted to the previous behaviour you can have it back by setting the default compatible to 3 Now the main thing to understand is that PARI GP is not a symbolic manipulation package although it shares some of the functionalities One of the main consequences of this fact is that all expressions are evaluated as soon as they are written they never stay in a purely abstract form As an important example consider what happens when you use a variable name before assigning a value into it This is perfectly acceptable to GP which considers this variable in fact as a polynomial of degree 1 with coefficients 1 in degree 1 0 in degree 0 whose variable is the variable name you used If later you assign a value to that variable the objects which you have created before will still be considered as polynomials If you want to obtain their value use the function eval see Section 3 7 3 Finally note that if the variable x contains a vector or list you can assign a result to x m i e write something like z k expr If x is a matrix you can assign a result to x m n but not to x m If you want
424. to 1 the modulus of the created result is put on the heap and not on the stack and hence becomes a permanent copy which cannot be erased later by garbage collecting see Section 4 4 Functions will operate faster on such objects and memory consumption will be lower On the other hand care should be taken to avoid creating too many such objects Under GP the same effect can be obtained by assigning the object to a GP variable the value of which is a permanent object for the duration of the relevant library function call and is treated as such This value is subject to garbage collection since it will be deleted when the value changes This is preferable and the above flag is only retained for compatibility reasons it can still be useful in library mode The library syntax is ModO z y flag Also available are e for flag 1 gmodulo z y e for flag 0 gmodulcp z y 3 2 4 Pol z v x transforms the object x into a polynomial with main variable v If x is a scalar this gives a constant polynomial If x is a power series the effect is identical to truncate see there i e it chops off the O X If x is a vector this function creates the polynomial whose coefficients are given in x with x 1 being the leading coefficient which can be zero 59 Warning this is not a substitution function It is intended to be quick and dirty So if you try Pol a y on the polynomial a x y you will get y y which is not a valid PARI ob
425. tor having two components the first one is the vector of eigenvalues of x the second is the corresponding orthogonal matrix of eigenvectors of x The method used is Jacobi s method for symmetric matrices The library syntax is jacobi z 3 8 44 qflll x flag 0 LLL algorithm applied to the columns of the not necessarily square matrix x The columns of x must however be linearly independent unless specified otherwise below The result is a transformation matrix T such that x T is an LLL reduced basis of the lattice generated by the column vectors of x If flag O default the computations are done with real numbers i e not with rational numbers using Householder matrices for orthogonalization as presently programmed slow but stable If flag 1 it is assumed that the corresponding Gram matrix is integral The computation is done entirely with integers and the algorithm is both accurate and quite fast In this case x needs not be of maximal rank but if it is not T will not be square If flag 2 similar to case 1 except x should be an integer matrix whose columns are linearly independent The lattice generated by the columns of x is first partially reduced before applying the LLL algorithm A basis is said to be partially reduced if v v gt v for any two distinct basis vectors v v This can be significantly faster than flag 1 when one row is huge compared to the other rows If flag 4 x is assu
426. tries correspond to the component numbers to be extracted in the order specified If y is a string it can be e a single non zero index giving a component number a negative index means we start counting from the end 151 e a range of the form a b where a and b are indexes as above Any of a and b can be omitted in this case we take as default values a 1 and b 1 i e the first and last components respectively We then extract all components in the interval a b in reverse order if b lt a In addition if the first character in the string is the complement of the given set of indices is taken If z is not omitted x must be a matrix y is then the line specifier and z the column specifier where the component specifier is as explained above v la b c d el vecextract v 5 mask 1 la c vecextract v 4 2 11 MN component list 72 d b al vecextract v 2 4 interval 3 b c d vecextract v 1 3 interval reverse order 4 e d c vecextract v 2 complement 5 a c d el vecextract matid 3 2 76 o 1 0 0 0 1 The library syntax is extract z y or matextract z y z 3 8 57 vecsort z k flag 0 sorts the vector x in ascending order using the heapsort method x must be a vector and its components integers reals or fractions If k is present and is an integer sorts according to the value of the k th su
427. trixqz x p xz being an m x n matrix with m gt n with rational or integer entries this function has varying behaviour depending on the sign of p If p gt 0 x is assumed to be of maximal rank This function returns a matrix having only integral entries having the same image as x such that the GCD of all its n x n subdeterminants is equal to 1 when p is equal to 0 or not divisible by p otherwise Here p must be a prime number when it is non zero However if the function is used when p has no small prime factors it will either work or give the message impossible inverse modulo and a non trivial divisor of p If p 1 this function returns a matrix whose columns form a basis of the lattice equal to Z intersected with the lattice generated by the columns of x If p 2 returns a matrix whose columns form a basis of the lattice equal to Z intersected with the Q vector space generated by the columns of z The library syntax is matrixqz0 z p 3 8 36 matsize x x being a vector or matrix returns a row vector with two components the first being the number of rows 1 for a row vector the second the number of columns 1 for a column vector The library syntax is matsize z 147 3 8 37 matsnf X flag 0 if X is a singular or non singular matrix outputs the vector of elementary divisors of X i e the diagonal of the Smith normal form of X The binary digits of flag mean 1 complete output if set
428. ts without ambiguity It s not a trivial problem to adapt automatically this regular expression to an arbitrary prompt which can be self modifying Thus in this version 2 2 7 Emacs relies on the prompt being the default one So do not tamper with the prompt variable unless you modify it simultaneously in your emacs file see emacs pariemacs txt and misc gprc dft for examples 2 1 22 prompt cont default a string that will be printed to prompt for continuation lines e g in between braces or after a line terminating backslash Everything that applies to prompt applies to prompt_cont as well 2 1 23 psfile default pari ps name of the default file where GP is to dump its PostScript drawings these will always be appended so that no previous data are lost Environment and time expansion are performed 2 1 24 readline default 1 switches readline line editing facilities on and off This may be useful if you are running GP in a Sun cmdtool which interacts badly with readline Of course until readline is switched on again advanced editing features like automatic completion and editing history are not available 2 1 25 realprecision default 28 and 38 on 32 bit and 64 bit machines respectively the number of significant digits and at the same time the number of printed digits of real numbers see p Note that PARI internal precision works on a word basis 32 or 64 bits hence may not coincide with the number of decimal dig
429. u want to preserve It cleans up the most recent part of the stack between 1top and avma updating all the GENs added to the argument list A copy is done just before the cleaning to preserve them so they do not need to be connected before the call With gerepilecopy this is the most robust of the gerepile functions the less prone to user error but also the slowest An alternative syntax obsolete but kept for backward compatibility is given by void gerepilemany pari_sp ltop GEN gptr int n which works exactly as above except that the preserved GENs are the elements of the array gptr of length n gptrEo gptr 11 gptr n 1 e More efficient but tricky to use is void gerepilemanysp pari_sp ltop pari_sp lbot GEN gptr int n which cleans the stack between 1bot and ltop and updates the GENs pointed at by the elements of gptr without doing any copying This is subject to the same restrictions as gerepile the only difference being that more than one address gets updated 4 4 2 Examples 4 4 2 1 gerepile Let x and y be two preexisting PARI objects and suppose that we want to compute x y This can trivially be done using the following program we skip the necessary declarations everything in sight is a GEN pl gsqr x p2 gsqr y z gadd p1 p2 The GEN z indeed points at the desired quantity However consider the stack it contains as unnecessary garbage p1 and p2 More precisely it contains in this orde
430. uct of cyclic groups of order n v clgp cyc and the corresponding generators of the class group of respective orders n v clgp gen 99 v 6 v reg is the regulator computed to an accuracy which is the maximum of an internally determined accuracy and of the default v 7 is deprecated maintained for backward compatibility and always equal to 1 v 8 v tu a vector with 2 components the first being the number w of roots of unity in K and the second a primitive w th root of unity expressed as a polynomial v 9 v fu is a system of fundamental units also expressed as polynomials If flag 1 and the precision happens to be insufficient for obtaining the fundamental units exactly the internal precision is doubled and the computation redone until the exact results are obtained The user should be warned that this can take a very long time when the coefficients of the fundamental units on the integral basis are very large for example in the case of large real quadratic fields In that case there are alternate methods for representing algebraic numbers which are not implemented in PARI If flag 2 the fundamental units and roots of unity are not computed Hence the result has only 7 components the first seven ones tech is a technical vector empty by default containing c c2 nrel borne nbpid minsfb in this order see the beginning of the section or the keyword bnf You can supply any number of these provided you giv
431. ultiplicative group Z nZ as a 3 component row vector v where v 1 n is the order of that group v 2 is a k component row vector d of integers di such that dfi gt 1 and dli d i 1 for i gt 2 and Z nZ TE Z d i Z and v 3 is a k component row vector giving generators of the image of the cyclic groups Z d i Z The library syntax is znstar n 3 5 Functions related to elliptic curves We have implemented a number of functions which are useful for number theorists working on elliptic curves We always use Tate s notations The functions assume that the curve is given by a general Weierstrass model y 017Yy agy x agr 0442 06 where a priori the a can be of any scalar type This curve can be considered as a five component vector E a1 a2 a3 a4 a6 Points on E are represented as two component vectors x y except for the point at infinity i e the identity element of the group law represented by the one component vector 0 It is useful to have at one s disposal more information This is given by the function ellinit see there which usually gives a 19 component vector which we will call a long vector in this section If a specific flag is added a vector with only 13 component will be output which we will call a medium vector A medium vector just gives the first 13 components of the long vector corresponding to the same curve but is of course faster to compute The following member functions a
432. umber of the form Major Minor patch where the last two components can be omitted i e 1 is understood as versio 1 0 0 This is true if GP s version number satisfies the required inequality 2 9 2 2 Commands After the preprocessing the remaining lines are executed as sequence of expressions as usual separated by if necessary Only two kinds of expressions are recognized e default value where default is one of the available defaults see Section 2 1 which will be set to value on actual startup Don t forget the quotes around strings e g for prompt or help e read some_GP_file where some_GP_file is a regular GP script this time which will be read just before GP prompts you for commands but after initializing the defaults In particular file input is delayed until the gprc has been fully loaded This is the right place to input files containing alias commands or your favorite macros For instance you could set your prompt in the following portable way self modifying prompt looking like 18 03 gp gt prompt H M eL imgp elm gt readline wants non printing characters to be braced between A B pairs if READL prompt H M A e 1m Bgp A e m7B gt escape sequences not supported under emacs if EMACS prompt H M gp gt Note that any of the last two lines could be broken in the following way if EMACS prompt H M gp gt since the preprocessor directive applies to the
433. umber of x to s This should be used with extreme care since usually the type is set otherwise and the components and further codeword fields which are left unchanged may not match the PARI conventions for the new type void setlg GEN x long s sets the length of x to s Again this should be used with extreme care since usually the length is set otherwise and increasing the length joins previously unrelated 209 memory words to the root node of x This is however an extremely efficient way of truncating vectors or polynomials void setlgef GEN x long s sets the effective length of x to s where x is a polynomial The number s must be less than or equal to the length of x void setlgefint GEN x long s sets the effective length of the integer x to s The number s must be less than or equal to the length of x void setsigne GEN x long s sets the sign of x to s If x is an integer or real s must be equal to 1 0 or 1 and if x is a polynomial or a power series s must be equal to 0 or 1 void setexpo GEN x long s sets the binary exponent of the real number x to s after adding the appropriate bias The unbiased value s must be a 24 bit signed number void setvalp GEN x long s sets the p adic or X adic valuation of x to s if x is a p adic or a power series respectively void setprecp GEN x long s sets the p adic precision of the p adic number x to s void setvarn GEN x long s sets the variable number of the polynomia
434. und anymore 2 1 2 colors default this default is only usable if GP is running within certain color capable terminals For instance rxvt color_xterm and modern versions of xterm under X Windows or standard Linux DOS text consoles It causes GP to use a small palette of colors for its output With xterms the colormap used corresponds to the resources Xterm colorn where n ranges from 0 to 15 see the file misc color dft for an example Legal values for this default are strings di 0 where k lt 7 and each a is either e the keyword no use the default color usually black on transparent background e an integer between 0 and 15 corresponding to the aforementioned colormap e a triple cy c c2 where co stands for foreground color c for background color and ca for attributes 0 is default 1 is bold 4 is underline The output objects thus affected are respectively error messages history numbers prompt input line output help messages timer that s seven of them If k lt 7 the remaining a are assumed to be no For instance default colors 9 5 no no 4 typesets error messages in color 9 history numbers in color 5 output in color 4 and does not affect the rest A set of default colors for dark reverse video or PC console and light backgrounds respectively is activated when colors is set to darkbg resp lightbg or any proper prefix d is recognized as an abbreviation for darkbg A bold variant of darkbg
435. user defined functions 2 2 23 um prints the definitions of all user defined member functions 2 2 24 v prints the version number and implementation architecture 680x0 Sparc Alpha other of the GP executable you are using In library mode you can use instead the two character strings PARIVERSION and PARIINFO which correspond to the first two lines printed by GP just before the Copyright message 2 2 25 w n filename writes the object number n n into the named file in raw format If the number n is omitted writes the latest computed object If filename is omitted appends to logfile the GP function write is a trifle more powerful as you can have arbitrary filenames 24 2 2 26 x prints the complete tree with addresses and contents in hexadecimal of the internal representation of the latest computed object in GP As for s this is used primarily as a debugging device for PARI and the format should be self explanatory a before an object typically a modulus means the corresponding component is out of stack However used on a PARI integer it can be used as a decimal hexadecimal converter 2 2 27 My n switches simplify on 1 or off 0 If n is explicitly given set simplify to n 2 2 28 switches the timer on or off 2 2 29 prints the time taken by the latest computation Useful when you forgot to turn on the timer 2 3 Input formats for the PARI types Before describing more sophist
436. utines or most of them return long integers then you do not need to do any garbage collecting This will probably be the case in many of your subroutines Of course the objects that were on the stack before the function call are left alone Except for the ones listed below PARI functions only collect their own garbage e It may happen that all objects that were created after a certain point can be deleted for instance if the final result you need is not a GEN or if some search proved futile Then it is enough to record the value of avma just before the first garbage is created and restore it upon exit pari_sp av avma record initial avma garbage avma av restore it All objects created in the garbage zone will eventually be overwritten they should not be accessed anymore once avma has been restored e If you want to destroy i e give back the memory occupied by the latest PARI object on the stack e g the latest one obtained from a function call you can use the function void cgiv GEN z where z is the object you want to give back e Unfortunately life is not so simple and sometimes you will want to give back accumulated garbage during a computation without losing recent data For this you need the gerepile function or one of its variants described hereafter GEN gerepile pari_sp ltop pari_sp lbot GEN q This function cleans up the stack between 1ltop and lbot where lbot lt ltop and returns the
437. ve its description for completeness It reads all remaining arguments in string context see Section 2 6 6 and sends a NULL terminated list of GEN pointing to these The automatic concatenation rules in string context are implemented so that adjacent strings are read as different arguments as if they had been comma separated For instance if the remaining argument sequence is xx 1 yy the s atom will send a GEN g amp a amp b amp c NULL where a b c are GENs of type t_STR content xx t_INT and t_STR content yy The format to indicate a default value atom starts with a D is Dvalue type where type is the code for any mandatory atom previous group value is any valid GP expression which is converted according to type and the ending comma is mandatory For instance DO L stands for this optional argument will be converted to a long and is O by default So if the user given argument reads 1 3 at this point long 4 is sent to the function via itos and long 0 if the argument is omitted The following special syntaxes are available DG optional GEN send NULL if argument omitted D amp optional GEN send NULL if argument omitted DV optional entree send NULL if argument omitted DI optional char send NULL if argument omitted Dn optional variable number 1 if omitted e Automatic arguments f Fake long C function requires a pointer but we do not use the resulting long p real precision
438. ver division by 0 invmoder impossible inverse modulo archer not available on this architecture or operating system typeer wrong type errpile the PARI stack overflows Omitting the error name means we are trapping all errors For instance the following can be used to check in a safe way whether install works correctly in your GP broken_install trap archer return 0S install addii GG trap USE if addii 1 1 2 BROKEN The function returns 0 if everything works the omitted else clause of the if OS if the operating system does not support install USE if using an installed function triggers an error and BROKEN if the installed function did not behave as expected 2 8 Interfacing GP with other languages The PARI library was meant to be interfaced with C programs This specific use will be dealt with extensively in Chapter 4 GP itself provides a convenient if simple minded interpreter which enables you to execute rather intricate scripts see Section 3 11 Scripts when properly written tend to be shorter and much clearer than C programs and are certainly easier to write maintain or debug You don t need to deal with memory management garbage collection pointers declarations and so on Because of their intrinsic simplicity they are more robust as well They are unfortunately somewhat slower Thus their use will remain complementary it is suggested that you test and debug your algori
439. ving strings are not handled in a special way even in string context the largest possible expression is evaluated hence print a 1 is incorrect since a is not an object whose first component can be extracted On the other hand print a 1 is correct two distinct argument each converted to a string and so is print a 1 since a 1 is not a valid expression only a gets expanded then 1 and the result is concatenated as explained above In case of doubt you can surround part of your text by parenthesis to force immediate interpretation of a subexpression print a 1 is another solution e Since there are cases where expansion is not really desirable we now distinguish between Keywords and Strings String is what has been described so far Keywords are special relatives of Strings which are automatically assumed to be quoted whether you actually type in the quotes or not Thus expansion is never performed on them They get concatenated though The analyzer supplies automatically the quotes you have forgotten and treats Keywords just as normal strings otherwise For instance if you type a b b in Keyword context you will get the string whose contents are ab b In String context on the other hand you would get a2 b All GP functions have prototypes described in Chapter 3 below which specify the types of arguments they expect either generic PARI objects GEN or strings or keywords or unevaluated expre
440. w they are actually used as well as some convenient tricks automatic concatenation and expansion keywords valid in string context As explained above the general way to input a string is to enclose characters between quotes This is the only input construct where whitespace characters are significant the string will contain the exact number of spaces you typed in Besides you can escape characters by putting a just before them the translation is as follows e lt Escape gt n lt Newline gt t lt Tab gt For any other character x Xx is expanded to x In particular the only way to put a into a string is to escape it Thus for instance a would produce the string whose content is a This is definitely not the same thing as typing a whose content is merely the one letter string a You can concatenate two strings using the concat function If either argument is a string the other is automatically converted to a string if necessary it will be evaluated first concat ex 1 1 1 ex2 a 2 b ex concat b a 2 ex2 concat a b 40 13 2Qex Some functions expect strings for some of their arguments print would be an obvious example Str is a less obvious but useful one see the end of this section for a complete list While typing in such an argument you will be said to be in string context The rest of this section is devoted to special syntactical tricks which can be u
441. went wrong Probably with the installation procedure but it may be a bug in the Pari system in which case we would appreciate a report including the relevant dif file in the Oxxx directory and the file pari cfg 224 Known problems e program the GP function install may not be available on your platform triggering an error message not yet available for this architecture Have a look at the MACHINES files to check if your system is known not to support it or has never been tested yet e If when running gp dyn you get a message of the form ld so warning libpari so xrx has older revision than expected xxx possibly followed by more errors you already have a dynamic PARI library installed and a broken local configuration Either remove the old library or unset the LD_LIBRARY_PATH environment variable Try to disable this variable in any case if anything very wrong occurs with the gp dyn binary e g Illegal Instruction on startup It doesn t affect gp sta 2 5 2 Some more testing Optional You can test GP in compatibility mode with make test compat If you want to test the graphic routines use make test ploth You will have to click on the mouse button after seeing each image There will be eight of them probably shown twice try to resize at least one of them as a further test More generaly typing make without argument will print the list of available extra tests among all available targets The make bench and mak
442. ws zet s local n not needed and possibly confusing see below sumalt n 1 1 n 1 n s 1 2 1 s This gives reasonably good accuracy and speed as long as you are not too far from the domain of convergence Try it for s integral between 5 and 5 say or for s 0 5 i x t where t 14 134 The iterative constructs which use a variable name forxxx prodrrx sumrxx vector ma trix plot etc also consider the given variable to be local to the construct A value is pushed on entry and pulled on exit So it is not necessary for a function using such a construct to declare the variable as a dummy formal parameter In particular since loop variables are not visible outside their loops the variable n need not be declared in the protoype of our zet function above zet s sumalt n 1 1 m 1 n s 1 27 1 s would be a perfectly sensible and in fact better definition Since local global scope is a very tricky point here s one more example What s wrong with the following definition first_prime_div x local p forprime p 2 x if x p 0 break 37 P first_prime_div 10 11 0 Answer the index p in the forprime loop is local to the loop and is not visible to the outside world Hence it doesn t survive the break statement More precisely at this point the loop index is restored to its preceding value which is 0 local variables are initialized to 0 b
443. x the operation is done componentwise For real x n is the number of desired significant decimal digits If n is smaller than the precision of x x is truncated otherwise x is extended with zeros For x a p adic or a power series n is the desired number of significant p adic or X adic digits where X is the main variable of z Note that the function precision never changes the type of the result In particular it is not possible to use it to obtain a polynomial from a power series For that see truncate The library syntax is precisionO z 7 where n is a long Also available are ggprecision x result is a GEN and gprec x n where n is a long 3 2 41 random N 2 gives a random integer between 0 and N 1 N can be arbitrary large This is an internal PARI function and does not depend on the system s random number generator Note that the resulting integer is obtained by means of linear congruences and will not be well distributed in arithmetic progressions The library syntax is genrand JN 3 2 42 real x real part of x In the case where x is a quadratic number this is the coefficient of 1 in the canonical integral basis 1 w The library syntax is greal x This returns a copy of the real part The internal routine real_i is faster since it returns the pointer and skips the copy 3 2 43 round z amp e If x is in R rounds x to the nearest integer and sets e to the number of error bits that is the bin
444. x is greater than the internal precision the real number is printed in e format note that in versions before 1 38 93 this was instead printed with a at the end Note also that in beautified format a number of type integer or real is written without enclosing parentheses while most other types have them Hence if you see the expression 3 14 it is not of type real but probably of type complex with zero imaginary part if you want to be sure type x or use the function type 21 UNIX 2 2 Simple metacommands Simple metacommands are meant as shortcuts and should not be used in GP scripts see Sec tion 3 11 Beware that these as all of GP input are case sensitive For example Q is not identical to q In the following list braces are used to denote optional arguments with their default values when applicable e g n 0 means that if n is not there it is assumed to be 0 Whitespace or spaces between the metacommand and its arguments and within arguments is optional This can cause problems only with w when you insist on having a filename whose first character is a digit and with r or w if the filename itself contains a space In such cases just use the underlying read or write function see Section 3 11 2 28 2 2 1 command GP on line help interface As already mentioned if you type n where n is a number from 1 to 11 you will get the list of functions in Section 3 n of the manual the list of sections bein
445. xels The plotting device imposes an upper bound for x and y for instance the number of pixels for screen output These bounds are available through the plothsizes function The following sequence initializes in a portable way i e independent of the output device a window of maximal size accessed through coordinates in the 0 1000 x 0 1000 range s plothsizes plotinit 0 s 1 1 s 2 1 plotscale 0 0 1000 0 1000 3 10 13 plotkill w erase rectwindow w and free the corresponding memory Note that if you want to use the rectwindow w again you have to use plotinit first to specify the new size So it s better in this case to use plotinit directly as this throws away any previous work in the given rectwindow 3 10 14 plotlines w X Y flag 0 draw on the rectwindow w the polygon such that the x y coordinates of the vertices are in the vectors of equal length X and Y For simplicity the whole polygon is drawn not only the part of the polygon which is inside the rectwindow If flag is non zero close the polygon In any case the virtual cursor does not move X and Y are allowed to be scalars in this case both have to There a single segment will be drawn between the virtual cursor current position and the point X Y And only the part thereof which actually lies within the boundary of w Then move the virtual cursor to X Y even if it is outside the window If you want to draw a line from xl yl to 2 y2 whe
446. xplained in Section 3 1 4 division will not be performed only reduction to lowest terms If you really want a reducible fraction under GP you must use the type function see Sec tion 3 11 2 26 by typing type x FRACN Be warned however that this function must be used with extreme care 2 3 5 Complex numbers type t_COMPLEX to enter x iy type x I y not x i y The letter I stands for y 1 Recall from Chapter 1 that x and y can be of type t_INT t_REAL t_INTMOD t_FRAC t_FRACN or t_PADIC 25 2 3 6 p adic numbers type t_PADIC to enter a p adic number simply write a rational or integer expression and add to it O p k where p and k are integers This last expression indicates three things to GP first that it is dealing with a t_PADIC type the fact that p is an integer and not a polynomial which would be used to enter a series see Section 2 3 10 secondly the prime p note that it is not checked whether p is indeed prime you can work on 10 adics if you want but beware of disasters as soon as you do something non trivial like taking a square root and finally the number of significant p adic digits k Note that 0 25 is not the same as 0 572 you probably want the latter For example you can type in the 7 adic number 2x7 1 3 4 7 2x772 0 773 exactly as shown or equivalently as 905 7 0 773 2 3 7 Quadratic numbers type t_QUAD first you must define the default quadratic order or field in wh
447. y The value of flag is only significant for matrices If flag 0 the method used is essentially the same as for computing the adjoint matrix i e computing the traces of the powers of A If flag 1 uses Lagrange interpolation which is almost always slower If flag 2 uses the Hessenberg form This is faster than the default when the coefficients are integermod a prime or real numbers but is usually slower in other base rings The library syntax is charpoly0 4 v flag where v is the variable number Also available are the functions caract A v flag 1 carhess A4 v flag 2 and caradj A v pt where in this last case pt is a GEN which if not equal to NULL will receive the address of the adjoint matrix of A see matadjoint so both can be obtained at once 3 8 3 concat x y concatenation of x and y If x or y is not a vector or matrix it is considered as a one dimensional vector All types are allowed for x and y but the sizes must be compatible Note that matrices are concatenated horizontally i e the number of rows stays the same Using transpositions it is easy to concatenate them vertically To concatenate vectors sideways i e to obtain a two row or two column matrix first transform the vector into a one row or one column matrix using the function Mat Concatenating a row vector to a matrix having the same number of columns will add the row to the matrix top row if the vector is x i e comes first an
448. y Gide o See ede h 175 CYC cad Bow ae Ee eae ek tid 98 CYCLE AA 137 CY SWAIN os Se A As 13 D datapath a Bye ee ck thee 17 ADICTO Di 179 215 debug cis a ae ee 17 23 195 debugfiles 17 23 debugging o 195 debuglevel 81 DEBUGEEVEL v 2000 a e as 195 196 debugmem 17 23 DEBUGMEM v3 6 a eve eg a Bote ee es 195 debugmem sos ss mor o 195 decodemodule 100 decomposition into squares 148 Dedekind 72 108 128 135 default precision 0 4 9 A TS aie tere es hse as 42 165 DEFAUETPREG oi bra one ole Yaa ds 174 defaults 2 4 bose yk ea 14 23 definite binary quadratic form 190 detTe kata Saka d Bake Bee aes 189 degree co ooo ooo co coc 137 189 delete varie A id 192 ANOM aa a A ee 64 denominator 35 64 COPIV oo boon bea HR Ee PASE Le 135 136 derivpol on ans eb ee ae ee es 136 destruction 004 180 debo pbk en ee Oe Wie 144 Get he fli A a Seay Pek 144 detit ak G a es ee A 144 diagonal 22 2 ioe la e Ree 144 Diamondi w aa ma be Pete Yee a 90 DAA 2 Bs Se ohh 8 do es rt Sb Se heeds Se 98 difference 00050 0005 54 GAPE PCL oon Sects BEES A 172 231 A Oe Pate ae Ba ee 72 ro En oe ea al de cra 78 direwler tht Wha wiht es BL eee 78 79 Dirichlet series 78 79 108 dirm l 2 50205 3 ee ee Bok he a a 79 GU Zet ak o a o Se ee Seats ee Se Se
449. y advise you to make sure you get this very useful library before configuring or compiling GP In fact with readline even line editing becomes more powerful outside an Emacs buffer 2 11 1 A too short introduction to readline The basics are as follows read the readline user manual assume that C stands for the Control key combined with another and the same for M with the Meta key generally C combinations act on characters while the M ones operate on words The Meta key might be called Alt on some keyboards will display a black diamond on most others and can safely be replaced by Esc in any case Typing any ordinary key inserts text where the cursor stands the arrow keys enabling you to move in the line There are many more movement commands which will be familiar to the Emacs user for instance C a C e will take you to the start end of the line M b M f move the cursor backward forward by a word etc Just press the lt Return gt key at any point to send your command to GP All the commands you type in are stored in a history with multiline commands being saved as single concatenated lines The Up and Down arrows or C p C n will move you through it M lt M gt sending you to the start end of the history C r C s will start an incremental backward forward search You can kill text C k kills till the end of line M d to the end of current word which you can then yank back using the C y key M y will rotate the kill ring
450. y also be what you want Local variables can be given a default value as the formal variables 36 Example For instance foo x 1 y 2 z 3 print x y z defines a function which prints its arguments at most three of them separated by colons This then follows the rules of default arguments generation as explained at the beginning of Section 3 0 2 foo 6 7 6 7 3 foo 5 17553 foo 1 2 3 Once the function is defined using the above syntax you can use it like any other function In addition you can also recall its definition exactly as you do for predefined functions that is by writing name This will print the list of arguments as well as their default values the text of seq and a short help text if one was provided using the addhelp function see Section 3 11 2 1 One small difference to predefined functions is that you can never redefine the built in functions while you can redefine a user defined function as many times as you want Typing u will output the list of user defined functions An amusing example of a user defined function is the following It is intended to illustrate both the use of user defined functions and the power of the sumalt function Although the Riemann zeta function is included in the standard functions let us assume that this is not the case or that we want another implementation One way to define it which is probably the simplest but certainly not the most efficient is as follo
451. y default To sum up the routine returns the p declared local to it not the one which was local to forprime and ran through consecutive prime numbers Here s a corrected version first_prime_div x forprime p 2 x if x p 0 return p Again it is strongly recommended to declare all other local variables that are used inside a function if a function accesses a variable which is not one of its formal parameters the value used will be the one which happens to be on top of the stack at the time of the call This could be a global value or a local value belonging to any function higher in the call chain So be warned Recursive functions can easily be written as long as one pays proper attention to variable scope Here s a last example used to retrieve the coefficient array of a multivariate polynomial a non trivial task due to PARI s unsophisticated representation for those objects coeffs P nbvar local v if type P t_POL for i 0 nbvar 1 P P return P v vector poldegree P 1 i polcoeff P i 1 vector length v i coeffs v il nbvar 1 If P is a polynomial in k variables show that after the assignment v coeffs P k the coeff cient of x 1 a in P is given by v n1 1 nz 1 What would happen if the declaration local v had been omitted The operating system will automatically limit the recursion depth dive n if n dive n 1 dive 5000 xxx deep
452. y slower formulation is to simply write 184 gerepileall ltop 2 z1 amp z2 so that the array gptr is in fact not needed Here we do not assume anything about the stack the garbage can be disconnected and z1 z2 need not be at the bottom of the stack If all of these assumptions are in fact satisfied then we can call gerepilemanysp instead which will usually be faster since we do not need the initial copy on the other hand it is less cache friendly Another important usage is random garbage collection during loops whose size requirements we cannot or do not bother to control in advance pari_sp ltop avma limit stack_lim avma 1 GEN x y while garbage x anything garbage y anything garbage if avma lt limit memory is running low half spent since entry gerepileall ltop 2 amp x amp y Here we assume that only x and y are needed from one iteration to the next As it would be too costly to call gerepile once for each iteration we only do it when it seems to have become necessary The macro stack_lim avma n denotes an address where 2771 27 1 1 of the remaining stack space is exhausted 1 2 for n 1 2 3 for n 2 4 4 3 Some hints and tricks In this section we give some indications on how to avoid most problems connected with garbage collecting First although it looks complicated gerepile has turned out to be a very flexible and fast garbage coll
453. z v 3 5 8 elleisnum E k flag 0 E being an elliptic curve as output by ellinit or alterna tively given by a 2 component vector w1 w2 representing its periods and k being an even positive integer computes the numerical value of the Fisenstein series of weight k at E namely 2im wa 1 2 1 k DO nk q 1 9 n gt 0 where q e w1 w2 When flag is non zero and k 4 or 6 returns the elliptic invariants g2 or g3 such that y 4a gar gs is a Weierstrass equation for E The library syntax is elleisnum E k flag 3 5 9 elleta om returns the two component row vector 71 12 of quasi periods associated to om w1 w2 The library syntax is elleta om prec 3 5 10 ellglobalred calculates the arithmetic conductor the global minimal model of E and the global Tamagawa number c Here E is an elliptic curve given by a medium or long vector of the type given by ellinit and is supposed to have all its coefficients a in Q The result is a 3 component vector N v c N is the arithmetic conductor of the curve v gives the coordinate change for E over Q to the minimal integral model see ellminimalmodel Finally c is the product of the local Tamagawa numbers c a quantity which enters in the Birch and Swinnerton Dyer conjecture The library syntax is globalreduction E 91 3 5 11 ellheight z flag 0 global N eron Tate height of the point z on the elliptic curve E The vector

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