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User's Guide to Pari/GP

1. 138 148 ideallist gs yoke gue ee we Ho 138 139 ideallistO sde aos oa be de ne a 139 ideallistarch 139 ideallllr d soi sa rio wb e 141 ideallog 4 6264 4 20 139 14 idealmin 140 ideslmil ssi sra we k ee eee we k 140 idealm lred s soa ck oe ek he kg 140 idealnorm gt s s sersa terranno 140 idealpoW 6 2 62 ba we ee E ew a 140 idealpowred 0 0 0 140 idealpows o o 140 idealprimedec 140 idealprincipal 141 idealred a sse antiaerei 141 idsalstar wsos vik Bee a a Be ee 141 idealetarO o o es ecni oa oe ee do 142 idealtwoelt 142 idealval oo o 142 ideal twoLelto ecu oeda we 142 UdElE wa a a e e Re WR e ta 119 ideleprincipal s ses s giia e siis 142 Ii ar Hos 6 ts bot anan amp a de woe E dt 204 TMAR ni Ge pte oS ot de engl A Aes ee 82 IMALE y pae deve se EG SO ER a 174 AMASESCOMPL s urie Gok ea we ee ae 174 IMAG Bee ee Ae De ee a 82 imprecise object 22 ANGEAM p gpg a a wok oe SPS oa Be Ss 91 Incgam0 voi e A cal UNCGAMG 2 ek o ee eS ee eee ee a 91 InGhISIVE OF fs oa ea ee wo 74 indek faa a ses a Re be ea we 121 Index oie ecb Boe pad Be Be de 121 indoxrank s i ee a Be eo a EEG 174 IndexXSOrt 4244844 14 0 181 infinite product 191 infinite SUM e lnea aa dock de 193 Infinity cs a e Sab osa ii 191 initell larisa ek ewe ts 115 ADIEZO A y poss k orok Re
2. 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 6 115 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 rnf111gram which deserves to be improved The library syntax is rnfpolredabs nf pol flag prec 3 6 150 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 e The library syntax is rnfpseudobasis nf pol 3 6 151 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 J not in HNF in general such that all the ideals of J except perhaps the last one are e
3. 178 LLO seers es Ge aha Ge rd a tas 178 LAVA 205 goa a Se a as 178 Inganna ses wk Soe Ge A ee wa gi local 325 Soe Pk Ea ee 37 11 42 44 TOS 2 54 A oe tos 56 60 61 91 208 logfile e ua ar ee ee wD 208 logfile e vicios ui 56 M makebigbnf 0 127 Mat seb edoa ee Pw RS 34 76 170 matad joints ais deo eid wih ee a ce Ss 172 227 matalgtobasis 142 matbasistoalg 142 Matcompanion iz Matdet sssi raa ES He ee a 172 Matdetidt s sesoses na oo ee ra 172 matdiagonal 173 MATCLBOM s mia ior o dee ue he Sg 173 Matextract 4 soe ee be ae ee we a 181 Matfrobenius mmm ae e 173 Math Pari e doros re el wh dg ee d 52 math ll s eea aeoo ee Re a 114 Mat hHESS js e eare ar a BS 173 mathilbert so e i aoe Bd a Gok weed 173 Mathnf pee a teiaa toane Ee 169 173 MathntO 3 4 ween ee eee eR 173 Mathnfmod cios ae wh aoe Boe 174 mathnfmodid 174 matid spe sakarya be Bes tak amp x 174 MACIMAGS e es iari oe wh a ee a 174 MatimageO o 174 Matimagecompl 174 Matindexrank our 174 Matintersect 174 Matinverseimage 174 matisdiagonal 175 Matker pees He be Rae ew we 175 MAatkerO a a weve th wow Wee BOR ke we 175 patkeribl du ira ge ke ho dd 175 MatkerintO 175 matmuldiagonal 175 matmultodiagonal 175 Matpascal sa sos e wosa ooo 175
4. This is not to be confused with the history of your commands maintained by readline It only contains their non void results in sequence Several inputs only act through side effects and produce a void result for instance a print statement a for loop or a function definition The successive elements of the history array are called 41 2 As a shortcut the latest computed expression can also be called the previous one the one before that and so on The total number of history entries is A When you suppress the printing of the result with a semicolon its history number will not appear either so it is often a better idea to assign it to a variable for later use than to mentally recompute what its number is Of course on the next line just use as usual This history array is in fact better thought of as a queue its size is limited to 5000 entries by default after which gp starts forgetting the initial entries So 1 becomes unavailable as gp prints 45001 You can modify the history size using histsize 2 2 3 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 to 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 hit
5. ap k X then the result is Y az X The library syntax is laplace z 3 7 30 serreverse x reverse power series i e x71 not 1 x 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 x 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 o 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 library syntax is gsubst x y z where y is the variable number 167 3 7 32 substpol x y 2 replace the variable y by the argument z in the polynomial expression x Every type is allowed for x but the same behaviour as subst above apply The difference with subst is that y is allowed to be any polynomial here The substitution is done as per the following script subst_poly pol from to 1 local t subst_poly_t M from t subst lift Mod pol M variable M t to For instance substpol x 4 x 2 1 x72 y 11 y72 y 1 substpol x 4 x72 1 x73 y 12 x 2 y x 1 substpol x 4
6. 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 6 Mod 5 22 x719 1 22 x714 123 22xx 9 9 11 x 4 x 20 25 x710 5 The library syntax is polcompositum0 P Q flag 3 6 113 polgalois x Galois group of the non constant polynomial x Q X In the present version 2 3 3 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 4 component vector n s k name with the following meaning n is the cardi nality of the group s is its signature s 1 if the group is a subgroup of the alternating group An s 1 otherwise and name is a character string containing name of the transitive group according to the GAP 4 transitive groups library by Alexander Hulpke 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 151 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 Sp
7. 2 5 5 Multivariate power series 2 6 User defined functions 2 6 1 Definition 2 6 2 Use 2 6 3 Recursive functions 2 6 4 Function which take functions as parameters 2 6 5 Defining functions within a function 2 6 6 Variable scope 2 7 Member functions 2 8 Strings and Keywords 2 8 1 Strings 2 8 2 Keywords 2 8 3 Useful examples 2 9 Errors and error recovery 2 9 1 Errors 2 9 2 Error recovery 2 9 3 Break loop 2 9 4 Error handlers 2 9 5 Protecting code 2 9 6 Trapping specific exceptions S 2 10 Interfacing GP with other pnemacos 2 11 Defaults 2 11 1 colors 2 11 2 compatible 2 11 3 datadir 2 11 4 debug 2 11 5 debugfiles 2 11 6 debugmem 2 11 7 echo 2 11 8 factor_add_primes 2 11 9 format 2 11 10 help 2 11 11 histsize 2 11 12 lines 2 11 13 log 2 11 14 logfile 2 11 15 new _galois_format 2 11 16 output 2 11 17 parisize 2 11 18 path 34 34 35 35 38 38 38 38 39 40 41 41 42 43 43 43 44 45 46 46 47 48 48 48 49 50 51 51 52 52 53 54 55 55 55 55 55 55 56 56 56 56 56 56 56 57 57 57 57 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 11 2 12 Simple metacommands 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 13 M1 2 12 2 12 2 12 2 12 2 12 2 12 2 12
8. E 89 lost 16 decimals Try higher m m intnumstep 5 7 the value of m actually used above tab intnuminit 0 oo 1 m 1 try m one higher intnum x 0 oo sin x 3 exp x tab 0 3 6 5 45 E 107 OK this time Warning Like sumalt intnum often assigns a reasonable value to diverging integrals Use these values at your own risk For example intnum x 0 loo I x 2 sin x 71 2 0000000000 Note the formula e sin 1 x dx cos rs 21 1 s a priori valid only for 0 lt R s lt 2 but the right hand side provides an analytic continuation which may be evaluated at s 2 Multivariate integration Using successive univariate integration with respect to different formal parameters it is immediate to do naive multivariate integration But it is important to use a suitable intnuminit to precompute data for the internal integrations at least For example to compute the double integral on the unit disc x y lt 1 of the function x y we can write tab intnuminit 1 1 intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y72 tab tab The first tab is essential the second optional Compare tab intnuminit 1 1 time 30 ms intnum x 1 1 intnum y sqrt 1 x72 sqrt 1 x 2 x 2 y72 time 54 410 ms slow intnum x 1 1 intnum y sqgrt 1 x72 sqrt 1 x 2 x 2 y72 tab tab time 7 210 ms faster However the intnuminit pr
9. Using flag computes a modified mt polylogarithm of x We use Zagier s notations let Rm denotes R or S depending whether m is odd or even If flag 1 compute D 1 defined for x lt 1 by m l k m 1 Rm oe e tie As i reset log 1 a k k 0 If flag 2 compute D 1 defined for x lt 1 by log z log x t a y ee m k 0 If flag 3 compute P a defined for z lt 1 by ml 9k B i m l Bm M Rm a i log z Lim 2 Mos Iz k 0 These three functions satisfy the functional equation fm 1 2 1 fm z The library syntax is polylog0 m x flag prec 3 3 40 psi x the function of z i e the logarithmic derivative I x T zx The library syntax is gpsi z prec 3 3 41 sin x sine of z The library syntax is gsin x prec 3 3 42 sinh x hyperbolic sine of x The library syntax is gsh z prec 92 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 2 4 2 41 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 are the value of the same variable o
10. cc C compiler DLLD Dynamic library linker LD Static linker 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 213 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 The contents of the following variables are appended to the values computed by Configure CFLAGS Flags for CC CPPFLAGS Flags for CC preprocessor LDFLAGS Flags for LD The contents of the following variables are prepended to the values computed by Configure C_INCLUDE_PATH is prepended to the list of directories searched for include files Note that adding I flags to CFLAGS is not enough since Configure sometimes relies on finding the include files and parsing them and it does not parse CFLAGS at this time LIBRARY_PATH is prepended to the list of directories searched for libraries You may disable inlining by adding DDISABLE_INLINE to CFLAGS and prevent the use of the volatile keyword with DDISABLE VOLATILE 3 3 Debugging profiling If you also want to debug the PARI library Configure g creates a directory Orxx dbg containing a special Makefile ensuring that the GP and PARI library built there is suitable for debugging If you want to profile GP or the library using gprof for instance Configure pg will create an Oxxx prf director
11. index of the power order in the ring of integers nf bnr bnf nf number field r bnr bnf nf the number of real embeddings r2 bnr bnf nf the number of pairs of complex embeddings reg bnr bnf regulator 121 roots bnr bnf nf roots of the polynomial generating the field t2 bnr bnf nf the T2 matrix see nfinit tu bnr bnf a generator for the torsion units tufu bnr bnf w u ur uz is a vector of fundamental units w generates the torsion units zk bnr bnf nf integral basis i e a Z basis of the maximal order 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 3 6 7 Class group units and the GRH 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 P is the def
12. o eee eee ee eee 111 35 1elladd e aae ie o ae A O AB ek A E ANA e Ge eee ee a LTT 3 2nellalko dar e pa A A LAS A A A AE A A a A a DD do ella LR A saa kt a ta e da A da E A ds de o a iZ OA Clap sated A AR er A E ALA A A A TE A AE A Ate AZ 3 020 ellbiL s o ae Salto a ea ii a A A E Bye Te ye te amp A o e 2 3 0 0 ellchang curve et ddim e ds da da e AZ So Cellchangepolnt a asu Gere lactate Sole ab ats Ker Se Bn Re Ged Beas we Bo ee Be ees stl te oy ey Be Gk ee ae Ear ae He OLED 3 5 8 ellconvertname ss Ge skeet GS hee co BY pa A oe oe A ee ee ee Sp es a AILS g o 9 elleisnumy Ls RR Gee A ee BO te a a a oo a LS Sco LO elleta Hear A ay Se eee e Behe wy hee A Ee te a ee ES 3 5411 ellgenerators actos a BeBe ete Boe a A Pe a Go dean ed on 18 30 12 ellglobalred os 202 4 aha ke ae Stl ahaa A ab ele Ae eb ee ae ab Aaa Boe AMS 35 13 ellheight a eins a as geod ge an Bl ea DR eis ale ca a I AR a ee eee ae me yp ce oe LTT 3 5214 ellheightmatrix a4 ne a A a A RAGE a A A ee See CTIA Sro l5cellidentify ce s os donar e a it Woke eM ee AOR eek eo a a ea te oh gpa a A BSO LEIDEN ee ea ee a ee See Aon Sue ae DA 30017 elisSOnCUurve ia bape doth a at arth G Bye e ated e ELO Broek SOU y sk 6 Aaa a 6 OR lee blk ae BS he ok At a 8 oe E Md oh oe Es 20 19 elllocalred pis Be ete Soy wet aR Balled al Wie BE dy A Se args De aes Bel ate wade a Fe Sy SE A Ge AA e TS 3 0 20 elllseries lt lt 8 3 Seo ar E Ge ete oe a Bie ced BE
13. In practice very few counter examples are known requiring unlucky random seeds No counter example has been reported for c2 0 5 which should be almost as fast as cg 0 3 and shall very probably become the default If you use c2 12 then everything is correct assuming the GRH holds You can use bnfcertify to certify the computations unconditionally 122 Remarks Apart from the polynomial P you do not need to supply the technical parameters under the library you still need to send at least an empty vector coded as NULL 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 nrpid you must give some values as well for c and cz and provide a vector c ca nrpid Note also that you can use an nf instead of P which avoids recomputing the integral basis and analogous quantities 3 6 8 bnfcertify bnf bnf being as output by bnfinit checks whether the result is correct i e whether it is possible to remove the assumption of the Generalized Riemann Hypothesis It is correct if and only if the answer is 1 If it is incorrect the program may output some error message or loop indefinitely You can check its progress by increasing the debug level The library syntax is certifybuchall bnf and the result is a C long 3 6 9 bnfclassunit P flag 0 tech this function is DEPRECATED use bnfinit Buchmann s sub exponential algori
14. The library syntax is matalgtobasis nf x 3 6 73 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 74 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 0 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 142 3 6 75 newtonpoly x 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 occur 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 2 1 on 32 bit resp 64 bit machines see Section 3 2 49 The library syntax is newtonpoly z p 3 6 76 nfalgtobasis nf x this is the inverse function of nfbasistoalg Given an object x whose entr
15. a 9 In particular x is principal if and only if v is equal to the zero vector In the latter case x aZx where a is given by v2 Note that if a is too large to be given a warning message will be printed and v2 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 The user is warned that these two last setting may induce very lengthy computations The library syntax is isprincipalall bnf x flag 3 6 17 bnfisunit bnf x bnf being the number field data output by bnfinit and zx being an algebraic number type integer rational or polmod this outputs the decomposition of x on the fundamental units and the roots of unity if x is a unit the empty vector otherwise More precisely ifuj u are the fundamental units and is the generator of the group of roots of unity bnf tu the output is a vector z1 r r 1 such that x uj ut r 1 The a are integers for i lt r and is an integer modulo the order of fori r 1 The library syntax is isunit bnf x 126 3 6 18 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
16. creates the maximum and minimum of x and y when they can be compared The library syntax is gmax z y and gmin z y 3 1 17 vecmax x if x is a vector or a matrix returns the maximum of the elements of zx otherwise returns a copy of x Error if x is empty The library syntax is vecmax z 3 1 18 vecmin x if x is a vector or a matrix returns the minimum of the elements of x otherwise returns a copy of x Error if x is empty The library syntax is vecmin z 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 75 3 2 1 Col x transforms the object x into a column vector The vector will be with one com ponent only except when x is a vector or a quadratic form in which case the resulting vector is simply the initial object considered as a column vector a matrix the column of row vectors com prising the matrix is returned a character string a column 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 o
17. f x 0 fO 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 For example a 2 1 1 a2 1 kill a h1 42 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 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 207 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 11 You could for instance do default logfile new tex default log 1 printtex result 3 11 2 20 quit exits gp 3 11 2 21 read filename reads i
18. where is as before a lt 1 The function f is assumed to decrease like X In particular a lt 2 is equivalent to no a at all e a two component vector o a where is as before a gt 0 The function f is assumed to decrease like exp aX In this case it is essential that a be exactly the rate of exponential decrease and it is usually a good idea to increase the default value of m used for the integration step In practice if the function is exponentially decreasing sumnum is slower and less accurate than sumpos or suminf so should not be used The function uses the intnum routines and integration on the line R s The optional argument tab is as in intnum except it must be initialized with sumnuminit instead of intnuminit When tab is not precomputed sumnum can be slower than sumpos when the latter is applicable It is in general faster for slowly decreasing functions 193 Finally if flag is nonzero we assume that the function f to be summed is of real type i e satisfies f z f Z which speeds up the computation Xp 308 a sumpos n 1 1 n73 n 1 time 1 410 ms tab sumnuminit 2 time 1 620 ms slower but done once and for all b sumnum n 1 2 1 n 3 n 1 tab time 460 ms 3 times as fast as sumpos a b 4 1 0 E 306 0 E 320 I perfect sumnum n 1 2 1 n 3t n 1 tab 1 a function of real type time 240 ms 12 1 0 E 306 twice
19. 0 Stark units are used and in rare cases a vector of extensions may be returned whose compositum is the requested class field See bnrstark for details The library syntax is quadhilbert D pq prec 3 4 58 quadgen D creates the quadratic number w a VD 2 where a 0 if x 0mod 4 a 1 if D 1 mod4 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 x 3 4 59 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 60 quadray D f lambda relative equation for the ray class field of conductor f for the quadratic field of discriminant D using analytic methods A bnf for x D is also accepted in place of D For D lt 0 uses the o function If supplied lambda is is the technical element A 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 and a vector of relative equations may be returned See bnrstark for more details The library syntax is quadray D f lambda prec where an omitted lambda is coded as NULL 3 4 61 quadregulator regulator of the quadratic field of positive discrimina
20. 1 2 5 Strings 1 2 6 Notes PRS eS ey la BO 1 3 Multiprecision kernels Portability 1 4 The PARI philosophy 1 5 Operations and functions 1 5 1 Standard arithmetic operations 1 5 2 Conversions and similar functions 1 5 3 Transcendental functions 1 5 4 Arithmetic functions 1 5 5 Other functions A Chapter 2 Specific Use of the gp Calculator 2 1 Introduction 2 1 1 Startup 2 1 2 Getting help 2 1 3 Input 2 1 4 Interrupt Quit 2 2 The general gp input lin 2 2 1 Introduction 2 2 2 The gp history 2 2 3 Special editing characters 2 3 The PARI types 2 3 1 Integers 2 3 2 Real numbers 2 3 3 Intmods 2 3 4 Rational numbers 2 3 5 Complex numbers 2 3 6 p adic numbers 2 3 7 Quadratic numbers 2 3 8 Polmods 2 3 9 Polynomials 2 3 10 Power series 2 3 11 Rational functions 2 3 12 Binary quadratic forms of positive or negative discriminant 2 3 13 Row and column vectors 2 3 14 Matrices 2 3 15 Lists 19 19 19 19 219 20 21 21 21 22 22 22 23 23 24 24 24 25 25 25 27 27 27 27 28 28 28 29 29 29 30 30 30 31 31 31 31 32 32 33 33 34 34 34 34 34 2 3 16 Strings 2 3 17 Small vectors 2 3 18 Note on output formats 2 4 GP operators 2 5 Variables and embole expressions 2 5 1 Variable names 2 5 2 Vectors and matrices 2 5 3 Variables and polynomials 2 5 4 Variable priorities multivariate objects
21. 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 2 12 be abd wegen 2 13 The preferences file 2 13 2 13 ina 2 14 Using GNU Emacs 2 12 19 prettyprinter 20 primelimit 21 prompt 22 prompt_cont 23 psfile 24 readline 25 realprecision 26 secure 27 seriesprecision 28 simplify 29 strictmatch 30 TeXstyle 31 timer 1 po 2 3M 4Na 5 b 6 c 7 d s e 9 g 10 gf 11 gm 12 h 4 m 5 o 6 p 7 ps 8 q 19 r 20 s 21 Xt 22 Xu 23 um 24 v 25 w 26 x 27 y 28H 29 1 Syntax 2 Where is it 2 15 Using readline 2 15 1 A too short introduction to readline 57 57 58 58 58 58 58 58 58 59 59 59 59 59 59 60 60 61 61 61 61 61 61 61 61 61 61 61 61 61 61 61 62 62 62 62 62 62 62 62 62 62 62 63 63 64 65 66 66 2 15 2 Command completion and online help di e a ue Chapter 3 Functions and Operations Available in PARI and GP 3 1 Standard monadic or dyadic operators O E 31 97 3 1 3 314 3 1 5 3 1 6 3 1 7 3 1 8 divrem 3 1 97 3 1 10 bittest 3 1 11 shift 3 1 12 shiftmul 3 1 13 Comparison and boolean operators 3 1 14 lex 3 1 15 sign 3 1 16 max 3 1 17 vecmax 3 1 18 vecmin Va E a A Sate es Lah a AAA aS 3 2 Conversions and similar elementary functions or commands 3 2 1 Col 3 2 2 Li
22. 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 p adic unit except when the number is zero in which case u is zero the significand having a certain precision of k significant words 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 X9 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 floating point 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 22 1 3 Multiprecision kernels Portability You can skip this section if you are not interested in hardware technicalities The PARI multip
23. In degree 5 Cs 5 1 1 Ds 10 1 2 Mao 20 1 3 As 60 1 4 S5 120 1 5 In degree 6 Cg 6 1 1 S3 6 1 2 De 12 1 3 A4 12 1 4 Gis 18 1 5 Ag x Ca 24 1 6 Sf 24 1 7 Sy 24 1 8 G3 36 1 9 G 36 1 10 Sa x Cz 48 1 11 As PSL2 5 60 1 12 Gro 72 1 13 Ss PGLa 5 120 1 14 As 360 1 15 Se 720 1 16 In degree 7 C7 7 1 1 D7 14 1 2 Mai 21 1 3 Mas 42 1 4 PSL2 M PSL3 2 168 1 5 A7 2520 1 6 S7 5040 1 7 152 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 polgalois x prec To enable the new format in library mode set the global variable new_galois_format to 1 3 6 114 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 dividi
24. Then display the whole drawing in a special window on your screen 3 10 12 ploth X a b expr flag 0 n 0 high precision plot of the function y f z 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 xmin zmaz 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 f1 X fk X and then all the curves y f X will be drawn in the same window The binary digits of flag mean e 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 sinusoidal curves 198 ploth X 0 2 Pi X X sin X cos X 1 will draw a circle and the line y zx e 2 Recurs
25. bnrstark bnr returns the ray class field of Q v3 modulo 5 Usually one wants to apply to the result one of rnfpolredabs bnf pol 16 compute a reduced relative polynomial rnfpolredabs bnf pol 16 2 compute a reduced absolute polynomial The library syntax is bnrstark bnr subgroup where an omitted subgroup is coded by NULL 3 6 37 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 132 3 6 38 factornf x t factorization of the univariate polynomial x over the number field defined by the univariate polynomial t z 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 5 4 However if non rational number field elements occur as polmods or polynomials as coefficients of xz the variable of these polmods must be the same as the main variable of t For example factornf x 2 Mod y y 2 1 y 2 1 factornf x 2 y y 2 1 these two are OK factornf x 2 Mod z z72 1 y 2 1 xxx factornf inconsistent data in rnf function factornf x 2 z y 2 1 xxx factornf incorrect variable in rnf function The library syntax is polfnf x
26. curve name Examples ellconvertname 123b1 1 1128 1 1 ellconvertname 12 123b1 The library syntax is ellconvertname name 3 5 9 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 we 1 42 01 k So n a 1 q n gt 0 where q e w1 w3 When flag is non zero and k 4 or 6 returns the elliptic invariants g2 or g3 such that y da gat 93 is a Weierstrass equation for E The library syntax is elleisnum L k flag 3 5 10 elleta om returns the two component row vector 71 172 of quasi periods associated to om jwi wa The library syntax is elleta om prec 3 5 11 ellgenerators E returns a Z basis of the free part of the Mordell Weil group associated to E This function depends on the elldata database being installed and referencing the curve and so is only available for curves over Z of small conductors The library syntax is ellgenerators 3 5 12 ellglobalred calculates the arithmetic conductor the global minimal model of E and the global Tamagawa number c E must be an sell as output 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
27. divisors x divisors not an integer argument in an arithmetic function type x 2 t_POL type simplify x 13 t_INT Note that GP results are simplified as above before they are stored in the history Unless you disable automatic simplification with Xy that is In particular type 1 14 t_INT The library syntax is simplify lt 3 2 46 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 taille x returns the number of words instead 3 2 47 sizedigit x 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 Str x which is slower The library syntax is sizedigit 1 which returns a long 85 3 2 48 truncate x 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
28. entree ep is_entry i sum for a lt i lt b return somme ep a b i 2 gen_0 3 9 18 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 Use an algorithm of F Villegas as modified by D Zagier improves on Euler Van Wijngaarden method If flag 1 use a variant with slightly different polynomials 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 If the series already converges geometri cally suminf is often a better choice p28 sumalt i 1 1 i i log 2 time O ms 1 2 524354897 E 29 suminf i 1 1 i i x suminf user interrupt after 10min 20 100 ms p1000 sumalt i 1 1 i i log 2 time 90 ms 2 4 459597722 E 1002 sumalt i 0 1 i i exp 1 time 670 ms 3 4 03698781490633483156497361352190615794353338591897830587 E 944 suminf i 0 1 i i exp 1 192 time 110 ms 4 8 39147638 E 1000 faster and more accurate The library syntax is sumalt void E GEN eval GEN void GEN a long prec Also available is sumalt2 with the same arguments flag 1 3 9 19 sumdiv n X expr sum of expression expr over the positive divisors of n Arithmetic functions like sigma use the multiplicativity of the u
29. 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 complex e g acos 2 0 or acosh 0 0 Note also that the principal branch is always chosen e If the argument is an intmod 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 Note that in the case of a 2 adic number sqr x may not be identical to x x for example if x 1 0 2 and y 14 O 2 then 2x y 1 O 2 while sqr x 1 O 2 Here x z 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 xx y 4mod8 and sqr 1 4mod32 when both x and y are congruent to 2 modulo 4 However since intmod 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 c
30. gt 0 3 4 53 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 x is definite and for indefinite forms redreal x qfbred z rhoreal x qfbred z 1 redrealnod z sq qfbred z 2 isqrtD rhorealnod z sq qfbred z 3 isqrtD 107 3 4 54 qfbsolve Q p Solve the equation Q x y p over the integers where Q is a binary quadratic form and p a prime number Return x y as a two components vector or zero if there is no solution Note that this function returns only one solution and not all the solutions Let D discQ The algorithm used runs in probabilistic polynomial time in p through the computation of a square root of D modulo p it is polynomial time in D if Q is imaginary but exponential time if Q is real through the computation of a full cycle of reduced forms In the latter case note that bnfisprincipal p
31. i e such that Mz y the empty vector otherwise To get the complete inverse image 1t 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 28 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 29 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 pis 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 matkerO z flag Also available are ker x flag 0 keri x flag 1 3 8 30 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 2 instead The library syntax is matkerint0 x flag Also available is kerint x flag 0 3 8 31 matmuldiagonal z d product
32. 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 ezpr instead The library syntax is prodinf void E GEN eval GEN void GEN a long prec flag 0 or prodinf1 with the same arguments flag 1 191 3 9 16 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 method 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 3 9 17 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 12 9 094508852984436967261245533 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
33. inv list listarch log min mul 126 126 126 126 127 127 127 127 128 128 128 129 129 129 130 130 130 130 131 131 131 132 132 132 133 133 133 134 134 135 135 135 136 136 136 136 137 137 137 137 137 137 138 138 138 139 140 140 140 3 6 63 idealnorm 3 6 64 idealpow 3 6 65 idealprimedec 3 6 66 idealprincipal 3 6 67 idealred 3 6 68 idealstar 3 6 69 idealtwoelt 3 6 70 idealval 3 6 71 ideleprincipal 3 6 72 matalgtobasis 3 6 73 matbasistoalg 3 6 74 modreverse 3 6 75 newtonpoly 3 6 76 nfalgtobasis 3 6 77 nfbasis 3 6 78 nfbasistoalg 3 6 79 nfdetint 3 6 80 nfdisc 3 6 81 nfeltdiv 3 6 82 nfeltdiveuc 3 6 83 nfeltdivmodpr 3 6 84 nfeltdivrem 3 6 85 nfeltmod 3 6 86 nfeltmul 3 6 87 nfeltmulmodpr 3 6 88 nfeltpow 3 6 89 nfeltpowmodpr 3 6 90 nfeltreduce 3 6 91 nfeltreducemodpr 3 6 92 nfeltval 3 6 93 nffactor 3 6 94 nffactormod 3 6 95 nfgaloisapply 3 6 96 nfgaloisconj 3 6 97 nfhilbert 3 6 98 nfhnf 3 6 99 nfhnfmod 3 6 100 nfinit 3 6 101 nfisideal 3 6 102 nfisincl 3 6 103 nfisisom 3 6 104 nfnewprec 3 6 105 nfkermodpr 3 6 106 nfmodprinit 3 6 107 nfsubfields 3 6 108 nfroots 3 6 109 nfrootsofl 3 6 110 nfsnf 3 6 111 nfsolvemodpr 140 140 141 141 141 141 142 142 142 142 142 142 143 143 143 143 143 144 144 144 144 144 144 144 144 145 145 145 145 145 145 145 146 146 147 147 147 147 149 149 149 149 149 1
34. nfrootsofs sc deed FASE aE SE 150 DESMITA s s ooo coo 150 DES 2 25 8 4 bons Be ee Ke A 150 DESOLVOMO PE ice dow de aaa a 150 nfsubfield 133 nfsubtields e ia bo we ae 4 8 150 204 MO e Slaven Hae Be a ee moe Boe 121 NOM 2445 ee ed ead FA ere bake 82 norml2 ocre ee 82 THOU ici 6 eR ae ee ee Hee ee oK Beg 74 NUCOMP dira Ae a ees 107 MUdUpl gra we Pu ee ae Gwe ie a 107 HUMAI ascii 105 number field 32 DUMDPart 2 24 88 sdr disetir a 105 DUMATV goes pone Bae pi a oa Bee 105 D M cuca e eena ke BPE aos og 83 nu merat r 39 83 numerical derivation 36 numerical integration 183 mumtoperm ss ses srk eane 83 NUPOW cier 107 N ron Tate height 113 Dorta Ge Ge ee y A a a 162 OMEGA e ee eR ae HGS a a ees 8 109 OMGRA c Bt hk Be dew a Ga 105 111 ONCUTVE e eb be ee Le ee ee 115 OpPeTAuor So 408 on Wok ek ae eR Ee dw 35 DE pl o is Se te ae ce hs 74 OF So 54 te et oe bea SDE YS 19 Ordell a saner w temi pge as 116 OLA 2 neeesa a ei a e a 110 orderell ws ss palsa kesa E is 116 OTOTCO cia ai e a 153 OULPUG e Ge oe Be o e pia 56 61 P p adic number 20 21 31 PadicappE eii a a Gs 164 Padieprels ema a Go BS Gre 83 parametric plot saw sora o ooo 198 PariPython sie ee 52 parisiz s s eo ae we Ee Be wos 57 pari ra d s s ces ee pa a a a 84 pari_rand3i 84 Pascal triangle 175 Path oso ei e P oa
35. 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 matriz is a pair A I where A is a relative matrix and I 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 projective module generated by a pseudo matrix A J is the sum gt gt a A 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 J is the ideal 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 3 6 5 Class field theory A modulus in the sense of class field theory is a divisor supported on the non complex places of K In PARI terms this means either an ordinary ideal I as above no archimedean component or a pair 7 a where a is a vector with r 0 1 components corresponding to the infinite part of the divisor More precisely the i th component of a corresponds to the real embedding associat
36. ooo oo 164 Ppoldise soe e s ca e 164 poldis cOn tna oh de lA de A 164 poldiscreduced 164 poleval e e sb Bean heeds dee a 163 polin fie eee orden Gee Gr tes fee 133 polgalois 56 151 152 polhensellift 164 Ppolimt adi adrede cn a 165 polinterpolate 165 polisirreducible 165 Pollard Rho 95 101 pollead cessata Aw wR SO ee 165 pollegendre 165 DOMO Hi e be he ke ae we e 20 polmod 0 21 32 polmodrecip 142 POLFECID ria et eeu ee Gree Gye Bae 165 polred 2 223804 8445 440455 152 153 polredO se bee bbe aa 153 polredabs ua dee o hk Be 153 polredabsO 2 2 868s 8844 Bb eR EG 153 polredord 2 2 2 sie ete ee be resi 153 polresultant civ dows wd de a tw 165 polresultantO e se ccr ee Bae a 165 DOLTEV mosom woe ad fr Be arg polroots crsa rart ean rai 165 169 polrootsmod 110 166 polrootspadic 110 166 POLStUPM k ka dia Gow ee et bode ye Ge 166 polsubcyclo se sassa ee lt 0 0 166 polsylvestermatrix 166 POLSyM cies ele oe A a es 167 poltchebi 4 244444 48 4 6 4 a 167 poltschirnhaus 153 polylog 2 4 8s ee ta Po sa ea 91 POUVVOSO ceci she ii e ee we a OR g2 polynomial 20 21 32 PDOLZAS fo ec a ee es 167 polzagier o 167 polzagreel 167 PostScTAPL s s cco gw ee ee ew 197 powell rse
37. returns the i th component of vector x i j x j and x i respectively return the entry of coordinates i j the j th column and the i 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 a x 1 x 1 subst a x 1 Ak variable name expected subst a x 1 subst a x 1 1 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 x 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 a 1 1 member unary postfix member extracts member from structure x see Section 2 7 e Priority 5 unary prefix logical not x return 1 if x is equal to 0 specifically if gemp0 x 1 and 0 otherwise unary prefix cardinality x returns 1length z e Priority 4 unary prefix toggles the sign of its argument has no effect whatsoever e Priority
38. s s sxs eee Bas we e 113 Gll6tA ssec dae EER SO oe 113 Cllgeneratorses sesk aa omaa de a 113 ellglobalred 2 604454 eb ees 113 ellbSIgbt cas iee we aa 113 ellhelgbt0O cisco 113 ellheightmatriX 113 ellidentify pa son de 114 ellinit se a aaa oa GR eS 111 114 ellinitO gt as rep ee ee ee 115 ellisoncurve 115 Clark hoe ot eee Sa ee A 115 elllocalred pior eti esi gog ee ee eG 115 elliseries 24 aie a ani 115 116 ellminimalmodel 113 116 ellorder soros dance DR Be Boe 116 ellordinate i goss wok Boe pol a eK oe 116 ellpointtoZ 62 ee bag 20 1 0 116 ClIPOW Sack dee ee eA de Od Ge AS 116 ellrootnoO o sora os saos ee 116 ellsearch 117 ellsearchcurve 117 ellsigma sses srest oaan 116 117 Gllsubs corsa aa 117 elltaniyama 117 CLECOBS 2 se daas a aan S a a e aa 117 eLILoOESO vivas Hl A ke ae ee 118 SLIWP oso xe oo ah moe BG tes eae cee HR oe he 118 CllwpO sssaaa oe 244 tesira 118 ellzeta oo sessen esameno 118 ellzt point soe ss se eds sod e aes 118 EMACS poe E 64 Cone a EEE EEE 45 70 environment expansion 78 environment expansion 54 environment variable 78 EA ws 2 Ae wo A a Re ees 90 error handler 1 0 ee eee 51 error recovery 2 ee 49 error trapping 2 ee ee ee 49 OTTO sumar eee 47 50 205 eta soy gb eeoa oe ER ah EOS 90 111 ICH ses ia a Gree Gece os 101 Eu
39. several screens 2 1 4 Interrupt Quit Typing quit at the prompt ends the session and exits gp At any point you can type Ctrl C that is press simultaneously the Control and C keys the current computation is interrupted and control given back to you at the gp prompt together with a message like gcd user interrupt after 840 ms telling you how much time ellapsed since the last command was typed in and in which GP function the computation was aborted It does not mean that that much time was spent in the function only that the evaluator was busy processing that specific function when you stopped it 2 2 The general gp input line The gp calculator uses a purely interpreted language GP 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 and the main loop does not really execute but rather evaluates sequences of expressions Of course it is by no means a true LISP 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 only this one is evaluated 28 2 2 1 Introduction User interaction with a gp session proceeds as follows First one types a sequence of characters at the gp prompt see Section 2 15 1 for a description of the line editor When you hit the lt Return gt key gp gets you
40. 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 not prefix operator amp amp and operator or operator exist giving as results 1 true or O false 1 5 2 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 intmod or a polmod this is actually not quite true internally the format is 2 a where a and b are integers 24 1 5 3 Transcendental functions They usually operate on any complex number power series and some also on p adics The list is everexpanding and of course contains all the elementary functions plus many others Recall that by extension PARI usually allows a transcendental function to operate componentwise on vectors or matrices 1 5 4 Arithmet
41. use parentheses for grouping to that enclosed subexpressions be evaluated independently init x add y x y mul y x y 43 2 6 6 Variable scope Local variables should more appropriately be called temporary values since they are in fact local to the function declaring them and any subroutine called from within In the following example f O local a 3 EO gO y 1 gQ sees the y introduced in f True lexical scoping does not exist in GP See e g the difference between local and my in Perl In an iterative constructs which use a variable name forrxx prodrxx sumrxx vector matrix plot etc the given variable is also local to the construct A value is pushed on entry and poped on exit So it is not necessary for a function using such an iterator to declare the variable as local On the other hand if you exit the loop prematurely e g using the break statement you must save the loop index in another variable since its value prior the loop will be restored upon exit for instance for i 1 n if ok i break if i gt n return failure is incorrect since the value of tested by the i gt n is quite unrelated to the loop index Finally 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 above since the parameter would shadow the variable If spe
42. which saves time Hence bnr gen would produce an error If flag 1 as the default except that generators are computed The library syntax is bnrinitO bnf f flag 3 6 33 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 34 bnrisprincipal bnr xz 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 z a g If flag 0 outputs only vy 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 bnrisprincipal bnr x flag 131 3 6 35 bnrrootnumber bnr chi flag 0 if x chi is a not necessarily primitive character over bnr let L s x 30 4 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 s X W x A s X where A s x A x 7 s L s x is
43. word of precision which on a 32 bit machine actually gives you 9 digits 9 lt log 2 lt 10 default realprecision 2 realprecision 9 significant digits 2 digits displayed 2 11 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 11 27 seriesprecision default 16 number of significant terms when converting a polynomial or rational function to a power series see ps 58 2 11 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 Xy 2 11 29 strictmatch default 1 this is a toggle which ca
44. x y is a polynomial but y z is a rational function See Section 2 5 4 The library syntax is numer z 3 2 38 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 39 padicprec x 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 40 permtonum z 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 83 3 2 41 precision x 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 matrix 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 whe
45. x72 1 x 1 72 y 13 4 y 6 x y 2 3xy 3 The library syntax is gsubstpol z y z 3 7 33 substvec x v w v being a vector of monomials variables w a vector of expressions of the same length replace in the expression x all occurences of v by wi The substitutions are done simultaneously more precisely the v are first replaced by new variables in x then these are replaced by the w substvec x y x y Ly x 1 ly x substvec x y x y y x y 12 ly x yl not y 2 y The library syntax is gsubstvec z v w 3 7 34 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 expansion is transparent to the user in 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 35 thue inf 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 a modulo units of positive norm in the number field defined by P as computed by bnfisintnorm If the result is conditional on the GRH or some heuristic strenghtening a Warning is printed Otherwise the result is unconditional barring bugs For instance here s how to solve the Thue equation gis Baye 4 tnf
46. 11 1 1 L subgrouplist bnr IndexBound 1 vector L i galoissubcyclo bnr L i Setting L subgrouplist bnr IndexBound 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 3 6 46 galoissubfields G 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 47 galoissubgroups gal Output all the subgroups of the Galois group gal 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 Such subgroup can be used instead of a Galois group in the following command galoisisabelian galoissubgroups galoisexport and galoisidentify To get the subfield fixed by a subgroup sub of gal use galoisfixedfield gal sub 1 The library syntax is galoissubgroups gal 3 6 48 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
47. 11 2 28 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 29 write filename str writes appends to filename the remaining arguments and appends a newline same output as print 3 11 2 30 writel filename str writes appends to filename the remaining arguments with out a trailing newline same output as print1 3 11 2 31 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 5 4 of new variables set in this way remain the same preset variables retain their former pr
48. 115 K kbessel ceci a ae eb Sh ae Re 89 Kbessel2 a4 i wae de ea eA eee Bk sg 89 ker aims fae ees A a b 175 RETIRA ea 175 Kerimt nic Re a a 175 keyword o 46 KILL capa y egoe we eG ee ae 207 KOdaiTa 22 2 sona 68 pr a Pah E 115 Kronecker symbol 104 kronecker 4 104 Laplace i sa oia da a oe 167 VOM gk kad we A 104 leading term 165 leaves oc babe eae e kee Gabbe ees 22 leaves fe ce ROA OA ss 21 Legendre polynomial 165 Legendre symbol 104 legendi sor sa we A A we A aa 165 length 20 eb ee ee EM Ee ees 82 Lenstra a hd wed eee a a he ee 101 163 LS se at ee oe GG 8 ah ee ne Ge 74 LeEXCMP waa e Gok ow AO Ge ea 75 TOXSODE s fig ost hoes He els BS 181 LIDIA 2454448485585 64 4SG9 2 101 DIFE e e p aca o Wee Be ke a 80 82 LEDO cp 2 E ea ds 82 Limit s e saose w Ge RE Ge 43 linde 2025 Gee ee ae ess 169 171 lindep 2 e eee 2 bal le ah Pe ae ws 171 line editor avance 66 linear dependence sil DANES eno Bessy a hag Oe he Bre we we a a 56 LISO 2 eos See eee Gs ese ee og ee 53 stos cra rs a eo 20 34 List ocres a ed 75 LISTE cir Bb koe aon Be be 171 TiStINSELt 4 se eow ta GA oe aS Gs 171 listkill s re ee ewe a ee ee ew 172 LEStPUE oss ka eee a Boe eee es 178 LTSESOBE Libras cres ea 172 LLL acre 141 146 171 173 175 177 LEL coimas e ae we 178 MTV era A alan Ege oe ae wt bl 178 lllpramint 2 bee ae eee ae 178 lllgramkerim
49. 166 EOOTMOAZ eu Sx eas Gas ww a Re es 166 Yootpadic 2k es yp Rae is 166 TOOUS ae misce oh i cate de Ses 111 121 166 TOO SOL 2 4 4 54 24k aa we OS 150 EOOBSOLA s r s org ee o e i 166 round 2 a 143 round do 143 163 o PA we ee Be 84 TOW VEC OT gt s soe sasso Ee iak 20 34 S scalar product o o sowas sei awa e 71 Scalar type 4 acacia o a wag d 22 Schertz ei rra a ds 108 Schonsge cia bee edad aa od 165 scientific format 55 S CUre a 4 6 eck SOR wee a 58 Sello a one Gk E Dl ae ne 111 114 SCL 6 0 644 a aa a E ak Pw awe ae OS TT SOFCONVOL snas mia e s 167 seriesprecision 58 61 serlaplace 0 167 serreversSe 02 000 ae 167 SOG ed Ae Ge Salk Be aa a setintersect 0 179 setisset 0 04 180 SOtMINUS oo se we ABs ha a ee we 180 SOTA tor oe tl a ee E 84 208 setsearch 4 180 Setunion os s 6 664 hw ew ea Ys 180 Shanks SQUFOF 95 101 Shanks ss sss sesso 77 105 106 107 SHITE a le Gone eg la e be 74 SATE LM UL socia aa a a e agl 74 SIMA e us Re eed 98 109 110 193 SION errar a a ee a 75 SIGN A Pe a 75 Signat ae scs a ena bs we A ae a 179 SIQMUNITS ood s iosop Be Ee 127 simplefactmod 101 simplify 58 60 85 Sin gece koe ae eR RE He eH we 92 sindexlexsort 181 Sindexsort a s asos ea a o 181 SIND as maneni a e e a da 92 Sizebyte 26 22 ose usas 85 SIZOd BTE a ke e de
50. 2 11 4 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 11 5 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 11 6 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 the first functionality memory usage is disabled if you re not running a version compiled for debugging see Appendix A 2 11 7 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 or 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 59 UNIX 2 11 8 factor_add_primes default 0 if this is set the integer factorization machinery will call addprimes on primes factor that were difficult to find so they are automatically tried first in other factorizations If a routine is performing or has performed a factorization and is interrupted by an error or via Control C this let you
51. 27 bnrclassnolist bnf list bnf being as output by bnfinit and list being a list of moduli with units as output by ideallist or ideallistarch outputs the list of the class numbers of the corresponding ray class groups To compute a single class number bnrclassno is more efficient bnf bnfinit x 2 2 7 L ideallist bnf 100 2 H bnrclassnolist bnf L H 98 74 1 3 1 1 L 1 98 ids vector 1 i 1 i mod 1 5 98 88 O 1 14 0 0 7 98 10 O 1 The weird 1 i mod 1 is the first component of 1 i mod i e the finite part of the con ductor This is cosmetic since by construction the archimedean part is trivial I do not want to see it This tells us that the ray class groups modulo the ideals of norm 98 printed as 5 have respectively order 1 3 and 1 Indeed we may check directly bnrclassno bnf ids 2 16 3 The library syntax is bnrclassnolist bnf list 129 3 6 28 bnrconductor a a2 a3 flag 04 conductor f of the subfield of a ray class field as defined by a az az see bnr at the beginning of this section If flag 0 returns f If flag 1 returns f Cl H where Cl is the ray class group modulo f as a finite abelian group finally H is the subgroup of Cl defining the extension 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 su
52. 3 bitand 5 3 13 7 The library syntax is gbitand z y 3 2 18 bitneg z n 1 bitwise negation of an integer x truncated to n bits that is the integer n 1 y not 1 2 i 0 The special case n 1 means no truncation an infinite sequence of leading 1 is then represented as a negative number See Section 3 2 17 for the behaviour for negative arguments The library syntax is gbitneg z 3 2 19 bitnegimply x y bitwise negated imply of two integers x and y or not x gt y that is the integer Y x andnot y 2 See Section 3 2 17 for the behaviour for negative arguments The library syntax is gbitnegimply z y 3 2 20 bitor x y bitwise inclusive or of two integers x and y that is the integer So ai or yi 2 See Section 3 2 17 for the behaviour for negative arguments The library syntax is gbitor z y 79 3 2 21 bittest x n outputs the nt bit of x starting from the right i e the coefficient of 2 in the binary expansion of x The result is 0 or 1 To extract several bits at once as a vector pass a vector for n The library syntax is bittest x n where n and the result are longs 3 2 22 bitxor z y bitwise exclusive or of two integers x and y that is the integer 3 Xor yi 2 See Section 3 2 17 for the behaviour for negative arguments The library syntax is gbitxor z y 3 2 23 ceil x ceiling of x When z is in R the result is the smallest integer great
53. 3 6 101 nfisideal nf x returns 1 if x is an ideal in the number field nf 0 otherwise The library syntax is isideal x 3 6 102 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 xo a Tf y is a number field nf a much faster algorithm is used factoring x over y using nffactor Before version 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 6 103 nfisisom z 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 6 104 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 6 105 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 149 3 6 106 nfmo
54. BA A OM e A IN AE AE at ao et 120 31070 General se taa wet RPh Oo Rep Teil gt e BoE A ete on ee bed a HT 3 6 7 Class group units and the GRH 2 mok pua doa oils s doan a 122 33028 DiICentify e es e a a de a egy e da a de wel othe the a ies a oh oe 2123 3 6 9 bnfclassunit a A cre AA A A a ds cee a hee a AZ IA A A E eo Ay el A 24 3 6 11 bnidecodemodule z 4 aad a ce a e e te a e e o ZA 302 DR e sr a ds A A id ap age de Bote oe a Deen e pr 124 3 0413 bnifisintnorm da A lear AA BO A A eee ee A A a ZO 3 6 14 bnfisnorm 3 6 15 bnfissunit 3 6 16 bnfisprincipal 3 6 17 bnfisunit 3 6 18 bnfmake 3 6 19 bnfnarrow 3 6 20 bnfsignunit 3 6 21 bnfreg 3 6 22 bnfsunit 3 6 23 bnfunit 3 6 24 bnrL1 3 6 25 bnrclass 3 6 26 bnrclassno 3 6 27 bnrclassnolist 3 6 28 bnrconductor 3 6 29 bnrconductorofchar 3 6 30 bnrdisc 3 6 31 bnrdisclist 3 6 32 bnrinit 3 6 33 bnrisconductor 3 6 34 bnrisprincipal 3 6 35 bnrrootnumber 3 6 36 bnrstark 3 6 37 dirzetak 3 6 38 factornf 3 6 39 galoisexport 3 6 40 galoisfixedfield 3 6 41 galoisidentify 3 6 42 galoisinit 3 6 43 galoisisabelian 3 6 44 galoispermtopol 3 6 45 galoissubcyclo 3 6 46 galoissubfields 3 6 47 galoissubgroups 3 6 48 idealadd 3 6 49 idealaddtoone 3 6 50 idealappr 3 6 51 idealchinese 3 6 52 idealcoprime 3 6 53 id 3 6 54 id 3 6 55 id 3 6 56 id 3 6 57 id 3 6 58 id 3 6 59 id 3 6 60 id 3 6 61 id 3 6 62 id div factor hnf intersect
55. 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 3 3 If flag is equal to 1 uses another implementation of this function which is faster when x gt 1 The library syntax is kbessel nu x prec and kbessel2 nu x prec respectively 3 3 23 besseln nu x N Bessel function of index nu and argument z 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 1 hyperbolic cosine of x The library syntax is gch x 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 log2 1 gt gt 1 q n The library syntax is dilog x prec 3 3 28 eint1 x n exponential integral 2 dt x R If n is present outputs the n dimensional vector eint1 x einti 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 x prec 3 3 29 erfe x complementary error function 2 7 S e dt x R The library syntax is erfc 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 conver
56. D 4 where D is the common discriminant of x and y When zx and y do not have the same discriminant the result is undefined The current implementation is straightforward and in general slower than the generic routine since the latter take advantadge of asymptotically fast operations and careful optimizations The library syntax is nucomp z y The auxiliary function nudupl z can be used when ay 3 4 50 qfbnupow z 7 n th power of the primitive positive definite binary quadratic form x using Shanks s NUCOMP and NUDUPL algorithms see gfbnucomp in particular the final warning The library syntax is nupow z n 3 4 51 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 The library syntax is powraw x n where n must be a long integer 3 4 52 qfbprimeform z p prime binary quadratic form of discriminant x whose first coefficient is 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 p lt 0 is allowed and the distance component of the form is set equal to zero according to the current precision Note that negative definite t_QFI are not implemented The library syntax is primeform z p prec where the third variable prec is a long but is only taken into account when x
57. E 308 fast and perfect sumpos n 1 27 n 1 time 10 ms 12 2 78 E 308 also fast and perfect sumum n 1 2 2 n 1 kk sumnum precision too low in mpscl nonsense sumnum n 1 2 log 2 27 n omitted 1 1 of real type time 5 860 ms 13 1 5 E 236 slow and lost 70 decimals m intnumstep 14 9 sumnum n 1 2 log 2 2 n m 1 1 1 time 11 770 ms 5 1 9 E 305 now perfect but slow The library syntax is sumnum void E GEN eval GEN void GEN a GEN sig GEN tab long flag long prec 3 9 22 sumnumalt X a sig expr tab flag 0 numerical summation of 1 expr X the variable X taking integer values from ceiling of a to 00 where expr is assumed to be a holomorphic function for R X gt sig or sig 1 Warning This function uses the intnum routines and is orders of magnitude slower than sumalt It is only given for completeness and should not be used in practice Warning2 The expression expr must not include the 1 coefficient Thus sumalt n a 1 f n is approximately equal to sumnumalt n a sig f n sig is coded as in sumnum However for slowly decreasing functions where sig is coded as o a with a lt 1 it is not really important to indicate a In fact as for sumalt the program will often give meaningful results usually analytic continuations even for divergent series On the other hand the exponential decre
58. PARI expression including of course expressions using loops 182 Library mode Since it is easier to program directly the loops in library mode these functions are mainly useful for GP programming Using them in library mode is tricky and we will not give any details although the reader can try and figure it out by himself by checking the example given for sum On the other hand numerical routines code a function to be integrated summed etc with two parameters named GEN eval GEN void void E The second is meant to contain all auxilliary data needed by your function The first is such that eval x E returns your function evaluated at x For instance one may code the family of functions fi 2 x t via GEN f GEN x void t return gsqr gadd x GEN t One can then integrate f between a and b with the call intnum void stoi 1 amp fun a b NULL prec Since you can set E to a pointer to any struct typecast to void the above mechanism handles arbitrary functions For simple functions without extra parameters you may set E NULL and ignore that argument in your function definition Numerical integration Starting with version 2 2 9 the powerful double exponential univariate integration method is implemented in intnum and its variants Romberg integration is still available under the name intnumromb but superseded It is possible to compute numerically integrals to thousands of decimal place
59. RR RRA NR i teed ee LG DOS mats IE Ge A A A A ERA AN a Gee ces o ee e TO 3 8739 mMatsolVe ws ay eh ae Km a A a a a a A toe a a RA ETO 3 39 40 Mmatsolvemod vs 20 aie a RI a be IA AD ER ee RADA e Bate a ET 3 8 41 matsupplement s top m a fo ae A ii A Se Bote A a e TT 3 842 Mattranspose a a le e ye rs RIA A a AA a a TR SS 43minpolys oz ati ae a a a Boge BIR AS E AE ee Ge Be AAA a ee eo des EA 3 8 44 qfgaussred omo arae ROS LR Ho ie aR AO a Ge A A Se OT SB 45 qtjacobie doa ma Re te a We A a A ee a a E A a ee a a EELE SAGGI A ke ds Mie dig dy IE Sic Pee a OR Tyee Pyne ak eee E e Be OD a ae SOS S 8 47 qflllgram 3 3 lied a oa bie A amp oh Ga atid A hoe Te ate Ga oo delat Le CTS IGAS GfMINIM 2 asa 4 46 e ts dole Ahh ee late ab de a a buh ah EQ 3 8 49 Gi periection te io Gere anh we ls al ae ET Be Ge Gea E Be ee al a Ty Se GaP Ge aw ah Ser eer es LET SEGN arep lt 0 Seon SY He Se heer BS hae oe E eR BOR eel a oe ho A ee le oes AO S 8 51 qtsign 2 e A BoM Sy a A wR ee b kp a Bee Wit eo O e wy Veo Ee eda Doe GR Ss he Se Ah be Sie e hs SS he o 2 ESO AO CUE Ooh eB ee Be BE HE a Oe Se a oe ee ee Sr bee a ety es 180 Owe Sets a e a 4 2k E GG 8 eo ere RAR a hs Ue Bo ele Bk te ee L0 SO DO SO USCALC lt i gt oo Sr ch oe ASA tp A AA A OA See Bi ees o BO TACO PELUDA eG BR SON AAA AAA ar LSO IDOT TACOS aod a AA a alk o a de we a een ta a 80 DO CCE a IL A MOM A IO A e O O E A AO 1 AO VECES a ara e AAA a A Dele oe Se e atte o
60. 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 162 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 p e if p is an integer greater than 2 returns a p adic 0 of precision e In all other cases returns a power series zero with precision given by ev where v is the X adic valuation of p with respect to its main variable The library syntax is zeropadic p e for a p adic and zeroser v e for a power series zero in variable v which is a long The precision e is 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 The derivative of a scalar type is zero and the derivative of a vector or matrix is done componentwise One can use x as a shortcut if the derivative is with respect to the main variable of x By definition the main variable of a t_POLMOD is the main variable among the coefficients from its two polynomial components representative and modulus in other words assuming a polmod represents an element of R X T X the variable X is a mute var
61. a Z 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 bZk 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 137 3 6 56 idealintersect nf A B intersection of the two ideals A and B in the number field nf The result is given in HNF nf nfinit x 2 1 idealintersect nf 2 x 1 12 2 0 o 2 This function does not apply to general Z modules e g orders since its arguments are replaced by the ideals they generate The following script intersects Z modules A and B given by matrices of compatible dimensions with integer coefficients ZM_intersect A B local Ker matkerint concat A B mathnf A vecextract Ker Str HA The library syntax is idealintersect nf A B 3 6 57 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 58 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 itsel
62. 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 2 11 16 output default 1 there are four possible values 0 raw 1 prettymatriz 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 prettymatriz 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 3 3 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 the value of prettyprinter The default script tex2mail converts its input to readable two dimensional text Ind
63. 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 10 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 11 return x 0 returns from current subroutine with result x If x is omitted return the void value return no result like print 3 11 1 12 until a seq evaluates seg until a is not equal to 0 i e until a is true If a is initially not equal to 0 seg is evaluated once more generally the condition on a is tested after execution of the seq not before as in while 3 11 1 13 while a seq while a is non zero evaluates the expression sequence seq 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 204 UNIX 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 8 the latter don t get
64. ac ods a da 69 ELOOD occ nd e ai a 81 TOO sl Spee hy Pee 69 TOG wi a Gree eee Sea ee ee 202 Ford 5 54 se 2 aOR e 2a ah eh eS 143 FOEGIV Gb a E He OG eS 202 forell gee da bee Gor Rw ee we wes 203 formal integration 163 format pp kordak re e eer ak Bee 55 TOPE ns 203 TOTSTOD em aasa enea GEE ys 203 forsubgroup 161 203 TOLVES aveo dna ee a ra 204 EPM Ker 4 foe e ae 175 EEA isd ack ad ert ee Mee Sp de Roe Y 82 EU ee th cae eh ee I sy Sak Oe ess 121 fundamental units 109 121 123 124 fund nit i e ssr ua Boe oe E a Ow 109 G SaDS iia e de o es ee S 88 BACH wag ood a e Gees we a e aaa 88 SACOS Y Peak hae Bw Hae PO Oe 88 Gadd saod ee ate MS we De a ae al QUIOIS RA oe ae eA ae ae 45 Galois 125 145 146 150 151 158 204 galoisapply s s soe ece saote est s 146 gal isctonj o rrea nacta a oee nae a 147 galoisconjO a sa s s sca ae e oe eo 147 galoisc n 2 e osag dared aed ye 147 galoisconjles vii deh ew da 147 galoisexport 133 134 galoisfixedfield 133 204 galoisidentify 134 galoisinit 133 134 135 galoisisabelian 135 galoispermtopol 135 galoissubcyclo 132 135 136 166 204 galoissubfields 136 150 galoissubgroups 136 GAMMA ale Se is on So Be oe ate a a 90 Gammah sk ew es 90 gand oc fey hcl ee wa ER ae he Bo 74 Cate ay Se ote ate Gro a a ws 88 DOEN e oaa aa W
65. 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 3 3 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 s 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 a1 OK division by zero in gdiv gdivgs or ginv Starting break loop type break to go back to GP OK 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 0 explaining the problem We could have typed any gp command not only the name of a variable of course There is n
66. as fast no imaginary part c sumnum n 1 2 1 n72 1 tab 1 time 170 ms fast d sumpos n 1 1 n 2 1 time 2 700 ms slow d c time O ms 5 1 97 E 306 perfect For slowly decreasing function we must indicate singularities Xp 308 a sumnum n 1 2 n 4 3 time 9 930 ms slow because of the computation of n a zeta 4 3 time 110 ms 1 2 42 E 107 lost 200 decimals because of singularity at oo b sumnum n 1 2 4 3 n 4 3 omitted 1 of real type time 12 210 ms b zeta 4 3 3 1 05 E 300 better 4 3 Since the complex values of the function are used beware of determination problems For instance p 308 tab sumnuminit 2 3 2 time 1 870 ms sumnum n 1 2 3 2 1 n sqrt n tab 1 zeta 3 2 time 690 ms 1 1 19 E 305 fast and correct sumnum n 1 2 3 2 1 sqrt n 3 tab 1 zeta 3 2 time 730 ms 12 1 55 nonsense However sumnum n 1 2 3 2 1 n7 3 2 tab 1 zeta 3 2 time 8 990 ms 3 1 19 E 305 perfect as 1 nx yn above but much slower 194 For exponentially decreasing functions sumnum is given for completeness but one of suminf or sumpos should always be preferred If you experiment with such functions and sumnum anyway indicate the exact rate of decrease and increase m by 1 or 2 suminf n 1 2 n 1 time 10 ms 1 1 11
67. 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 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 11 3 datadir default the location of installed precomputed data the name of directory containing the optional data files For now only the galdata and elldata packages
68. 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 1 128 bnrL1 bnr2 2 returns the order and the first non zero terms of the abelian L functions Ls 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 25 bnrclass bnf ideal flag 0 this function is DEPRECATED use bnrinit bnf being as output by bnfinit the units are mandatory unless the ideal is trivial and ideal being a modulus computes the ray class group of the number field for the modulus ideal as a finite abelian group The library syntax is bnrclass0 bnf ideal flag 3 6 26 bnrclassno bnf 1 bnf being as output by bnfinit units are mandatory unless the ideal is trivial and J being a modulus computes the ray class number of the number field for the modulus J This is faster than bnrinit and should be used if only the ray class number is desired See bnrclassnolist if you need ray class numbers for all moduli less than some bound The library syntax is bnrclassno bnf I 3 6
69. bok dk Ba ace Ee 207 QUOTEN 2 a GA ae eae ER EHS HS EM n R PL E EE eee ene oy oe ae ce eee 121 2 ei oe eos veh at ed po ee 121 Fandom ow ei eas a 84 Tanks 65 44 2068 45 2b ee dak 2 175 rational function 20 33 rational number 20 21 31 raw format o 56 CAG 4h ee eG Gee ate ee oe dae tae ce et 61 read 4 67 ace Gone ay Ses 48 59 208 210 readline end Ba Ok ra Ee 66 readline s m Bb al es BA we dd 58 H6AdVeG 8s a ee a RR e 48 208 real number 20 21 30 Tale cham Saw base SRS ae HS 84 realprecision 30 58 61 real i snk tee ee ae 84 TOCA e eos eho ee eee a BG ae eee ee 167 recursion depth 43 TE6ECULSION Eo x 2 Weed wee bh a Ss 42 recursive P O 198 Tedimag sa s soe sorana a ee a 107 redrealb y a n da a onl a A es 107 redrealnod s ieas son eeoa a ak Ei a 107 reduceddiscsmith 164 reduction 106 107 reference card 0004 60 TOS oe bk eb ee eae Ree os 121 ROLULA eds edo a Gee a er 109 regulatorn soei ae ca ae 127 removeprimes 109 reorder 39 80 208 resultant as a a Seek Ae wee e 165 ESTU ws 526 cack e bee EE AS 50 204 ThOreaL go bie sn aon hp a a OK ES 107 rhorealnod 107 Riemann zeta function 42 94 TI ho 6 hi rt ee a a ee el Be BG G 120 rnfalgtobasis 154 THEDASIS espe Gow we wk eae wok dog 154 231 rmnfbasistoale s swa p Hk
70. can be used as a decimal hexadecimal converter 2 12 27 y n switches simplify on 1 or off 0 If n is explicitly given set simplify to n 2 12 28 switches the timer on or off 2 12 29 prints the time taken by the latest computation Useful when you forgot to turn on the timer 62 2 13 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 aliases 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 13 1 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 13 1 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 boo
71. defined over R or not e When F is defined over R F roots is a vector whose three components contain the roots of the right hand side of the associated Weierstrass equation y a12 2 az 2 g x 114 If the roots are all real then they are ordered by decreasing value If only one is real it is the first component Then w E omega 1 is the real period of E integral of dx 2y a 2 a3 over the connected component of the identity element of the real points of the curve and wa E omega 2 is a complex period In other words E omega forms a basis of the complex lattice defining E with r a having positive imaginary part FE etais a row vector containing the corresponding values 7 and n2 such that 17 w2 N201 ir Finally 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 H 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 parametrization E tate yields the three component vector u u q in the notations of Tate If the u component does not belong to Q it is set to zero E w is Mestre s w this is technical 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 ell
72. e Ges 154 rnfcharpoly 154 ERECONQUCLOL sr 4 be wR A A eo 154 rnfdedekind 154 mfdet 2 264 c 064608 2 BG Reo Hs t55 ED AISC 2 sh aa ee sa pa s Bees 155 TNTAISC sia Se oe eo oe ee e 155 rnfelementabstorel 155 rnfelementdown 155 rnfelementreltoabs 165 rnfelementup 155 rnfeltabstorel 155 rnteltdown cidad ais dle 155 rnfeltreltoabs 159 rnfeltup swara k ee PR es 155 rnfequation 155 rnfequationO 156 rnfhnfbasis 156 rnfidealabstorel 156 rnfidealdown 156 rnfidealhermite ss s sa osima ss 156 rnfidealhnf 156 rnfidealmul 156 rnfidealnormabs 156 rnfidealnormrel 157 rnfidealreltoabs L57 rnfidealtwoelement 157 rnfidealtwoelt 157 rnfidealup 0 157 PNTANIG e bos ee a ewe 157 InfinitalB ss ei oe eS 158 TOTIS TOS les s s e a Bowe a ld e we 158 TH TISHOXDO esca 158 159 rnfisnorminit 158 159 TOTKUMMCOO 4 2x4 8544 242 64 159 161 rntlligram vs 000200200022 159 IN NOTMEFDUD aca rad Goad do 159 rnfpolred 244 88 44 4 eek ede 5 160 rnfpolredabs 160 rnfpseudobasis 160 rnfsteinitZ 2 466444 88 6 es 160 Roblot 2 2 4 82 665 24 45 6 eee 143 TOOtUMOd Bae e wae ae e
73. e a n e a a e ae ee as 172 detint massa as da BA ws 172 diagonal as soale u a a ae ae iia Diamond 204 J12 diff A BA ew ae Aa A PA 121 difference ee 70 AILO A ee bees SE oo 90 CAE cas Ge Ea BAR Gece ow d 98 Giveuler s sosia a g RMR He ee eS OS 98 Dirichlet series 98 132 Girmul ae See Be Re ee eS 98 dirzetak os hs rara a Beles 132 disc adas den d a 6a Be 111 121 disti r tes ota a ao a won SO 144 AISSE 3 4 5 sado i SS ew ES we 164 GiviSoOrs ar 98 202 A 40 72 divs oos e roa ee dra de 193 E E E ee ee E E 67 222 E OCHO cea e a EE 55 61 BOM 2 38 oe es we ten gt a ee E 95 101 editing characters 29 GI CER ox pag Ga ot eee es ek He 173 Sinti segat goa ee exit Pte hy Gees 90 element_div 144 element_divmodpr 144 element_mul 144 element_mulmodpr 144 element_pow 144 element_powmodpr 144 element_reduce 145 element Val ls ke a a ek BS 145 Cll a aes ee Sosa oe a 45 111 114 ell ars 44 Shas SR ow aS SG 114 elladd 2 46 eee we we 111 ellak 3 5 4 serna hoe ee eS 112 Ellan e Wak is deed wd Goa E 112 Cllap 34424 2 SH wG oan Ba ee 112 CLVADO coses Wek eS a tee te 112 ellbil esas be ewe ewe ee eo 112 ellchangecurve css 95 hws 112 ellchangepoint 112 ellconvertname 112 1183 elldata 112 113 114 117 203 elleisntm
74. ee aa 57 Paulin s rosso 143 PET ce utes be tee ee ee ae ee 179 Perl fa od ai ehai na a a es ee 53 Ppermtonum e seere enra 83 Phi sis sica a e a ae es 98 Pi rr al eR a 88 Plot sf we eo Se a ee eos 197 PIO TDOR p boi shares a he a ae ak a das 197 plotclip 2 4 soseen a aa e os 197 plotcolor s scce nee sa tenias 198 plotcopy 197 198 Ploteursor s sa be cross 198 plotdliW ca s dcs hip a era Gog et 198 PLOT sold adi oe 70 198 plothrawW lt e so opos eds 199 PpLOTISIZES e iii a 199 PLOG DIE e s racte onae e ae eos 199 plotkill sssaaa s sia e 00 200 plotlines oss cd am 200 plotlinetype 200 plotmove sa sraa asd o 200 plotpointa a res sesa ie Tia sa ai 200 plotpointsize 200 plotpointtype ss sss ee ra ee 200 229 Plotrbok sa s s re vias ss ee 200 plotrecth is Siac eh cd eee 199 201 plotrecthraW x seess ee ee 201 plotrline co sose bbe a ee 505 201 plotrmove sa serina we eS Oe a 201 plotrpoint gt s iesse sa ee a bee ds 201 plotscale ica ire dodo wes et 199 201 plotstring coi o oss 47 201 Plotterm eos eco ee we e 47 A kt eal Re ees ee 97 pointch A 112 pointell i sessy ospa ssaki 118 POUNLER eck kok dae le amp eee 1H a GOace dow 70 POL toute ae ek ee a oe Be aes 76 POLCOCEE seein Sb ee ee es 80 164 polcoettO 2 gps ke We a ai Goa ia 164 polcompositum 150 polcompositumO 151 polcyelo ui aeni GS oe a ak ee Se e 164 poldegree o
75. for a different value of the variable z giving the transform preventing us to use a function such as intmellininvshort On the other hand using the exponential type of integral we obtain less accurate results but we skip expensive recomputations See intmellininvshort and intfuncinit for more explanations Note If you do not like the code 1 for 00 you are welcome to set e g oo 1 or INFINITY 1 then using 00 00 INFINITY etc will have the expected behaviour We shall now see many examples to get a feeling for what the various parameters achieve All examples below assume precision is set to 105 decimal digits We first type p 105 oo 1 for clarity 187 Apparent singularities Even if the function f x represented by expr has no singularities it may be important to define the function differently near special points For instance if f x 1 exp x 1 exp x x then fy f x de y Euler s constant Euler But f x 1 exp x 1 exp x x intnum x 0 00 1 x Euler 1 6 00 E 67 thus only correct to 76 decimal digits This is because close to 0 the function f is computed with an enormous loss of accuracy A better solution is f x 1 exp x 1 exp x x F truncate f t O t77 expansion around t 0 g x if x gt le 18 f x subst F t x note that6 18 gt 105 intnum x 0 00 1 g x Euler 2 0 E 10
76. form Note this routine is implemented using repeated independent 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 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 z The library syntax is hbessell 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 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 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 jbessel nu x prec 3 3 21 besseljh n x J Bessel function of half integral index More precisely besseljh n x computes J 1 2 2 where n must be of type integer and x is any element of C In the present version 2 3 3 this function is not very accurate when x is small The library syntax is jbesselh n x prec 89 3 3 22 besselk nu x flag 0 K
77. 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 1 with the same exceptions e for any x G there exists a unique family e1 e such that no exceptions for 1 lt i lt g we have 0 lt e lt 0 g 9 95 If present den must be a suitable value for gal 5 The library syntax is galoisinit gal den 3 6 43 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 This command also accepts subgroups returned by galoissubgroups The library syntax is galoisisabelian gal fl where fl is a C long integer 3 6 44 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 45 galoissubcyclo N H fl 0 v computes the subextension of Q C fixed by the subgroup H C Z nZ By the Kronecker Weber theorem all abelian number fie
78. 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 cp a quantity which enters in the Birch and Swinnerton Dyer conjecture The library syntax is ellglobalred E 113 3 5 13 ellheight E z flag 2 global N eron Tate height of the point z on the elliptic curve E defined over Q given by a standard minimal integral model E must be an ell as output by ellinit flagselects the algorithm used to compute the archimedean local height 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 If flag 2 use Mestre s AGM algorithm The latter is much faster than the other two both in theory converges quadratically and in practice The library syntax is ellheight0 E z flag prec Also available are ghell E z prec flag 0 and ghell2 F z prec flag 1 3 5 14 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 el1bi1 4 x 1 x 31 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 n
79. gscalsmat x n which is the same when zg is a long 3 8 23 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 x flag Also available is image x flag 0 3 8 24 matimagecompl x gives the vector of the column indices which are not extracted by the function matimage Hence the number of components of matimagecompl x plus the number of columns of matimage x is equal to the number of columns of the matrix zx The library syntax is imagecompl z 3 8 25 matindexrank 1 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 26 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 y 174 3 8 27 matinverseimage M y gives a column vector belonging to the inverse image z of the column vector or matrix y by the matrix M if one exists
80. 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 This bug is difficult to correct because the mathematical problem itself contains some amount of imprecision and it is not easy to design an intuitive generic interface for such beasts 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 an 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
81. 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 19 bnfnarrow bnf bnf being 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 narrow 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 bnrinit The library syntax is buchnarrow bnf 3 6 20 bnfsignunit bnf bnf being as output by bnfinit this computes an r x r ra 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 d4 2 i j if S i j lt 0 1 0 S concat
82. increase precision etc The function used to manipulate these values is called default 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 Xd 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 8 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 a
83. 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 Note that all the components are exact integral or rational except for the roots in v 5 Note also that 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 13 bnfisintnorm bnf x computes a complete system of solutions 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 125 3 6 14 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
84. library syntax is idealadd nf x y 3 6 49 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 e x i for 1 lt i lt k and ree eli 1 The library syntax is idealaddtooneO nf x y where an omitted y is coded as NULL 136 3 6 50 idealappr 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 p such that the valuation of x at gp is non zero we have vola Ve x and vela gt 0 for all other p 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 such that for all prime ideals p occurring in x vp 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 nf x flag 3 6 51 idealchinese nf x y x being a prime ideal factorization i e a 2 by 2 matrix whose first column contain prime
85. limit lim Z p Z via its finite quotients and Q as its field of fractions Like real numbers the codewords contain an exponent giving the p adic valuation of the number and also the information on the precision of the number which is 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 0 mod 4 and w 1 Vd 2 when d 1mod 4 where d is the discriminant of a quadratic order Complex numbers correspond to the important special case w y 1 Complex numbers are partially recursive the two components a and b can be of type t_INT t_REAL t_INTMOD t_FRAC or t_PADIC and can be mixed subject to the limitations mentioned above For example a bi with a and b p adic is in Q 1 but this is equal to Qp when p 1 mod 4 hence we must exclude these p when one explicitly uses a complex p adic type Quadratic numbers are more restricted their components may be as above except that t_REAL is not allowed 21 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 In the present version 2 3 3 of PARI there are bugs in the
86. long as one pays proper attention to variable scope Here is an example used to retrieve the coefficient array of a multivari ate polynomial a non trivial task due to PARI s unsophisticated representation for those objects coeffs P nbvar if type P t_POL for i 1 nbvar P P return P vector poldegree P 1 i coeffs polcoeff P i 1 nbvar 1 If P is a polynomial in k variables show that after the assignment v coeffs P k the coefficient of xi p in P is given by v n1 1 ngt1 The operating system automatically limits the recursion depth dive n if n dive n 1 dive 5000 deep recursion if n dive n 1 There is no way to increase the recursion limit which may be different on your machine from within gp To increase it before launching gp you can use ulimit or limit depending on your shell and raise the process available stack space increase stacksize 2 6 4 Function which take functions as parameters Use the following trick neat example due to Bill Daly calc f x eval Str f x If you call this with calc sin 1 it will return sin 1 evaluated 2 6 5 Defining functions within a function The first idea init x add y x y mul y x y does not work since in the construction f seq the function body contains everything until the end of the expression Hence executing init defines the wrong function add The way out is to
87. looks much nicer than the built in beautified format output 2 2 11 20 primelimit default 500k 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 The maximal value is a little less than 2 2 resp 2 4 on a 32 bit resp 64 bit machine 57 EMACS 2 11 21 prompt default a string that will be printed as prompt Note that most usual escape sequences are available there e 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 H 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 11 1 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 results without ambiguity It is not a trivial problem to ad
88. matapascal s s vacs a ao se 175 patranka la a a we 175 Matrice 2644 gwa Se os Ske oe SS 175 Matrix v4 6 ea E ea 20 21 34 48 Matrix ih keer ded Geko Gdok be e 175 MATL XAZ is aa Aw Gg Gack ww g 175 atriz sse Ga es hoe a a e 176 MA SIZO skk se goal Se ead gb go st cog he Hd 176 Matent osease 176 MatsnfO se sa kaspe tad ea nekt 176 MAtsOLVe e eo ea eS e dc S 176 mMatsolvemod 176 matsolvemod e peace ds bea ws 177 matsupplement 177 Mattranspose 177 MAX erpe Soaehs oy em wb es dng eek 75 member functions 45 111 121 MIO ni e A ee as 75 Minideal doses e Gs madres 140 MINIM s es 48 eee ee ea 179 MINIMO Las a E ai Rae we 179 minimal model 113 116 minimal polynomial Er MINPOLY esoo aa r Mod 2 eb ee eR ee eR ee 76 HOPE e foe we bee ae eh a ed we eS 149 MO TOVETSO aoaaa 142 MOdULar gc oo 102 MOGUIUS sora a Re eed 120 Moebius 95 104 105 moebius 95 105 Mordell Weil group 113 114 116 117 peulet s dui aa Ae dow a E 87 Mpfact sa eee ea e a 100 Mpfactr 2 ic Sees i beeen te 100 MPP se ee A wt e Ae Me cee 88 MPOS 609 grs sa ae ge a 95 101 MU gosh cane a Gee dee do Seige a 105 multivariate polynomial 42 N nbessel 90 NewWtonpoly o 142 new_galois_format 55 56 152 NOME ach ey Sse wets Boy Gee ee ee 50 204 DEXTPIIDMO 2 2 cs sesa ee ms 105 Wi aaa eee ee 45 1
89. no operator to obtain it in GP divrem 1 2 3 2 1 1 2 1 2 3 42 2 divrem Mod 2 9 3 2 kkk 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 z n is powering If the 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 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 x1 is identical to x PARI is able to rewrite the multiplication x x of two identical objects as x 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 72 As an exception if the exponent is a rational number p q and x an integer modulo a prime or a p adic number return a solution y of y x if it exists C
90. of expr X cos 272X from a to b in other words Fourier cosine transform from a to b of the function represented by expr a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfouriercos void E GEN eval GEN void GEN a GEN b GEN z GEN tab long prec 3 9 3 intfourierexp X a b z expr tab numerical integration of expr X exp 27zX from a to b in other words Fourier transform from a to b of the function represented by expr Note the minus sign a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfourierexp void E GEN eval GEN void GEN a GEN b GEN z GEN tab long prec 3 9 4 intfouriersin X a b z expr tab numerical integration of expr X sin 27zX from a to b in other words Fourier sine transform from a to b of the function represented by expr a and b are coded as in intnum and are not necessarily at infinity but if they are oscillations i e 1 a1 are forbidden The library syntax is intfouriersin void E GEN eval GEN void GEN a GEN b GEN z GEN tab long prec 3 9 5 intfuncinit X a b expr flag 0 fm 0 initalize tables for use with integral transforms such as intmellininv etc where a and b are coded as in intnum expr is the function s X to wh
91. of the en closing braces Both lists of variables are comma separated and allowed to be empty 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 ki11 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 Arguments are passed by value not as variables modifying a function s argument in the function body is allowed but does not modify its value in the calling frame In fact a copy of the actual parameter is assigned to the formal parameter when the function is called Uninitialized formal variables are 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 preceding function parameters a default value for x may involve x for j lt i A
92. ph BO AA a a ee Ge de ces oe Me ee a HT 335 27 ellminimalmodel 2 4 2 2 sran a e ae Bye a ke OO tee el an Boa a A ELG Sora 2rellorden ss Me los eae O aie eh ae eB a oe et AA A E RR aed ee Bate me ay ALO S O 23vellordinate set ie Bote ee a See A be dieters be eR Ee Bote ee ele LO 350 24 Ell pOinttozZ aoa k le we ba ae ae A RA Ge Be ek AA ee Be N 30 20 CLP OW aa ee ik all ity Se ass es Woe ER Pe ghee E in Gags We CH salty Mh sa atte cae he ag phy A ee ae A oe ge te Ge a ee STG 3 9 208 lO NO ee we RO A i A a Ae Aa e A 3 00 21 ellsigma Aa an a a a We ee a a E a ee ke PEAT 3 0128 ll SCAR Le sere ds Mh dip dy E AA EA A AE ee Be OS Bale es ALS 3 0 2 Que llSub o 0 al a adas Pe a aa E Aye dr ds e IT 3 90 30 CltaniaMa d se te a e od e A e a a e A Boor o A RR E RA ete Wl oy Sy Se GOP He ae ea ere 2 LES 30 92 WD aan E hen Se eter BS hae oe E pe BOR eek ea oe ho A Ge ee Wo hes a LS S20 A A Sy ace Ha a ake Sy a ee eh oe ol kp Boe be eh was bos Se eo a E 310 302 eClIZtOPOINGs se w per Se St che Mee ot tod MN Bo SS e a SE ad te el Sad HO i OPE a EG ede LS 3 6 Functions related to general number fields 2 119 3 6 1 Number field structures s s e a e mne ee 119 3 6 2 Algebraic numbers and ideals 6 ww ww ee 119 3 6 3 Finite abelian groups re s e eva SoS ee A ee ee ee er A 1120 3 6 4 Relative extensions 2 8 2 drar dh a a wk ee owe a a ee eee we a ara 120 3 025 Class Held theory m i e MEM
93. plotting routines enabled The PostScript output is written in a file whose name is derived from the psfile 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 you probably want to remove this file or change the value of psfile in between plots On the other hand in this manner as many plots as desired can be kept in a single file 3 10 4 And library mode 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 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 5 plot X a b expr Ymin Ymaz 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 6 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 12 y2 are opposite corners Only the part of the rectangle which is
94. preferences file or gprc to be discussed in Section 2 13 can be found gp then read its and execute the commands it contains This provides an easy way to customize gp The files argument is processed right after the gprc A copyright message then appears which includes the version number and a lot of useful technical information After the copyright the computer writes the top level help information some initial defaults and then waits after printing its prompt which is by default Whether extended on line help and line editing are available or not is indicated in this gp banner between the version number and the copyright message Consider investigating the matter with the person who installed gp if they are not Do this as well if there is no mention of the GMP kernel 2 1 2 Getting help To get help type a and hit return A menu appears describing the eleven main categories of available functions and how to get more detailed help If you now type n with 1 lt n lt 11 you 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 12 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 t
95. 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 can 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 can 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 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 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 is no risk of breaking them by installing a new pari library 216 4 1 Extra packages The following optional packages endow PARI with some extra capabilities only two packages for now e elldata This package contains the elliptic curves in John Cremona s database It is needed by the functions ellidentify ellsearch and can be used by ellinit to initialize a curve given by its standard code e galdata The default polgalois function can only compute Galois groups of polynomials of degree less or equa
96. psplothraw gt s sess esed eses sti 201 Python 24 24 64654 40 ee ey bag 53 Q QED csc haa eee sisas 77 QtbO e av aoge m y eee oe Boe de 77 qfbclassno 105 106 108 A ae oo GAs a 106 qfbcompraw 000 106 qfbhclassno 04 106 GEDNUCOMP as se a SE Ga we de ws 106 QfOnUpoW a os 6 se eee e 107 QibpowrawW bss es he e 107 qfbprimeform 107 dfbred 2i 4 i 8 544 58 844 045 107 dfbred0 p o ea saioe ss 107 qfbsolve 02 107 108 qleVval 2204 ba see va ds ede ad 163 Gteaussreds noria 177 GET x oe a ae Fe oe Sy ee a eG ne ie Gf jacobs iso oe eye eae Eyi ae 177 Qf111 cocos ramas e es 169 177 A eS Be dee oe eu 178 A see Re a el oe a a 178 qflllgram0 4 sosro e 46 Sa oa 44 s 178 QimMinim cocinas oe oS 178 179 qiminimO 0 24 6 ee ee aa 179 qfperfection v 2 2 4 179 GE aut bee eee oe St eee T GErep o Beak amp aoe ON E 179 GfirepO idii ape ss ae a ee Ges 179 OSI gas be Go Saw Be ae 179 quadclassunit 108 quadclassunitO 108 quaddiSC r och a ae ae we GP and 108 GUAZEN e 0 360 200 5 544 31 109 quadhilbert 108 109 Quadpoly s casa e ee Pinia s 109 quadpolyO 2 46 se ee eb etess 109 quadratic number 20 21 31 quadray ss sassa eee ee ee we 109 quadregula 6 8 2 eee ee 108 quadregulator 109 quadunit 000 109 UT ase adsl ur alas a Sloe Hee ke BP a AE 61 208 QUOUG 2 Bs
97. recover the prime factors already found 2 11 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 2732 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 11 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 7 request see Section 2 12 1 or M H under readline see Section 2 15 1 2 11 11 histsize default 5000 gp keeps a history of the last histsize results computed so far which you can recover using the notati
98. scalar this simply returns the absolute value of x if x is rational t_INT or t_FRAC and either 1 inexact input or x exact input otherwise the result should be identical to gcd x 0 The content of a rational function is the ratio of the contents of the numerator and the de nominator In recursive structures if a matrix or vector coefficient x appears the gcd is taken not with x but with its content content 2 4 matid 3 11 2 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 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 nmax 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 The 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
99. syntax is sqred z 3 8 45 qfjacobi x x being a real symmetric matrix this gives a vector 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 177 3 8 46 qflll x flag 0 LLL algorithm applied to the columns of the matrix x The columns of x must be linearly independent unless specified otherwise below The result is a unimodular transformation matrix T such that T is an LLL reduced basis of the lattice generated by the column vectors of x If flag 0 default the computations are done with floating point numbers using House holder matrices for orthogonalization If x has integral entries then computations are nonetheless approximate with precision varying as needed Lehmer s trick as generalized by Schnorr If flag 1 it is assumed that x is integral The computation is done entirely with integers In this case x needs not be of maximal rank but if it is not T will not be square This is slower and no more accurate than flag 0 above if x has small dimension say 100 or less If flag 2 x should be an integer matrix whose columns are linearly independent Returns a partially reduced basis for z using an unpublished algorithm by Peter Montgomery a basis is said to be partially reduced if v v gt v for any two distinct b
100. t 3 6 39 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 1 2 3 4 5 6 1 4 2 6 3 5 extern echo IdGroup s gap q 43 6 1 galoisidentify G 44 6 1 This command also accepts subgroups returned by galoissubgroups The library syntax is galoisexport gal flag 3 6 40 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 gx 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 of gal pol see Section 2 5 4 Exampl
101. te cee Be os 162 Input om me a Be a Bch eS 206 install voir ae ee A eG 48 52 206 ME CUS yoi i adan ae a cH o he Gere Sake he 183 INGER cra cogos ee es 163 INCEPE A Boa eee ca Ki 20 21 30 integral basis 143 internal longword format 62 internal representation 62 interpolating polynomial 165 226 Intersect esa ra ee a 174 intformal urgen Goa abe i 163 intfouriercos s 2 Gas a ah e GG 183 intfourierexp 184 intfouriersin 184 nttUncinit sica la ga ee e 184 intlaplaceidY ee cose wor 6 im 184 185 intmellininy escitas sida 185 intmellininvshort 185 186 intmod hoa a e E E a a a E ER h 20 o a es aa e h a a aa o 21 30 AHtnHUM sia aaa pes 183 186 190 193 intnuminit s s e se e srs e 190 intnumromb 190 191 intnumstep 191 INVES s s see Rae ee a ee RG 73 inverseimage 175 isdia g al esate a at de So 175 TSEXACEZCKO veia Rona BE Aen wee 74 isfundamental lt s sss g eee 102 Side ain a ala 149 IBpoOwer o ea cgo ee we eaa a 102 IBpring oo caa Sa Boh a Be 102 103 ispri cipalall minis o eH 126 ispseudoprime 102 103 105 ASSQUATC sso so te ie a eG 103 issquarefree 95 104 TSUN ui Geen oe Bee hE ae ge A eg 126 J Dt oS ee ea oe ee 111 JAacobE sa ta e Ee ot a he 177 MDCSSOM lkus a eae te ec ed 89 A ara gee ee de eee Ghee A 89 JELL sn os se ee sodas ER akg
102. 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 heap for quicker computations next time the function is called 3 3 5 abs x absolute value of x modulus if x is complex 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 11 1 abs 3 7 4 7x 1 12 5 7 abs 1 1 13 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 cos7 ie 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 x 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
103. the discriminant of the number field not of the polynomial x and an omit ted fa is 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 78 nfbasistoalg nf x 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 79 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 143 3 6 80 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 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 nfdiscf0 z flag fa where an omitted fa is input as NULL You can also use discf x flag 0 3 6 81 nfeltdiv nf x y given two elements x and y in nf computes their quotie
104. the enlarged L function associated to L The generators of the ray class group are needed 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 V 145 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 3 6 36 bnrstark bnr subgroup bnr being as output by bnrinit 1 finds a relative equation for the class field corresponding to the modulus in bnr and the given congruence subgroup as usual omit subgroup if you want the whole ray class group The routine uses Stark units and needs to find a suitable auxilliary conductor which may not exist when the class field is not cyclic over the base In this case bnrstark is allowed to return a vector of polynomials defining independent relative extensions whose compositum is the requested class field It was decided that it was more useful to keep the extra information thus made available hence the user has to take the compositum herself 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 Here is an example bnf bnr bnfinit y 2 3 bnrinit bnf 5 1 pol
105. the file misc color dft for an example Accepted values for this default are strings 4d1 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 co c1 c2 where co stands for foreground color c for background color and cy 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 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 Tech
106. thueinit x 13 5 thue tnf 4 41 1 11 Hence the only solution is z 1 y 1 and the result is unconditional On the other hand 168 tnf thueinit x 3 2 x7 2 3 x 17 thue tnf 15 thue Warning Non trivial conditional class group May miss solutions of the norm equation 42 1 1 This time the result is conditional All results computed using this tnf are likewise conditional except for a right hand side of 1 The library syntax is thue tnf a sol where an omitted sol is coded as NULL 3 7 36 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 If the conditional computed class group is trivial or you are only interested in the case a 1 then results are unconditional anyway So one should only use the flag is thue prints a Warning see the example there 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 Since PARI does not have a strong typing system scalars live in unspecifi
107. understood as contfrac z nmaz The library syntax is contfracO z b nmax Also available are gboundcf x nmax gef 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 ag a an 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 bo b1 6n and ao a1 n this is then considered as a generalized continued fraction and we have similarly pn qn 1 bo0la9 b1 a1 bp 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 core0 n 0 and core2 n core0 n 1 97 3 4 12 coredisc n flag if n is a non zero integer written as n df with d fundamental discriminant 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 b
108. v2 gives generators for these cyclic groups Ef must be an ell as output by ellinit E ellinit 0 0 0 1 0 elltors E 1 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 0 use Doud s algorithm bound torsion by computing E 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 an sell The library syntax is elltors0 E flag 3 5 32 ellwp E z x flag 0 Computes the value at z of the Weierstrass 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 z o z The library syntax is ellwp0 E z flag prec precdl Also available is weipell E precdl for the power series 3 5 33 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 ell
109. variable for which you supply no default value is initialized to the integer zero For instance foo x 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 41 7 3 foo 5 5 3 foo 209 NON O o list of local variables is the list of additional temporary variables 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 may also be what you want See Section 2 6 6 for details Local variables can be given a default value as the formal variables 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 KK user function f variable x declared twice 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 2 Use Once the function is
110. vectorv d 1 i 1 S add 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 bnf tufu ex tpuexpo bnf vector ex 1 i factorback vu ex i 1 ex 1 is 1 The library syntax is signunits bnf 3 6 21 bnfreg bnf bnf being as output by bnfinit computes its regulator The library syntax is regulator bnf tech prec where tech is as in bnfinit 127 3 6 22 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 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 23 bnfunit bnf bnf being as output by bnfinit outputs the vector of fundamental units of the number field This funct
111. we ee Rea es 163 A Geiss aged ae ae ae cede ae 90 BLOOR e 6 6 ce Rh o dep GE Tg che A 81 BITAG oops Sew Herd a 82 Geen asa tee eee Go Grd Ges amp 90 SEAMMA Fae HRS Sa ep Ea ee as 90 GSCds ke See va Pb Aa ee a a 102 A a E decal eae at Gece ae 74 ggprecision 84 SEE aga ta pag a ee Ea eas 74 poral hc Bk Gh GS oe we oe ae tae 86 SHO 1D op bh aise a e ie A AR Gea idee Ah 113 phell 2h eee bee ee ew Ee a wg 113 GAMA so ae ae dacs on Se ae dee at as E 82 gisfundamental 102 gisirreducible 165 SISPEIMO eyd E dre Sach gs Gis Gana a 103 gispseudoprime 103 issquare sra ey ee we es 104 gissquarefree 104 gissquarerem 104 glambdak soss soaa ee ee ee ew 162 GUGM Be Bee Gr ee Sv eG tee he ee G 105 Ele musica e Eee a ee we 74 glength 22 6 eb pee E a ss 82 glngamma 0 dl global sos acs seras Phe ee KS s 44 global 22 5 eens POR eB eS 37 44 206 GlOG ck eo hie a Grete era Gree ake eas gi El enpenga e aa a koai e des 74 paal css serias Bows 81 A A Me kas She mca gh a we cea she ak 75 EM N casara rr GRR Es 75 a A 72 emodulos ui a 76 MUD a a e See a a oG a gmull bk ee eck do e 74 ONG Scene ae ce eee cee a A 74 ONCE suis ms GA a 70 BNOMM era be be ee eee be eee ek y 82 SNOMMI2 al ora Ge hd whe GE eg 83 EDOT ey a a ee as 74 GOL oe a et a gh BSS Hack ee eee 74 DP caseras ach E ee ee ae 19 GB oe gee Agro a a Ay ee ane tsi oe ee 1
112. x and y i e such that lem x y ged x y abs a x y If y is omitted and x is a vector returns the lcm of all components of x When z and y are both given and one of them is a vector matrix type the LCM is again taken recursively on each component but in a different way If y is a vector resp matrix then the result has the same type as y and components equal to 1cm x y il resp 1cm x y i Else if x is a vector matrix the result has the same type as x and an analogous definition Note that for these types lcm is not commutative Note that 1cm v is quite different from 1 v 1 for i 1 v 1 1cm 1 v i Indeed 1cm v is a scalar but 1 may not be if one of the v i is a vector matrix The computa tion uses a divide conquer tree and should be much more efficient especially when using the GMP multiprecision kernel and more subquadratic algorithms become available y vector 10 4 i random 104 lcm v time 323 ms 1 v 1 for i 1 tv 1 1cem 1 v i time 833 ms The library syntax is glem x y 3 4 37 moebius x Moebius y function of x must be of type integer The library syntax is mu zx the result 0 or 1 is a long 3 4 38 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 rigoro
113. z n 116 3 5 26 ellrootno F p 1 E being an sell as output by ellinit this computes the local if p 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 27 ellsigma E z flag 0 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 If flag 1 computes an arbitrary determination of log o z If flag 2 3 same using the product expansion instead of theta series The library syntax is ellsigma E z flag 3 5 28 ellsearch N if N is an integer it is taken as a conductor else if N is a string it can be a curve name 11a1 a isogeny class 1la or a conductor 11 This function finds all curves in the elldata database with the given property If N is a full curve name the output format is N a1 a2 a3 a4 46 G where a a2 a3 a4 ag are the coefficients of the Weierstrass equation of the curve and G is a Z basis of the free part of the Mordell Weil group associated to the curve If N is not a full curve name the output is the list as a vector of all matching curves in the above for
114. 0 18 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 19 plotmove w x y move the virtual cursor of the rectwindow w to position x y 3 10 20 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 21 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 22 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 taken into account by the gnuplot interface 200 3 10 23 plotrbox w dx dy draw in the rectwindow w the outline of
115. 00000000 y 1 0 1 If a series with a zero leading coefficient must be inverted then as a desperation measure that coefficient is discarded and a warning is issued C 0 y 0 y72 1 C k Warning normalizing a series with O leading term 72 y 1 0 1 The last result could be construed as a bug since it is a priori impossible to deduce such a result from the input 0 may represent any sufficiently small real number But it was thought more useful to try and go on with an approximate computation than to raise an early exception In the first example above to compute A 1 B the denominator of A was converted to a power series then inverted 33 2 3 11 Rational functions types t_RFRAC as for fractions all rational functions are automat ically reduced to lowest terms 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 cre ates the binary form x 2xy 3y It is imaginary of internal type t_QFI since 2 4 x 3 8 is negative Although imaginary forms could be positive or negative definite only positive definite forms are implemented 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 distan
116. 1 The components of P should be accessed by member functions P p P e P f and P gen returns the vector p a The library syntax is primedec nf p 3 6 66 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 67 idealred nf J vdir 0 LLL reduction of the ideal J in the number field nf along the direction vdir Tf vdir is present it must be an r1 r2 component vector rl and r2 number of real and complex places of nf as usual This function finds a small a in J 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 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 T
117. 1 3 11 Programming in GP 3 11 1 Control statements A number of control statements are available in 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 seq will always denote a sequence of expressions including the empty sequence Caveat in constructs like for X a b seq the variable X is considered local to the loop leading to possibly unexpected behaviour n 5 for n 1 10 if something_nice break a5 at this point n is 5 If the sequence seq modifies the loop index then the loop is modified accordingly for n 1 10 n 2 print n 3 6 9 12 3 11 1 1 break n 1 interrupts execution of current seq 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 evaluates seq where the formal variable X goes from a to b Nothing is done if a gt b a and b must be in R 3 11 1 3 fordiv n X seq evaluates seq where the formal variable X ranges through the divisors of n see divisors which is used as a subroutine It is assumed that factor can handle n without negative exponents Inst
118. 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 z 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 15 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 16 bnfisprincipal bnf x flag 1 bnf being the number field data output by bnfinit and 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 v2 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
119. 1 5 30 200 right in 3 4s WAAMA A PSLQ p200 algdep 2 1 6 37 1 5 30 3 AN failure in 14s p250 algdep 2 1 6 3 1 5 30 3 AN right in 18s Proceeding by increments of 5 digits of accuracy algdep with default flag produces its first correct result at 205 digits and from then on a steady stream of correct results Interestingly enough our PSLQ also reliably succeeds from 205 digits on and is 5 times slower at that accuracy The above example is the testcase studied in a 2000 paper by Borwein and Lisonek Appli cations of integer relation algorithms Discrete Math 217 p 65 82 The paper conludes in the superiority of the PSLQ algorithm which either shows that PARI s implementation of PSLQ is lacking or that its LLL is extremely good The version of PARI tested there was 1 39 which succeeded reliably from precision 265 on in about 60 as much time as the current version The library syntax is algdepO z k flag prec where k and flag are longs Also available is algdep z k prec flag 0 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 I A if A is a square matrix If A is not a square matrix it returns the characteristic polynomial of the map multiplication by A if A is a scalar in particular a polmod E g charpoly I x72 1 The value of flag is only significant for matrices If flag 0 the
120. 128 Points_too 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 13 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 listz 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 14 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 199 3 10 15 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 pixels The plotting device imposes an upper bound for x and y for instance the number of pixels for screen output The
121. 18 ME sos a Bb Hose Bape ae Be Roe ene 121 nfalgtobasiS 143 nibasis 4 e464 44 44 4 ome 143 148 nibasisO ca iaw ite ead be GA 143 nfbasistoalg 143 es ss sasa aeai Rede Ss 143 NEGISG hd Wide hee Se etd Ge Se Ged 143 HEGISClO es oi dR Lee Hire dee ee 143 MEdiVEUC maduron bots we Oe BSG as 144 nidivrem Los corso rs 144 NEGLI cala ru aan ak ee ees 144 nteltdiveuoe va bb aw a a Era e ees 144 nfeltdivmodpr 144 nfeltdivrem 144 nfeltmod o 144 nfeltmul o 144 nfeltmulmodpr 144 nfeltpow bg ia Geek we ek ee ee 144 228 nfeltpowmodpr 144 nfeltreduce 145 nfeltreducemodpr 145 nteltval 3 46 6 aa wo os de dt mon fod 145 nffactor 454 i424 244 99 132 145 149 nffactormod 145 Nigaloisapply suhag ow w dare we Ss 145 nfgaloisconj 134 146 nfhermite 147 nfihermitemod sumas ey 147 MEHPLHEPE ea mak es do me ce oh Bee a 147 nfhnf wscwdanw ey wee A 147 nfhnfmod ee ius 6 on eee foe oS 147 HEANIt cido ura 118 134 147 152 153 a E ce als Se thy GG 149 ntisideal crisis ow 149 nfisinci pes Wie She Ae a 149 NfIUSISOM 444 xen ea He eS GS 149 nfkermodpr 0 149 MiMOd sas seka e a Ee eS 144 MPMOUPLAINIG s ii o ed ds 144 149 mfnewprec 2454 2 Pa bad we es 148 149 nfreducemodpr 145 NETOOUS ea ee gos ae Ee Ake EE i 150
122. 210 225 H Hadamard product 167 hashing function 45 hashtable 0 4 45 hbessell vastas es bah ees 89 hbessel2 unreal as wee 89 HCVASSUNO hera a 106 CADA ase ie o akc gee e ia e 61 A sasaa Bo da Gap ee We a ce 56 Hermite normal form 120 137 147 173 174 Nes cara 173 DEL ene tht a ta E 102 Hilbert class field 108 Hilbert matrix 173 Hilbert symbol 102 147 SS 102 History gos Fe bee wa Re ee de als 48 histsize aus sea GG wae Bs 29 56 Ya oe Bite ee oe he es ee Bee a 173 Dafal gt si e a e 173 hnfmod 0 000 eee 174 HnfmodId ane aca ee we Ae ee ew ee 174 hafeyval s 2 2 s gn fe ee EG oe eee aw a 163 Hurwitz class number 106 hyper css a ee BR GE es 90 I Dek wee ee ede eR oe ES OL BF IDESSOL ii ae HOP A SG Oe Gg 89 ideal lista eo x Se aa Be ae x 120 OCU a BOS ee ee OS ee G 119 idealadd oe st Bw se che i eee Bd 136 idealaddtoone 136 idealaddtooneO 136 1dealappr feted eb ee amp oe ate ee 136 idealapprO 2 5 pais wa a 137 idealchinese 137 idealcoprime 137 idealdiV s si lt 2 4 eti e a Laer UWdealdivO ham a ee Rone He a 137 idealdivexact 137 idealfactor s scs aii eae a ww HS aes 137 idealhermite 137 idealini ans dci a aa 137 156 idealhnfO 137 idealintersect 137 138 174 idealinv
123. 3 multiplication exact division 3 2 3 2 not 1 5 4 Euclidean quotient and remainder i e if qy r with 0 lt r lt y if x and y are polynomials assume instead that degr lt deg y and that the leading terms of r and x have the same sign then x y q x y r 36 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 1 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 O 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 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 test 1 x 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 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 while i l
124. 50 150 150 150 150 150 3 6 1 12 PolcoMpostUMi ci e RO iit a Boge E A Goa em a Beware da a LO Y 3 6 113 polgalols o eii pat A A E a A A a ap ee e TAS OLI poltred cas o A A A A A a A OS 36 115 polredabs e T Lada da ts e e Oe a e dd a E o J o OS 370 116 polredord aean A E ALA Dae a SA EA E A A A OS AR OS 3 60 11 7 poltschirnhaus 2 o iras o a a a ri g aioi Se 4 3 6 118 rnfalgtobasis s d 4 s 4 dadka e de a a o TA SLO Orrntbasise sete Be t te ica a ls a BEST Be o a A Wad oy a GOP ar de a e DA 3 6 120 rntbasistoale e ac e dat aS hee Howe a E E ee So ine amp 54 3 6 121 rnfcharpoly 4 voto foe ea ee A a a Oe A Un Mad Boe a Son OA 3 0 1 22rntconductor de nist A wee RA AE eR e Eee wy Sah AD ae ay 2 A 3 6 123 rnfdedekinds ura pr eect BE Bee a tae ele Ree o a dee bo ak EOD 36 124 nf db s s a 402 F Soa Bw a a BR ahe a aa a a goa ae 190 SrOul2orrntdise E Mein wesc aac cae te ae pics El a ea ae A sor ete cae Cas em A ee es Cae ae ete ge e ey es A IS 3 6 126 rnfeltabstorel nrs w do meta Ee aT Re A a rr OO 3 01 27 rnfeltdown zas ko 40 daa de we a aa a a a A aa OO 3 6 128 Tnfeltreltoabs amp 2 ar wos yeah why Sic Bw Soe a A AA ee A Woe Ge ee SO 320 129 TnfeltuD ra or a lalate at ad G Bye o e OO 3 6 130 rnfequatiom s a 6 edad a buk ca a a e POC SO L3olrnthnibasis ease Soh SO AR Balled E A A De aes Ad O e Ad e S 3 6 132 rnfidealabstorel E a o tri Bee bo a BO ee RA A AN a oe Te es a D6 3 6 133rnfidealdo
125. 6 perfect It is up to the user to determine constants such as the 10718 and 7 used above True singularities With true singularities the result is much worse For instance intnum x 0 1 1 sqrt x 2 1 1 92 E 59 only 59 correct decimals intnum x 0 1 2 1 1 sqrt x 2 2 0 E 105 better Oscillating functions intnum x 0 oo sin x x Pi 2 1 20 78 nonsense intnum x 0 00 1 sin x x Pi 2 2 0 004 bad intnum x O oo0 I sin x x Pi 2 3 0 E 105 perfect intnum x 0 oo I sin 2 x x Pi 2 oops wrong k 4 0 07 intnum x 0 o00 2 I sin 2 x x Pi 2 75 0 E 105 perfect intnum x 0 loo 1 sin x 3 x Pi 4 6 0 0092 bad sin x 3 3 sin x sin 3 x 4 47 0 x 17 We may use the above linearization and compute two oscillating integrals with infinite endpoints loo I and oo 3 I respectively or notice the obvious change of variable and reduce to the single integral 3 Jee sin x x dx We finish with some more complicated examples intnum x 0 oo0 I 1 cos x x 2 Pi 2 1 0 0004 bad intnum x 0 1 1 cos x x 2 intnum x 1 oo 1 x 2 intnum x 1 oo 1 cos x x 2 Pi 2 42 2 18 E 106 OK 188 intnum x 0 oo 1 sin x 3 exp x 0 3 43 5 45 E 107 MM OK intnum x 0 oo 11 sin x 3 exp x 0 3 4 1 33
126. 6 131 rnfhnfbasis bnf x given 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 0 otherwise The library syntax is rnfhnfbasis nf x 3 6 132 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 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 6 133 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 6 132 returns the ideal of K below z i e the intersection of x with K The library syntax is rnfidealdown rnf x 3 6 134 rnfidealhnf rnf x rnf being a relative number field extension L K as output by rn finit and z 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 vie
127. 81 82 82 82 82 82 83 83 83 83 83 83 84 84 84 84 85 85 85 86 86 86 86 87 87 88 88 88 88 88 88 88 88 88 89 89 89 89 89 89 89 89 89 89 90 90 90 90 OD COLA DR So RR E e e AA a So ee Sr See Tek tare GR Eas ce 2 GU 32 MOG is e atl iy SA oat ese ais ea irae ae Ca Sy Tey ss ede as he wk hep SP ans cee Tic AR Be eee de ay ee 9 B33 280eCINGT ca re ee A BO ee AA tae ae a ee D0 3002 9 CTICs La AS sh hee erat Sia Ae owed Oe oo te te Ge Oe Ae das a a ee ekg I oA er Mote 0 UNE e A BR er he gd ota oe See wa oa Dee EA ae Be ee A e 290 DL Re og tye eg e oa id at a S fue Te ale tee os ds S O 2332 Pamma de ase 68 le op doi a eb oh ald a O 33 09 GAMMA ae Ke Bee Se a RRA aE EA ae ihe BEE RSE Ge Gees aT Dae Re e ae HE A Ty Se Ge Ge Mw Pie ere 4 ay YL 33 34 hyperW 2 8 ra Gee heel BS hee Go Hea EH Ret wea oe ON Bee BE ee Spee e GOL Bro OO INGRAM e Go Se Sy ace HOR es O Ay ae A OM EO kp Bo A eh Bak hos a eo 2 gL 3 39 90 INCBAMG a fk a A wR RO A a ay O Ae O a ee OL A JOB es wo Bee ae Bee we OB BP Ba eet RO bee ar a debe Pade Sede ap Go ap a go be aOR 3 338 lngamma es apa a aha 8 4 ae Sel elas es wb lela A ek be ae ab ee tle a SOL 33239 Polyloge cc a we ath ee ee ae a EO Bh AO cay Se eee Ge ie ca ge Be ee as Cae ee ee a OS e DZ E ae E Be et ee A te a es de a oe at QZ FSAL Sines p foe a kn A te MON we ei ae
128. 9 BD Soe cael es say God a OE GH sty Se es Bde 2I Ephelp eaa eati aene a 60 Bpolvar se coros eh woa E wR moa a 86 EPOW ciara sa a E eee 73 87 EDIC o aei e Se e See e a 27 da 0 162 GPRG s mean arae e a A a e Y 64 PLEC doa dikon EG Oe da e a we 84 Ae ma a a 92 gp default ss tresana as ee 205 greal 2 fiw dad ye Boe i i ao y 84 GRH 108 122 123 125 126 158 169 PYNdtOd rasca a ee ee 84 BrOUNd fe we eee we Yok A ee we 84 gscalmat s 663 bbe be oa we os 174 pscalsmat sosai so ee db e 174 OSM sr a Be he ke ce E dz A sala oth he a wee ate Ba ae 74 PSION 4 ox gti woe Be es At 75 Ola e a e ee a E ee E E hs 92 A Be es es 71 92 A ccc he hog hh e ee ae es ee 93 ESATEN os ea ea eG eS 94 OSUD on es Pa adead eae be ae ae 71 BSUDS gt 244 v8 4 cok 4 Ree eee ae 167 gsubstpol saree ste aden eee Ge 168 SUDStVeC gt srs eee Ge Ra ea ws 168 gSUMdIVK 25 kee eee eh REE ss 110 Stal escacs owe e a a 94 PCR wee hate he dae tatu e ba ae 94 PLOCOL reido ah e he Pe ae 75 gtomat magr i p e a ii a i e a A Y 76 ECOPOL Y eien a a ew TT gtopolyrev TI ETOSOT or roaa dia ss 77 BOSE dns d onos ee nS eee ee es BLOVEC 2oiteik ee eee a Be eee a 78 gtovecsmall 78 B LACO eosa sesi ae eee eke Bs 180 trans 24 baa eg ew ee Soa be ee 177 GtEMUNE pes ge we eee Ye ee eee es 86 GVO Ahk ce Geir etm A we ts 86 A are ee tans Ge ee os tok Get E 94 eze bale mm daa rl a Oe EG 162 DZP a tea ts Ge Grek dee ae 61
129. ESR OP VOCE y aia lr e rt aa e a eo SUS 318161 vectorsmialll terio Woo ds Bee a da E act ee Rte hs Bea ay o a Ge kyl ad a a tte Seog a th po BZ a a 2 SR RAN E BS ee E A 3 9 Sums products integrals and similar functions 182 Sta aL Ge he ios dae a ee A Ae ech e Uh Ae a A Ghe OS 3 9 2 INT OULIENCOS gt iag or ts A A a ae E a Dior o 184 9 37intfourierexp asata 86 lee he ae ea BO A A A A a ts SA BORA IMtLOUTIETSIN mc A oR A ns hoe A E a Wea Team ape Gh Se a e de a ASA SD UNCI Gi a att iy e a ei Ea E y ys a E a hep SSP a cee A AR a a ee IBA 3 9 6intlaplaceinv se 2 io A A A Be AA te ae a Ge SA 3 9 7 dntmellininive e A ae at e a a O a e ae a E A o SO 3 9 8 intmellininvshort sx 0 ti A E AA Bone Ga AA ae BR A A Ga a SO 3 9 9 TN os al e ala de 3 a at a S fue Te ale tee a ds gt LSO JQ lO ATG i sa eb ta rr ts ble a e A a O 3 LIADO UM rombo ia as Seles a A A a GOP Ge age rd ra 90 3 9 12 intiumstep da A AAA E A ARA AAN Bee is a Mees OL BIOs os A RO 3914 prodeulen He ia A IA ay AE AAA e A a he A A a a SLOT 39 19 prodint gt aa rra BP a A A See Bete a a ao DOM S 9 16 Solveus s t a adale a a a a aa a ar aaa 92 TASA a A a a ea eae A e A o E ae Ses A ae as ede ge en ey ee LOL SOLLSISUMalt Aasna AA AA A AAA A A a Ee 192 SOLO SUMGIVE a ar de a a a MON we a A a 2 LS SOSUN 2 A RA NA why E Bae E A AAA nt 98 3 9 21 SUmMNUM a ar a a a a ar E Oe Te ate e a da e LS 3 9 22sumnumalt 2 0 6 6 daa a Oh ke oe da a a
130. Ee ah acco ds a e a Gea ee oh ee ira tas 4 a 1200 3510 23 plote bok eh a Tead E n Se seeds aS Bie ce Ye ee ee A eo Gated a oes OW ee e OL 3 10 24 plotrecth a uta Hm ea BU a eS a Se ke Be ae el an Boa a Ge A 201 3 10 29 plotrecthraw lt 2 5 4 i fe ot Jol Ba Jb we RO Be A Bk ee Be Be Ae a Ee Uk ee a ZOOL 3 10 26 plotrline sco di Bote ds oe ew A ee ee SE de Bate ae A ae ee Geo bot a Sager ZO 3 10 27 plotrimove asa ta A ewe ht et ie A A e A a A oe Be el e201 31028 PIOPPO aP arai A E A A Aes Doge A Eph a Ga dead Bate ght Ma oe ge 2 OL 3 10 29plotscale s Ls sta sa oe ai ae te ho Gi a E Ca sy Se pos ed a UE a he a eee A AR Be eee ge ay ae dz 201 3 10 30 plotstring a a ae ACS A A A ee A a A ee ee Se Le OT DICIR psdraWwe 2 ve Sho eat aa a wl ede e a he o a a ee a e eA eel a 201 34032 pSplothiyy ise 9s Re Se A A AA A a Pe REA e Se Bee Be HO Fee S20 3 10 33 psplothraw iS oy e Ge AC Ene Oe rd sr aR ane tle ot ele Be RL sii Prosi an Ob Sh Mato A ate ot Xe oii HE Mat atk de fe eat DA 3 111 Control statements e d eos te a BE a ne a A eR a A a ce A a A ae er a a 202 3 11 2 Specific functions used in GP programming oo oh a AAS we a ot i A AA oh a ae 208 Appendix A Installation Guide for the UNIX Feoi A a AA e a Md A A a path Che ra NA IA a A NA a EN ZN 18 Chapter 1 Overview of the PARI system 1 1 Introduction PARI GP is a specialized computer algebra system primarily aimed at number theorists but can be us
131. I data on the old stack is moved to the new one and the old stack is discarded If x 0 the size of the new stack is twice the size of the old one Although it is a function allocatemem cannot be used in loop like constructs or as part of a larger expression e g 2 allocatemem Such an attempt will raise an error The technical reason is that this routine usually moves the stack so objects from the current expression may not be correct anymore e g loop indexes The library syntax is allocatemoremem z where x is an unsigned long and the return type is void gp uses a variant which makes sure it was not called within a loop 3 11 2 4 default key val returns the default corresponding to keyword key If val is present sets the default to val first which is subject to string expansion first Typing default or Xd yields the complete default list as well as their current values See Section 2 11 for a list of available defaults and Section 2 12 for some shortcut alternatives Note that the shortcut are meant for interactive use and usually display more information than default The library syntax is gp_default key val where key and val are char 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 For instance error n n is not squarefree 3 11 2 6 extern str the string str is the name of an exter
132. LEX T i Complex numbers type t_PADIC Qp p adic numbers type t_QUAD Qlu 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_VEC a Row i e horizontal vectors type t_COL T Column i e vertical vectors type t_MAT Mangal Matrices type t_LIST TE Lists type t_STR Character strings and where the types T in recursive types can be different in each component The internal type t_VECSMALL holds vectors of small integers whose absolute value is bounded by 2 resp 263 on 32 bit resp 64 bit machines They are used internally to represent permu tations polynomials or matrices over a small finite field etc 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 20 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 mu
133. O eA ghee doe Ak BR Se ete oo 132 tapes 6 ob SA oe dak eae oS 48 49 208 truecoeff sse 6 eee ee ee ee 81 164 EEUNCATEO g sice sa goag n hee 82 85 tSChirnhals s ire aipg e a 154 DU a a He eee A a 121 LU a er Gee cx ts Gere 12 tutorial a cas fee Be RA aa 60 LP scene are Hod Saale Ada Als 48 209 bypeO 2 see eek bee ee ee we ws 210 ECOL ogg os ha we Be a ee SS 20 34 ELCOMPLEX gee es oe Se ee eS 20 31 tLFRAC unn ded bee we OA 20 31 CINT rios ee ok Ae o ea a 20 30 t_INTMOD lt lt 20 30 o A oe ee wee cee 20 34 233 GMAT rr dep snc tant ap Et Sy Ge R e 20 34 PPM so cos Headend ede d 20 31 O e e ae eee a a oe ee a ee 20 32 t POLMOD sg se kc od ks he se dd 20 32 TQ nist sls o de ee So A 20 33 BER e a o cto Ree o WS 20 33 Gi QUAD oo rra awe g 20 31 CREAG ies oia dees a dk Gow amp 20 30 HEREC ra OSHS e hed 20 33 G2SERS bie he Ge Oe ew A da 20 33 CSIR aer poe ee he Bk 20 34 VEG a Seat SA le 20 34 GoVECSMALL 4 ses Sk Hla ceca Se 20 34 U WISMT veais esa 43 PACIL a our a ak de ES 204 user defined functions 41 V valuation aoaaa aa 86 van Hoeij 99 132 Van Wijngaarden 192 variable priority 32 38 variable 4 e x He oh Oh eS 32 00 OF variable ames rasa e 86 VeC ie se eS 34 78 vecbezout lt s s macka sa boa eie a 96 vecbezoutres aoaaa aa a 96 veceintl o 4 6 642 540 68 aoud p et 90 VOCOXTYACE so pi heed whee 174 180 V
134. OCM X airis a WA a GS a 75 VGCMIM beste th wart Ger Oa Ae et ens a 75 vecsmalll acerco dais BS eS 20 Vecsmall 78 VOCSO dba loro Skee a s 181 VECSOTtO 245 68 ass 181 VeCt6Uur 2 o s bee es SEY ee Bee es 182 vecteursmall 182 VECLOL suisa caw Re Be ER OM ore G 21 VECUOr dasi s sin ao 8 a 181 182 yectorsmall 182 V CUOLV 5 4 45 lara Sd a a 182 version number 62 Vil bas e da A oe aes A 66 VVECtCUT s aoo a anu se a dc a 182 W WEDGE Nx a a a ada 94 WeberO ae id bis a ee eS 94 Weierstrass g function 118 Weierstrass equation 110 Weil curve T7 weipell 2 22 6 4 ee eee e tura in 118 WOrbert ratos aaa 94 WOTDer 1 02 048 94 WOE Der 2 z sa e a e a ee 94 WA DON es se p aooo a aE kaa o 48 210 WALLS as esas ma a amp He SOS 204 Wiles ne e un a ll ee a 112 WELLE a go aeria a a don a 48 59 62 210 Writel e cc aa Ga eia 44 8 65 4 4 4 210 WIITODIO e soos s sem me evis 209 210 WILCECEK soi ced ls e e cs OE E 210 X LO 8 264 ws air a a 81 o E 81 Cll oo mii dao ee ee 81 I eS Sore a SS a Be 81 Z Zassenhaus 100 163 ZOYeNt 2 6 a KK eb ee al we OR 191 ZOU coimas A A Bs Eee 116 TOTO ams 44 OG Se ERR EK wD 22 Z ropadi sssr s Be ee awne 163 ZOLOBOLO ii Site Bee eR amp e di 163 zeta function 42 Zotta 3 4 6 den a E a Ree ees 94 ZOLaK 2 as Mask a mk et 161 Zet
135. Read the file emacs pariemacs txt in standard distribution for details 2 15 Using 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 f you have compiled gp with the GNU readline library If you do not have Emacs available or cannot stand using it we really 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 15 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 singl
136. ST 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 34 2 3 18 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 that particular zero When the integer part of a real number x is not known exactly because the exponent of x is greater than the internal precision the real number is printed in e format 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 or polynomial of degree 0 to be sure use x or
137. 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 expression sequences In the keyword case only a very small set of words will actually be meaningful the default function is a prominent example 47 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 and readvec system all the printzzx functions all the writexrrz functions 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 8 3 Useful examples The function Str converts its arguments into strings and concatenate them Coupled with eval it is very powerful The following example creates generic matri
138. TOEXEC 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 _gprce 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 among 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 64 2 14 Using 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 autoload gp script mode pari nil t autoload gp pari nil t autoload gpman pari nil t setq auto mode alist cons gp script mode auto mode alist gp gp P which autoloads functions from pari el See also pariemacs txt These files are included in the PARI distrib
139. UDS sidra Bbw we 167 169 SUDStpoOl su as a ad ai 167 SUDSEVEC ca aera a p as Oe ek ee 168 SUTIL sa dee a a mt o e ca cms He Eee 70 SUM 2 4 04 45 94 a oa does 182 192 sumalt 25 0 Beene be bs 189 192 193 SUMALC e soor Goad oe Ge oe w dak RE Y 193 SIMA si a tn ee Se Gee Ge 110 193 SUMINE 2 6 few dcx Go amp whe A 192 193 194 SUMMUM fois aa oe cat oe de Rte ee ete a 193 195 sumumalt 4 e242 264 e ba eG 195 SUMNUMINIG s s isy ee Rd eee we 196 SUMPOS r xo wie be wed Bo 193 194 196 SUMPOSZ oc ceso 196 SUPPL cs esas rs a da Hee 177 sylvestermatrix 167 symmetric powers 167 So abe doe 48 58 206 208 T C2 ne RR A eee 121 taille ce sa be eee renens 85 GAC 2 cea as oa eee os 85 Tamagawa number 113 115 DAM er a be ee en Se 94 tanh 6 6 eh oa oe a 0 EA 94 Taniyama Weil conjecture 112 Wate 2 6 ora A A 110 bateria dos a a ee a Ga a a 111 A AA 168 Taylor series n Taylor sm aea e e oe ee a we eR a a 112 taylor esye Pee cas Be eo 168 ECHSD pues o Be ee As 167 DOTE e da a de ee Be 94 teichmuller 94 tex2mail 57 TeXstyle o ooo ooo 56 59 theta s eosa Be ge ee ee ow ee 94 thetan llk s snuro oa eiii a da e 94 Th s as e gani aea i Save E Ses 168 169 Ch einit netre ate Gi aaan Heist pas 169 time expansion 53 GOMER a SA ks ects oe ee et ee Rae Ga 59 GLACE e be gon Hn He a oe He 180 TA
140. User s Guide to PARI GP version 2 3 3 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 Copyright 2000 2006 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 2006 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 Important note How to get the latest version How to report bugs 1 2 The PARI types 1 2 1 Integers and reals 1 2 2 Intmods rational numbers P adic numbers polmods and rational functions 1 2 3 Complex numbers and quadratic numbers 1 2 4 Polynomials power series vectors matrices and lists
141. W a k ee ke ee EE ee 106 COMPTESS e ka li E 61 CONCA 244 22 2445 24 46 170 171 CONMQUELOL se ione eee ee hae eb ed 129 COM e tie a ree Gree ae de 81 CONJVEOC Li oops 81 Conrad as aee k ata te e 112 Content bX dat a ke 39 40 96 97 CONT FAC 2254446844 2 a 97 CONT TACO e 4 ne 4 be be ar 97 contr acond ges oe o ke ee a 97 continued fraction 97 Control statements 202 CONVOL Lidia a Me Oe aa RE Ee A 167 do 344 a 644 Pte Hae a L12 COPE i064 rr a a Be E 97 COLOCO i waked wie a E So ee a E g 97 GOROD aopa a e a a a we Ss 97 COLCdISC ssena regga a O 97 COrediscO s s ye ew Oe we ee we 97 COLECdISCD Wir a ee 97 GOs s dl Bie ice ce Bos ee ke ae ee ee he 90 COS 2 o dows Be aK ae ee Be a 90 G tan sas 2 koe be SEAS ES 90 CPU time a22 oS haa ee we See ee 59 CYC sed beh tg eb eed es Ee es 121 chido Ko ea cr AE ee eee es 164 D datadir 64 s soso Ss PAS eH aa 55 debug isos 55 61 debugfiles s 2 6 ssk esati 55 61 d b gl vel o rasca ad o a a a 101 deb gm m sa asos s ka we paka 55 61 decodemodule 124 decomposition into squares Leg Dedekind 90 132 154 161 162 default precision 23 default rumora 47 48 205 defaults c so sa ta eia aa 020084 53 61 degree saco oes poenae ee 164 Genom sos cocs s eee RR ee RR i 81 denominator 39 40 81 QETIY gh wt Oe RA Ee Bw G 163 Gerivpol Vs cris 163 A 172 GOG2 mea
142. WI bnt wp Ke A a a a a A A a a a A LO 3 0 134srnfidealhnt arrra aie a ee a a A A A Eee RR RARA e cd 10 30 139 Tnfidealmul e be a A A A A et A a a 256 3 6 136 rnfidealnormabs ile yk re AA A a A A a IS 316 134 1rntidealRormtel sua e o A Cay ME e Uc hy Bee A a a da VS 3 6 138 Tnfidealreltoabs os o mos e A Aa e DOF 316 139 rnfidealtwoelt ho car lr o a a a a a a Aa ee ES 30 LAO TOS a ts Bie e E Sic Pee A A E A E e Be EOS Bk ee O SG ES a AA E Oke lala td amp bs dbl acti e Ave Tete te amp bo leet ee Se OF 3 6 142 TO IS TES e dca a bus e a e A ee date Ae he late ab a a bt TOS BOLAS FO SOOTIA ie Be Gert Geet a Ew Se al ate ee A Gea wt Bee Berd ee ale ee oY Be a A E eer 2 LOS 3 6 144 rnfisnorminit arg Ao A ee A a oe a A e Wp three a ED 30145 IM UMMES is Ka as Aa ap eo he a a ep ee 99 3 6 146 rntlleram 3 E ada ARI dl a E a B59 3 6 14 7 rninormgroup o pop a Sree ho a e A a Bode a Ga ri a 60 3 6 148 rnfpolred e a 4 ide ae ar aa a a me GeO A a 160 36 140 ntpolredabsSta i ts e A a a EA A A a Se a sa PO 3 6 150 Tntpseudobasis n e oe Se me e A A AA aa a GO 36 151 rnfsteinitg S is da AA a a da a a a aa 60 3 0192 UbDegrouplsta da ion A A A O e dk O A LAA ss TL SOLO Zeta a ia e AAA a o a A sr ee OL 36 154 Z takinit ai a tr a a a a A doe ae 62 3 7 Polynomials and power series o ee 168 SS sD AETV lao al be By ae A A Sk a Sy eh a Ae et A A a A a a or 168 Bn oreval e a ae e A A a wt Soh AMI As
143. a 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 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 information 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 6 142 rnfisfree bnf x given bnf as output by bnfinit and either a polynomial x with co efficients 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 6 143 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 x in the extension L K The output is a vector x q where a Norm x 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 t
144. a A 85 smallfact ce Lio as e 99 smallinitell 115 Smith normal form 121 125 127 141 150 176 203 SMIECH gt ea ae doom a A 176 SOLVE yle e dorg o nd ob 191 somme nonor pd oened ee e 192 232 SOEC e e a a e iy eo eg 181 SQL vid laa a rth 92 SQred A LIT BQT ooa cae aaae ae ee 93 SQEL ico a a AR 110 SQPtint 2 62 has we ea HG ee ew ee 110 SQL oa ka ert ae Gv E wees ee eee 93 SUELE e o a e Bee wl 110 STECA cios ye eke a e ee 102 SUCK e Gece ces ee he en gee Go date Goda 57 61 stacksize 04 43 Stark units s s soca genr h naw 109 132 Star e e do iras e Grains Gee ee a 62 SteimitZ class a s sor 466 amp bade bee a a 160 GE caes ea a we eo we E 46 47 77 SLECHD 4 4 45 600 e ek ee e aco Be 78 Strexpand o o oo o 78 SLE LIMO 4 6 ba a aa ee ee 53 58 StrictmatCh s och aoa cp oe de ee 59 string contezt s i a 2 46 SUING iaa Gch ko ae g a 20 34 46 SUrtex us te ee ede a a Ee A 78 StrtoGEN 26 065 Ree aa ee k a ed SCUM sgis s eoii aap Be g SoMa ee SE 166 SUUIMpart ga se dopa a ne ea 166 SUDCYCLO daster A Ge we 166 SUDEEL eos ie He a 117 subfield o 150 subfields o so e e so eaeoe necra 150 SUDITOUD 2 4 62 a 8 eR a E a Re 120 S DETOUD i k a ei ow o wt de ee od 203 subgrouplist 160 203 subgrouplist0 161 BUDES iso e rl A we 165 SUDEESOX cs air ak A a 96 subresultant algorithm 102 164 165 B
145. a a IA As A OS EA AS 3 8 10 Mattrobenlls e egos ds A a A a A o a A TS SOL MathEss gt citada de le a rs A A A A A A ah oe ts TT 3 90 18 Mathlbert rata E ee ay A A es Rog A ee a Wa Team ape Gh Se a dy AS Sec LOpmath nt a Es att ey Sa os ai set ei ea E CB Sy Tel ss ede as Ee a he SP an cee aa RR Be ee de La we ee ETS 38 20 amathnimod te s are a oe A ee A a ae a Ge TA 3 90 21 mathnftmodid la a ae ws ele Baa a a a a a e el ae a A 39 22 Mati e ees je BR A ota oe Se a ae Dee a Cy ee ee Be A A SIA 3 0239 matinag Ere ria a eke Bh esl med A rr he Se Ti Se CS 3 89 24 matimagecompl a e y dadra e eh oh Ble e Slee a ke tlt ak le a at ELTA 3 89 29 Matindexvanke ss li act ase Seles a ike BET RN Get Bec a De he ee ade a Sy Se Ge Ge awh a eer A 3 8 260 IDOTIM CTSESO su ac A WS Mae So HK A MOS ee ea oS a Bee a e AA 3 8 27 matinverseimage co foe wa Oe Aea A a arid Ok A en a a Ta Bo AS 3 3 28 matisdiagonal als his a me ae E a A AA eae ee A ee e ee Om a a O 3 90 29 matke e ao di Bete sr eect BE Be a Stair ele Rw SieBot oe Gh eee bl ak O 3 8 30 matkering 2 2 4 ade 4 soa 8 4 a a Bb ahe a aa ar a ego ae ae TO 3 831 matmuldiagonalsc ana a acids gi en ee ae a aa e Ge ea ee a a ae ee ee a ee ep ee a SES 3 8 32 matmultodiagonal gt s S u Gp e a ee ROE a e A a A A 3 839 matpascal ss hos aan de a oe a a a fe oa a a a LO ODE MAA 3 vin RE A A why E A A AA A e O 3 830 MAX AI E E O ES 30230 MATIA Z Y Eras alar rr da iaa do el a ae e ETB A A A
146. a a a es 90 12 SUI E A RO ADE el AO BE E RO Sel A A O e O AAA O 392A ODO oN e A Be Se AAA A A A A ARA A eo AA is a DA amp O 3 10 Plotting functions ee 196 3 10 1 High level plotting functions a a a a a a 196 3 10 2 Low level plotting functions o 7s e teine E kee p eurie aeaa a di e 1966 3 10 3 Functions for PostScript output e saod and PE y dle e p k m ala e a 197 3104 And library mode t aia eit E A E A E od ede ey Y O AI a ee ee ae eo de OR 310 5 plot do Seg e A A A a AAA A A a e SLOT LO plotboX s a a eS Se dr a ek a oe aaa a A a a e OT SiO plotelipa pe A ds Bie ig dy IE ic Pee A A A rae AE e Be OS A ee 98 3 10 8ploteolor e Elda o ras ie a aa GS Ave dr do leet A 298 STO PIOLCOPY s esa a kus e es ad a a LS 3 LOLOSplOtCUESOL de eevee A WB al A A A A ee a da rt e TOS SLOT plotdraw s ox o E Hee eth A A AA A AA AAN A A Wo ie 198 3 10412 ploth is a A A a a ds a a a 98 3 LO 3 gt plothraw 2320224 das Bos Jb IA dl RA Eb POS 3 10 14 plothsizeS Au ur ds rr A e a A Geo Bea ae Gh a ar OO STO LS Dlotinit si dhana e Qh alan A A A a A A e e 2D 3 LO LO plotkill 12 EA AA ee A EA A a E e is eae 1200 3 10 17 plotlines sc Lo ta A A AA AA a E 200 3 10 18 plotlinetyDe s koai e drar kh AA a a a a a A a a 2 200 S LOS O plotmove gt e AA A A A e A PI AS A DON 200 3 10 20 plot pots Aia ra e AA a o ha ds o rr 2 200 3 10 21 plotpointsize 6 ie do la a She a a ds a a a o 4200 3 10 22 plotpointty pe wola er e hE he
147. a prime power polrootsmod or polrootspadic will be much faster X must be smaller than exp log B deg P log N The library syntax is zncoppersmith P N X B where an omitted B is coded as NULL 3 4 67 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 101 The library syntax is znlog z g 3 4 68 znorder z o 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 If optional parameter o is given it is assumed to be a multiple of the order used to limit the search space The library syntax is znorder x 0 where an omitted o is coded as NULL Also available is order z 3 4 69 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 70 znstar n gives the structure of the multiplicative 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 d i such that d i gt 1 and
148. a single non zero index giving a component number a negative index means we start counting from the end 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 180 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 a b c d el vecextract v 5 mask 1 la c vecextract v 4 2 1 component list 72 d b a 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 o 0 1 The library syntax is extract x y or matextract z y z 3 8 59 vecsort z k flag 0 sorts the vector x in ascending order using a mergesort 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 subcomponents of the components of x Note that mergesort is stable hence is the initial ordering of equal entries
149. a th O A a e a OS 3d factor padi nyag ie Bot ds A A a A o a A bg 168 3 20 INC OTMAL vaa i ee A A A A A a A oe e amp GA SP FOZ Pad ICAPPE ss aer E A 8 A A Meg Bote A Se ph a E rade A gh te a a oy de OA Sobel POlGOeE e ie aba AR he Bo Gi BP E Ca sy Se ss ed a UE wa he SP a cee Tic AR Be eee de Le es AIGA IM pold gree x2 as A A ON Se ee ROE A aha Be A OA SEI polcycloc 2s 8 ja SL o ae Ce eh edhe A a a IA da a A ee o AS te oe A SGA Sa LO POLIS A Bei A oh ban oe Set a oe Tee a Ge ee ee Be ee ee e AG 3 71 poldis rediuced terio ara he a e a a ro o a4 31412 polhensellift d e teu deara 4 o es e o ad a 60 31 13 pOlMterpolate 2 di aca A fa ad BET RN Get Gece a De Be e de a a da ag A eee OO 3 7 14 polisirreducible Lir E da A A A a ee ee So hee e 6D grlo pollead i A a d ikana a O Apa A a BO a a a A o 65 3 7416 pollegendre ena i Aa a a AE AID A AA a A ee O e SOS SL POlbCGIpy Doce rs BP Lk eee eB a A A Si Bete da ra a e LO 3 7 18 polresultant s s ade s a amp a a a ob abe a pof ER ee oa poit botomat k ee aa a 165 3 LO POITOOUS Lis ai te as A A E ea ae A e E Ca a BI AE ee as A a ee ay wee A 66 Sf 20polrootsmiod m oa A AAA Aa rr 166 33 21 polrootsp dic koai e 6 sean de we a aa AO wk a e one aa 66 312227 DO LS LULA AA why E A E AAA A a nt 8166 31 23 polsubcycl i aisat rr A a a ad E iat e Goto anihi e GG 37 24 polsylvestermatnix is doala e ke da a buk ca aa a 6 ZO PO LS YA reis PERE SE SO RAR LA E NE a De ae A
150. accuracy we have zeta 3 1 1 202056903159594285399738161 zeta 3 1e 20 2 1 202056903159594285401719424 zetak zetakinit x 3 1e 20 13 1 2020569031595952919 5 digits are wrong zetak zetakinit x 3 1e 28 4 25 33411749 junk e As the precision increases results become unexpectedly completely wrong p100 zetak zetakinit x 2 5 1 1 30 1 7 26691813 E 108 perfect p150 zetak zetakinit x 2 5 1 1 30 2 2 486113578 E 156 MAN perfect p200 zetak zetakinit x 2 5 1 1 30 43 4 47 E 75 more than half of the digits are wrong p250 zetak zetakinit x 2 5 1 1 30 4 1 6 E43 junk The library syntax is glambdak znf x prec or gzetak znf x prec 3 6 154 zetakinit x computes a number of initialization data concerning the number field de fined by the polynomial x so as to be able 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
151. ag where bound must be a C long integer Also available is ideallist nf bound corresponding to the case flag 4 3 6 59 ideallistarch nf list arch list is a vector of vectors of bid s as output by ideallist with flag 0 to 3 Return a vector of vectors with the same number of components as the original list The leaves give information about moduli whose finite part is as in original list in the same order and archimedean part is now arch it was originally trivial The information contained is of the same kind as was present in the input see ideallist in particular the meaning of flag bnf bnfinit x 2 2 bnf sign 72 2 0 two places at infinity L ideallist bnf 100 0 1 L 98 vector 1 i 1 i clgp 44 42 421 36 6 6 42 42 La ideallistarch bnf L 1 1 add them to the modulus 1 La 98 vector 1 i 1 i clgp 76 168 42 2 2 144 6 6 2 2 168 42 2 2 Of course the results above are obvious adding t places at infinity will add t copies of Z 2Z to the ray class group The following application is more typical L ideallist bnf 100 2 units are required now La ideallistarch bnf L 1 1 H bnrclassnolist bnf La H 98 6 25 12 2 The library syntax is ideallistarch nf list arch 139 3 6 60 ideallog nf x bid nf is a number field bid a big ideal as output by idealstar and x a non necessaril
152. akinit lt se ooe p ecra e e wie Eikas 162 Zideallog 2 baie se eed eg eee es 139 ZN ae cytes oe ee ek ee ey es ae By es 121 ZOCOPppersmith 110 Z0 lOG eco e GP be a R wes 110 ZNOrder Luisa HA HOR RR OG wo 110 ZOPYIMTOO ee ee a 110 ZUOSTAL e es ga aa Get Gat 110 234
153. al 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 This function uses a fast variant of Adleman Lenstra s algorithm The library syntax is ffinit p n v where v is a variable number 101 3 4 27 ged x y creates the greatest common divisor of x and y x and y can be of quite general types for instance both rational numbers If y is omitted and zx is a vector returns the gcd of all components of x i e this is equivalent to content x When z and y are both given and one of them is a vector matrix type the GCD is again taken recursively on each component but in a different way If y is a vector resp matrix then the result has the same type as y and components equal to gcd x y il resp gcd x y i1 Else if x is a vector matrix the result has the same type as x and an analogous definition Note that for these types gcd is not commutative 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 The library syntax is ggced z y For general polynomial inputs srged x y is also available For univariate rational p
154. ant and d is the relative discriminant considered as an element of nf nf 2 The main variable of nf must be of lower priority than that of pol see Section 2 5 4 The library syntax is rnfdiscf bnf pol 3 6 126 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 6 127 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 coefficients computes x as an element of K as a polmod assuming x 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 3 6 128 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 rnfeleme
155. ant of f The library syntax is reduceddiscsmith z 164 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 zxa ya vu x 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 xa ya v ze where e will contain an error estimate on the returned value 3 7 14 polisirreducible pol pol being a polynomial univariate in the present version 2 3 3 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 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 other
156. antor The library syntax is factcantor z p 3 4 21 factorff x p a factors the polynomial 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 contains the irreducible factors of x and the second their exponents 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 factorff z p a 100 3 4 22 factorial x or x factorial of x The expression z gives a result which is an integer while factorial x gives a real number The library syntax is mpfact x for x and mpfactr x 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 factoring engine called by most arithmetical functions flag is optional its binary digits mean 1 av
157. apt automatically this regular expression to an arbitrary prompt which can be self modifying Thus in this version 2 3 3 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 11 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 11 23 psfile default pari ps name of the default file where gp is to dump its PostScript drawings these are appended so that no previous data are lost Environment and time expansion are performed 2 11 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 11 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 digits you input For instance to get 2 decimal digits you need one
158. ase must be indicated tab is as in intnum but if used must be initialized with sumnuminit If flag is nonzero assumes that the function f to be summed is of real type i e satisfies f z f Z and then twice faster when tab is precomputed Xp 308 tab sumnuminit 2 omitted 1 abcissa o 2 alternating sums time 1 620 ms slow but done once and for all a sumnumalt n 1 2 1 n 3 n 1 tab 1 time 230 ms similar speed to sumnum b sumalt n 1 1 n n 3 n 1 time 0 ms infinitely faster a b 195 time O ms 1 1 66 E 308 perfect The library syntax is sumnumalt void E GEN eval GEN void GEN a GEN sig GEN tab long flag long prec 3 9 23 sumnuminit sig m 0 sgn 1 initialize tables for numerical summation using sumnum with sgn 1 or sumnumalt with sgn 1 sig is the abcissa of integration coded as in sumnun and m is as in intnuminit The library syntax is sumnuminit GEN sig long m long sgn long prec 3 9 24 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 For regular functions the function sumnum is in general much faster once the initializations have been made using sumnuminit If flag 1 use slig
159. asis vectors v vj This is significantly faster than flag 1 esp when one row is huge compared to the other rows Note that the resulting basis is not LLL reduced in general If flag 4 x is assumed 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 but x may have polynomial coefficients The library syntax is qflll0 z flag prec Also available are Ml x prec flag 0 Ulint x flag 1 and Ulkerim zx flag 4 3 8 47 qflllgram G flag 0 same as qf111 except that the matrix G x x is the Gram matrix of some lattice vectors x and not the coordinates of the vectors themselves In particular G must now be a square symmetric real matrix corresponding to a positive definite quadratic form The result is a unimodular transformation matrix T such that x T is an LLL reduced basis of the lattice generated by the column vectors of zx If flag 0 default the computations are done with floating point numbers using House holder matrices for orthogonalization If G has integral entries then computations are nonetheless approximate wi
160. ass 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 module 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 6 148 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 3 3 this is slower and less efficient than rnfpolredabs The library syntax is rnfpolred nf pol prec 3 6 149 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
161. bgroup 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 29 bnrconductorofchar bnr chi bnr being a big ray number field as output by bnrinit 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 30 bnrdisc al a2 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 as the default case except that if the modulus is not the exact conductor corre sponding to the L no data is computed and the result is 0 If flag 3 as case 2 but output relative data The library syntax is bnrdiscO al a2 a3 flag 3 6 31 bnrdisclist bnf bound arch bnf being as output by bnfinit with units computes a list of discriminants of Abelian extensions of the number f
162. bove expansion has been performed e time expansion the string is sent through the 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 str time at your shell prompt to get a complete list This is applied to prompt psfile and logfile For instance see http www fermigier com fermigier PariPython see http modular fas harvard edu sage see http clisp cons org x KKK 53 EMACS 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 many shells by itself stands for your home directory and user is expanded to user s home directory This is applied to all filenames 2 11 1 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
163. brary 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 33 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 3 2 34 lift x v lifts an element x 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 z 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 11 2 1ift 3 0 379 12 3 lift Mod x x 2 1 3 Xx 1ift x Mod 1 3 Mod 2 3 14 x 2 lift x Mod y y 2 1 Mod 2 3 5 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 x v where v is a long and an omitted v is coded as 1 Also available is lift x 1ift0 x 1 82 3 2 35 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 matric
164. ce 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 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 u v yields the empty 0x0 matrix The function Mat can be used to transform any object into a matrix see Chapter 3 wi 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_LI
165. ceeding if you want the functionalities they provide All of them are free e GNU MP library This provides an alternative multiprecision kernel which is faster than PARI s native one but unfortunately binary incompatible To enable detection of GMP use Con figure with gmp You should really do this if you only intend to use GP and probably also if you will use libpari unless you have backwards compatibility requirements 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 com mandtools yet another reason to dump Suntools for X Windows 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 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 211 2 Compiling the library and the GP calculator 2 1 Bas
166. ces 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 Two last examples hist 10 20 returns all history entries from 10 to 20 neatly packed into a single vector histlast 10 returns the last 10 history entries hist a b vector b a 1 i eval Str a 1 i histlast n vector n i eval Str i 1 2 9 Errors and error recovery 2 9 1 Errors There are two kind of errors syntax errors and errors produced by functions in the PARI library Both kinds are fatal to your computation gp will report the error perform some cleanup restore variables modified while evaluating the erroneous command close open files reclaim 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 kkk expected character instead of factor x 2 1 a 48 possibly enlarged to a full arrow given enough trailing context if siN x lt eps do_something k expected character instead of if siN x lt eps do_something 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
167. ciated to the various values of flag the following functions are also available werberf x prec werberf1 x prec or werberf2 x prec 94 3 3 52 zeta s For s a complex number Riemann s zeta function s X 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 For s a p adic number Kubota Leopoldt zeta function at s that is the unique continuous p adic function on the p adic integers that interpolates the values of 1 p7 C k at negative integers k such that k 1 mod p 1 resp k is odd if p is odd resp p 2 The library 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 3 3 all arithmetic functions in the narrow sense of the word Euler s totient function the Moebius function the sums over divisors or powers of divis
168. clidean quotient 71 Euclidean remainder 71 Euler product 98 105 191 Euler totient function 95 98 Euler des aio e 192 EULBE aos Eo sa be a E 87 Euler Maclaurin 94 eulerphi o 95 98 al mers g a ke Be a 48 77 163 exact ObJeCts oh aac id oak Gow he Hw 22 OP fae at eo tk ue ee Gee 90 expression sequence 28 expression s ss saso ee 28 exterided gcd sacs 2 2 4 6 ett ad 96 EXCO Foe eura ene foe D 47 58 205 CXTOTDO v gaai aon RE de ER a 205 external prettyprilt o o o o o 57 OXTTACt rocosos sima e a 181 F Lactec anto s wisc Sanr Ee e 100 factmod boi aon Ke a Sat wk a 2s 101 factor 244 saoe e 5586 25 98 99 factor0 2 2 5 6 eee ee 99 factorback 0 99 100 factorbackO 100 fACtOrCantoOr an eh ewe ee ek eed 100 223 factoredbase 143 factoredpolred 153 factorii i 6a wk we We ewe OE a A 99 100 Factorial ss wok ew we a 100 factorin da i oe de e el a 98 101 factormod 99 101 factOrnt 4a 4548 244 ba bb eS 99 132 factorpadic 2 sew es 163 factorpadic4 s sa s si e sois 163 factor_add_primes 55 TELDE e e E E e A 101 FIDO a a y Se Gs 101 TIDONACC sa s t oema RA Ee a a Ea 101 field discriminant 143 filevianie poi 5 a sinos ee ee A kw a 54 finite field ao s a e tke a ee wk Mw 32 fixed floating point format 55 flage do at
169. computations considerably The components of a bnf or sbnf are technical and never used by the casual user In fact never access a component directly always use a proper member function However for the sake of completeness and internal documentation their description is as follows We use the notations ex plained 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 9 1 lt i lt r bnf 2 contains the matrix B i e the matrix containing the expressions of the prime ideal factorbase in terms of the go It is an r x c matrix 124 bnf 3 contains the complex logarithmic embeddings of the system of fundamental units which has been found It is an r r2 x r 712 1 matrix 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 7 or bnf nf is equal to the number field data nf as would be given by nfinit 8 bnf 8 is a vector containing the classgroup bnf clgp as a finite abelian group the regulator bnf
170. conjvec z prec 3 2 29 denominator x denominator of x The meaning of this is clear when z is a rational number or function If x is an integer or a polynomial it is treated as a rational number of function respectively and the result is equal to 1 For polynomials you probably want to use denominator content x instead As for modular objects t_INTMOD and t_PADIC have denominator 1 and the denominator of a t_POLMOD is the denominator of its minimal degree polynomial representative If x is a recursive structure for instance a vector or matrix the lcm of the denominators of its components a common denominator is computed This also applies for t_COMPLEXs and t_QUADs 81 Warning multivariate objects are created according to variable priorities with possibly surprising side effects 1 y is a polynomial but y x is a rational function See Section 2 5 4 The library syntax is denom z 3 2 30 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 x returns the euclidian quotient of the numerator by the denominator The library syntax is gfloor z 3 2 31 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 32 imag z imaginary part of x When x is a quadratic number this is the coefficient of w in the canonical integral basis 1 w The li
171. d i dli 1 for i gt 2 and Z nZ I _ Z d 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 110 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 ayxy agy ot agr 047 a6 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 initalizes and returns an ell structure by default If a specific flag is added a shortened sell for small ell is returned which is much faster to compute but contains less information The following member functions are available to deal with the output of ellinit both ell and sell al a6 b2 b8 c4 c6 coefficients of the elliptic curve area volume of the complex lattice defining FE 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
172. d as NULL 3 5 22 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 3 3 this is implemented only for elliptic curves defined over Q The library syntax is orderell F z 3 5 23 ellordinate F 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 E x 3 5 24 ellpointtoz F 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 o t z 2 In other words this is the inverse function of ellztopoint More precisely if w1 w2 are the real and complex periods of E t is such that 0 lt R t lt w1 and 0 lt S t lt S w2 If E has coefficients in Qp then either Tate s u is in Qp in which case the output is a p adic number t corresponding to the point z under the Tate parametrization or only its square is in which case the output is t 1 t E must be an ell as output by ellinit The library syntax is zell E z prec 3 5 25 ellpow 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 F
173. d primes The library syntax is primes x x must be a long 105 3 4 46 qfbclassno D flag 0 ordinary class number of the quadratic order of discriminant D In the present version 2 3 3 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 10 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 Important warning For D lt 0 this function may give incorrect 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 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 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 rea
174. d width of the virtual window called a rectwindow in the sequel At all times a virtual cursor initialized at 0 0 is associated to 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 The ones sharing the prefix plotr draw relatively to the current position of the virtual cursor the others use 196 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 The rectwindows are preserved so that further drawings using the same windows at different positions or different windows can be done without extra work 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 In addition to initializing the window you may use a scaled window to avoid unnecessary conversions For this use the function plotscale b
175. defined using the above syntax you can use it like any other function see the example with fun above 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 follows zet s local n not needed and possibly confusing see below sumalt n 1 1 m 1 n s 1 27 1 8 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 42 2 6 3 Recursive functions Recursive functions can easily be written as
176. derstood 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 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 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 s
177. di O e swe da O Sf 26poltcheDit y y Sey nc Ge A A A BE ph BO ADA AN a Ge ek Do o e OT SLI PO ZABier s ia o in EO a A A a By 2 ate lke A ts ee a OA TO 3128 ISC CON VOL o nE AAA A RR AA AV ER A eel e Bae tle a 22 O 3 29 serlaplacoio de Bote a A AA A A A A Se Bote A E o O o E A RR OI SUSU A Aa ee a A A A AA E E sa a O E A Se A a a LO 337 32 SUDSEDOL 0 io A A a A A Aa e 168 33133 SUBSIVEC Lsa a a rc a ek a oe RO aa A a a GS Sirsa taylor Ae 3 A e 2 he Oh See Be Re AA E AN A e e a VOS NO CHU ao by Bi 4 ra a A amp Aye dr o 68 3 f36 thueinit ela ca A 3 8 Vectors matrices linear algebra and sets 2 2 ee eee ee ee ees 169 SS Laledep 2 8 yaar BG eke A A BE ee ea ee eR ee Ge ee Soe a S169 3 8 2 Gharpoly i ay a key ace a BOR a A a gl kp eo ds ee hn a ea a ee O DUB 2 4 DIA as das AAA dadas a e wed Beha ie oe a RRO ER A 3 8 0 listcreate oda 26 4 ar A A A a me A A a Bk ee ee OZ SILO MStIOSCTE La SB tk EA AA A A E AAA A A a E A eee AL S820 listkill eyed A te e A AA AA A a TZ S S 831istput 2 Lar tp fee Week a a VA a ee A A al OZ BLOG NStSOrt 2 LP A Ae A A A AA A A IN SAS E A BED 38 10 matadjoint it A e A a o A Talento A a RO Ooi l1 matce mpaniom ile dale e ral aa e e a e A 3012 matdotri Circo CI a Me acu oe hy ta a Es a da E ti ode tle Snipa a th en EZ 3 58 13 matdetint ox ao E Me Sheds A cee A e A A O oes DO ee a AS 3 814 matdiagonal otr tard ae da ee ss ee a a a eS 3 09 TO yMmatelgen 2723 1724 ds
178. dic operators 3 1 1 The expressions x and zx refer to monadic operators the first does nothing the second negates x The library syntax is gneg x for 2 3 1 2 The expression x y is the sum and zx y is the difference of x and y Among the prominent impossibilities are addition subtraction between a scalar type and a vector or a matrix between vector matrices of incompatible sizes and between an intmod 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 intmod 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 qfbnucomp and qfbnupor The library syntax is gmul x 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
179. divisors z 3 4 17 eulerphi z Euler s totient function of x in other words Z xZ x must be of type integer The library syntax is phi z 98 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 01 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 101 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 prime factors p lt lim or up 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 fi
180. dprinit nf pr transforms the prime ideal pr into modpr format necessary for all operations modulo pr in the number field nf The library syntax is nfmodprinit nf pr 3 6 107 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 subfields 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 Kliiners 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 6 108 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 5 4 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 lt x 3 6 109 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 compone
181. e G galoisinit x74 1 galoisfixedfield G G group 2 2 12 x72 2 Mod x73 x x 4 1 x72 y x 1 x 2 y x 1 computes the factorization zf 1 x y 2x 1 x 22 1 The library syntax is galoisfixedfield gal perm flag v where v is a variable number an omitted v being coded by 1 133 3 6 41 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 This command also accepts subgroups returned by galoissubgroups 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 3 6 42 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 s
182. e A IN E AI AO 9 34d TAdirealer a ra AA e A AAA sr amp AOS EA AA E ee Bot A E E JA LO 2 dIVISOES Per Senter da Be tae hy e G arl ar a a e a a wal othe the Seay Ah hs da AOS O o A E ee E ON ee a 98 SANS factor coes a ae ee a A en E a A As a A a OA 90 SA TO factorbacie x a ot del e IA tes E Be CPS E Ae a 99 3 4 20TactorcaldtOr e tro ds A A td ap age e a Pe a 00 A o A A A A A 0 22 factorial 23 factorint 24 factormod 25 fibonacci 26 ffinit 27 gcd 28 hilbert 29 isfundamental 30 ispower 31 isprime 32 ispseudoprime 33 issquare 34 issquarefree 35 kronecker 36 lcm 37 moebius 38 nextprime 39 numdiv 40 numbpart 41 omega 42 precprime 43 prime 44 primepi 45 primes 46 qfbclassno 47 qfbcompraw 48 qfbhclassno 49 qfbnucomp 50 qfbnupow 51 qfbpowraw 52 qfbprimeform 53 qfbred 54 qfbsolve 55 quadclassunit 56 quaddisc 57 quadhilbert 58 quadgen 59 quadpoly 60 quadray 61 quadregulator 62 quadunit 63 removeprimes 64 sigma 65 sqrtint 66 zncoppersmith 3 4 3 4 3 4 3 4 67 znlog 68 znorder 69 znprimroot 70 znstar 101 101 101 101 101 102 102 102 102 103 103 103 104 104 104 105 105 105 105 105 105 105 105 105 106 106 106 107 107 107 107 107 108 108 108 109 109 109 109 109 109 110 110 110 110 110 110 110 110 3 5 Functions related to elliptic curves
183. e Re Ae ak ae a 88 BASIN eee ee ra 88 CatLan ppe bik de poe Ba ae fe ic Be ds 88 Gath pec dan eevee dee ee daw 89 PalSS ese hye ei gie Boe a ee a ee 176 gaussmodulo ok ke a Rw 177 gaussmodulo2 177 GDEZOUE e sosa a s ee ee 96 ars A E mide ae ak E ae 19 BDITDOB fe gbitnegimply 79 EDLLOL ee ae as tam ate Bi te a er ie GDITKOM session Ok ee ae 80 Sboundcl gees ea oie ee ee ee 97 SCO A te cos Bk hd he co Be R 101 SCCll 2 daw da te OA ae ee ea we 80 BCL aw toe eho a eg ee BS ee eee ae 97 A oe ee ee eee 97 BCU ae e e Oe Sat a BR eB a A 90 ECP sc Pew woe a we ea oe BO ws 74 PCMpO vss ho ela a he aoe ae atk Ba Be 74 ECMP 6644 owe ag die PR ee eG ws 74 PEMP L gos eka ae a eae ee a 74 GCOCEE wgs tint a Gri Ga ew 81 ECON se kee eave eee a ede es 81 BEOS co cis a 90 SCOUAM a aa Be ce ok es 90 BCU OL sioc aa resan a ERR EE Gs 86 GOIN ees Bee gee ee i are ee 71 POIVONT a e wh a ae Gs 71 gdiyentres sgor e c bos nae ma nps 72 pdivround ociosos ds oe we 71 gen member function 181 GEN scocca s ras w dopad ea oe dapa 20 BENGT 2 rse koe bee EG e bee e 110 generic MAWIX s ess kg gpa be ee eyed 48 224 SODA ici eG a a es 84 GENTOStT a oak be ee OR a BO we as 77 geg bbb A acess ww Oe a e 74 geqlal 444 pae ee wR eS S 74 Setheap cir we we ee 205 gettando os base ea Bes te 84 206 SCtStack e io gia oe Boh ae st Ae ee eg 206 Settime ori SU bw aR SO ws 206 Geval copos
184. e 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 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 C h backward delete char e C h backward kill word C xd dump functions CE C v C b can be annoying when copy pasting 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 66 Note By default and are bound to the function pari matched insert which if electric paren
185. e ideal pr The library syntax is element_mulmodpr nf x y pr 144 3 6 88 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 z k 3 6 89 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 z modulo the prime ideal pr The library syntax is element_powmodpr nf x k pr 3 6 90 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 91 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 x modulo pr The library syntax is nfreducemodpr nf x pr 3 6 92 nfeltval nf zx pr given an element x in nf and a prime ideal pr in the format output by idealprimedec computes their the valuation at pr of the element x 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 93 nffactor nf x factorization of the univariate polynomial x over the number field nf given by nfinit z has coefficien
186. e 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 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 56 setunion z y union of the two sets x and y The library syntax is setunion z y 3 8 57 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 58 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 entries correspond to the component numbers to be extracted in the order specified If y is a string it can be e
187. e obtained as intnumstep However this value may be off from the optimal one and this is important since the integration time is roughly proportional to 2 One may try consecutive values of m until they give the same value up to an accepted error The endpoints a and b are coded as follows If a is not at 00 it is either coded as a scalar real or complex or as a two component vector a a where the function is assumed to have a singularity of the form x a at a where e indicates that powers of logarithms are neglected In particular a a with a gt 0 is equivalent to a If a wrong singularity exponent is used the result 186 will lose a catastrophic number of decimals for instance approximately half the number of digits will be correct if a 1 2 is omitted The endpoints of integration can be 00 which is coded as 1 or as 1 a Here a codes the behaviour of the function at 00 as follows e a 0 or no q at all ie simply 1 assumes that the function to be integrated tends to zero but not exponentially fast and not oscillating such as sin x z e a gt 0 assumes that the function tends to zero exponentially fast approximately as exp az including reasonably oscillating functions such as exp x sin x The precise choice of a while useful in extreme cases is not critical and may be off by a factor of 10 or more from the correct value e a lt 1 assumes that the
188. e present implementation cannot treat a group G if one of its p Sylow subgroups has a cyclic factor with more than 231 resp 263 elements on a 32 bit resp 64 bit architecture 3 11 1 8 forvec X v seq flag 0 Let v be an n component vector where n is arbitrary of two component vectors a b for 1 lt i lt n This routine evaluates seq where the formal variables X 1 X n go from a to b from an to bn ie X goes from a an to b1 bn with respect to the lexicographic ordering 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 3 11 1 9 if a seq1 seq2 evaluates the expression sequence seq1 if a is non zero otherwise the expression seq2 Of course seg 1 or seg2 may be empty 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 if a seq evaluates seq 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
189. e that the computations can now be very lengthy when x has many minimal vectors The library syntax is perf x 3 8 50 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 qfminim 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 3 8 51 qfsign x signature of the quadratic form represented by the symmetric matrix z The result is a two component vector The library syntax is signat 1 179 3 8 52 setintersect x y intersection of the two sets x and y The library syntax is setintersect z y 3 8 53 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 1 and this returns a long 3 8 54 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 8 55 setsearch z y flag 0 searches if y belongs to the set x If it does and flag is zero or omitted returns th
190. e to logfile and switch log mode on 2 12 14 m as a but using prettymatrix format 2 12 15 o n sets output mode to n 0 raw 1 prettymatrix 2 prettyprint 3 external prettyprint 2 12 16 p n sets realprecision to n decimal digits Prints its current value if n is omitted 2 12 17 ps n sets seriesprecision to n significant terms Prints its current value if n is omitted 2 12 18 q quits the gp session and returns to the system Shortcut for the function quit see Section 3 11 2 20 61 2 12 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 31 is read using this command it is silently loaded without cluttering the history Assuming gp figures how to decompress files on your machine this command accepts com pressed 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 themse
191. ead of n it is possible to input a factorization matrix i e the output of factor n This routine uses divisors as a subroutine then loops over the divisors In particular if n is an integer divisors are sorted by increasing size To avoid storing all divisors possibly using a lot of memory the following much slower routine loops over the divisors using essentially constant space FORDIV N local P E P factor N E P 2 P PL 1 forvec v vector E i 0 E i X factorback P v Woo 33 202 for i 1 10 5 FORDIV i time 3 445 ms for i 1 10 5 fordiv i d time 490 ms 3 11 1 4 forell E a b seq evaluates seq where the formal variable E ranges through all elliptic curves of conductors from a to b Th elldata database must be installed and contain data for the specified conductors 3 11 1 5 forprime X a b seq evaluates seq where the formal variable X ranges over the prime numbers between a to b including a and b if they are prime 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 e NUN 1 3 11 1 6 forstep X a b s seq evaluates seq where the formal variable X goes from a to b in increments of s Nothing is done if s gt 0 and a gt b or if s lt 0 anda lt b
192. eal in HNF form associated to nf nfinit rnf pol idealgentoHNF nf y mathnf Mat nfalgtobasis nf y The library syntax is rnfidealreltoabs rnf x 3 6 139 rnfidealtwoelt rnf x rnf being a relative number field extension L K as output by rnfinit and z 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 3 6 140 rnfidealup rnf x rnf being a relative number field extension L K as output by rnfinit and x being an ideal of K gives the ideal 1Z 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 XX return y rnfidealup rnf as an ideal in HNF form associated to nf nfinit rnf pol idealgentoHNF nf y mathnf Mat nfalgtobasis nf y The library syntax is rnfidealup rnf x 3 6 141 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 5 4 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
193. ecifically 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 S2 2 1 1 In degree 3 43 C3 3 1 1 S3 6 1 1 In degree 4 Cy 4 1 1 Va 4 1 1 Da 8 1 1 44 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 A4 12 1 1 Gig 18 1 1 Sg 24 1 1 A4 x C2 24 1 2 S 245471 Ge 86 1 1 G 86 1 1 S4 x Co 48 1 1 As PSLa 5 60 1 1 Gro 72 1 1 Ss PGLa 5 120 1 1 As 360 1 1 Se 720 1 1 In degree 7 C7 7 1 1 D7 14 1 1 Mai 21 1 1 Mas 42 1 1 PSL2 7 PSLs 2 168 1 1 Ay 2520 1 1 7 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 Sp 2 1 1 In degree 3 A3 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
194. ed to the i th real root of K roots That ordering is not canonical but well defined once a defining polynomial for K is chosen For instance 1 1 11 is a modulus for a real quadratic field allowing ramification at any of the two places at infinity A bid or big ideal is a structure output by idealstar needed to compute in Z x 1 where I is a modulus in the above sense If is a finite abelian group as described above supplemented by technical data needed to solve discrete log problems 120 Finally we explain how to input ray number fields or bnr using class field theory These are defined by a triple al a2 a3 where the defining set al a2 a3 can have any of the following forms bnr bnr subgroup bnf module bnf module subgroup e bnf is as output by bnfinit where units are mandatory unless the modulus is trivial bnr is as output by bnrinit This is the ground field K e module is a modulus f as described above e subgroup a subgroup of the ray class group modulo f of K As described above 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 the subfield of the ray class field of K modulo f fixed by the given subgroup 3 6 6 General use All the functions which are specific to relative extensions number fields Buchmann s number fields Buchmann s number rays share the prefix rnf nf bnf bn
195. ed by anybody whose primary need is speed Although quite an amount of symbolic manipulation is possible PARI does badly compared to systems like Axiom Macsyma Maple Mathematica or Reduce on such tasks e g multivariate polynomials formal integration etc On the other hand the three main advantages of the sys tem are its speed 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 PARI is used in three different ways 1 as a library which can be called from an upper level language application for instance written in ANSI C or C 2 as a sophisticated programmable calculator named gp whose language GP contains most of the control instructions of a standard language like C 3 the compiler GP2C translates GP code to C and loads it into the gp interpreter A typical script compiled by GP2C runs 3 to 10 times faster The generated C code can be edited and optimized by hand It may also be used as a tutorial to library programming The present Chapter 1 gives an overview of the PARI GP system GP2C is distributed sepa rately and comes with its own manual Chapter 2 describes the GP programming language and the gp calculator Chapter 3 describes all routines available in the calculator Programming in library mode is explained in Chapters 4 and 5 in a separate booklet User s Guide to the PARI lib
196. ed commutative base rings It is very difficult to write robust linear algebra routines in such a general setting The developpers s choice has been to assume the base ring is a domain and work over its field of fractions If the base ring is not a domain one gets an error as soon as a non zero pivot turns out to be non invertible Some functions e g mathnf or mathnfmod specifically assume the base ring is Z 3 8 1 algdep x 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 In fact it is not even guaranteed to be irreducible One can check the closeness either by a polynomial evaluation use subst or by computing the roots of the polynomial given by algdep use polroots Internally lindep 1 x x flag is used If Lindep is not able to find a relation and returns a lower bound for the sup norm of the smallest relation algdep returns that bound instead A suitable non zero value of flag may improve on the default behaviour VAASAN LLL p200 algdep 2 1 6 37 1 5 30 wrong in 3 8s algdep 2 1 6 37 1 5 30 100 wrong in 1s algdep 2 1 6 37 1 5 30 170 right in 3 3s algdep 2 1 6 37 1 5 30 200 wrong in 2 9s 3 p250 169 algdep 2 1 6 37 1 5 30 right in 2 8s algdep 2 1 6 37
197. ed is of the essence and an object is large e g a bnf a huge matrix it should be declared global not passed as a parameter since this saves an expensive copy It is possible to declare it local and use it as a global variables from relevant subroutines but global is safer It is strongly recommended to explicitly declare all global variables at the beginning of your program and all local variable used inside a given function with the possible exception of loop indexes which are local to their loop 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 and is in general not what you want to do So be warned Coming back to our previous example zet since loop variables are not visible outside their loops the variable n need not be declared in the function protoype zet s sumalt n 1 1 n 1 n s 1 27 1 s would be a better definition One last example what is wrong with the following definition FirstPrimeDiv x local p forprime p 2 x if x p 0 break P J FirstPrimeDiv 10 1 0 44 Well the index p in the forprime loop is local to the loop and is not visible to the outside world Hence it does not survive the break statement More precisely at this point the loop index
198. ee eR a 89 Bernoulli numbers 89 94 bernreal saa lt a 244 eh ha we ad s 89 DOTOVEC 462 6 ee we ee we a 89 besselhit o e 89 besselh2 s eid a oS ho eee we 89 besseli 2 o o 89 besselj i i 4 eek ee wee eo Ree rea 89 DESSCI GM iba eeu en ge at ae He 89 besselk o 89 besseld o o 90 DestappPE ora s i ear ee Gr Ge a 95 bestapprO ces rerea a 95 B ZOUG 2 5 perete wh BAe a ees 96 DeZOUTLES u doa Gee a oes ew 96 DI na hae ES 45 120 Bad olaa va aaa as oot eee eS 121 DIBOMEBA ce eek Gh ahd e toe He ee Be oh ae ee 96 bilhell se e s wee boi Goa a ew ws 13 Linares aaa 78 binary Messa s aon anta ag en a DE E G 210 binary file i bee bak ae taas 61 208 binary flag o o 69 binary quadratic form 20 33 14 Binary Few a ake a ono a Be a we 78 binomial coefficient 96 binomial s soe caw be oe ee Bd 96 Birch and Swinnerton Dyer conjecture 113 bitand cosmos ge ed ee Bd 74 79 DICHE sara eer he aa e 79 bitnegimply 0 T9 DICO aha aa Se 74 79 DICCESG Es e bo we de 73 79 bitwise and 74 79 bitwise exclusive or 79 bitwise inclusive or 79 bitwise negation 79 DILWISESOP apre is a Awe 74 DICXOL bie do e eee E 79 ONE oe a co es ida 45 119 BAT Culata Bo eee See da S 121 a e y ea oh ahs Wa oes we at Gs eG 123 BHECLASSUNIE aor eirg Gee Se ee G Gee S 123 bn
199. ee nf galoisconj This is a prerequisite for most of the galoisxxx routines For instance P x76 108 G galoisinit P L galoissubgroups G vector L i galoisisabelian L i 1 vector L i galoisidentify L i The output is an 8 component vector gal gal 1 contains the polynomial pol gal pol 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 Sg of G expressed as a vector of permutations of L gal gen gal 8 contains the relative orders 01 0 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 01 0 ends by 2 2 3 o if G H S4 then 01 0g ends by 2 2 3 2 e else G is super solvable 134 e for 1 lt i lt g the subgroup of G generated by s1 Sg is normal with the exception of i
200. een its name and its body Error messages from the library will usually be clearer since by definition they answer a cor rectly worded query otherwise gp would have protested first Also they have more mathematical content which should be easier to grasp than a parser s logic For instance 1 0 OK division by zero 2 9 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 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 precision loss in truncation e g when trying to convert 1E1000 known to 28 digi
201. eing 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 3 4 14 direuler p a b expr c computes the Dirichlet series associated to 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 local factor expr p The series is output as a vector of coefficients If c is present output only the first c coefficients in the series The following command computes the sigma function associated to s s 1 direuler p 2 10 1 1 X 1 p X 11 1 3 4 7 6 12 8 15 13 18 The library syntax is direuler void E GEN eval GEN void GEN a GEN b 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 divisors of x The factorization of x as output by factor can be used instead By definition these divisors are the products of the irreducible factors of n as produced by factor n raised to appropriate powers no negative exponent may occur in the factorization If n is an integer they are the positive divisors in increasing order The library syntax is
202. elds factornf for the algorithms over number fields Over 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 2xt 1 1 t 2 1 content t 2 5 2 t 1 2 1 2 See also factornf and nffactor The library syntax is factorO z 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 The integer 1 corresponds to the empty factorization If the last argument is of number field type e g created by nfinit 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 ca
203. elow As long as this function is not called the scaling 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 3 3 all plotting functions both low and high level are written for the X11 window system hence also for GUP s based on X11 such as Openwindows and Motif only though little code remains which is actually platform dependent It is also possible to compile gp with either of the Qt or FLTK graphical libraries A Suntools Sunview Macintosh and an Atari Gem port were provided for previous versions but are now obsolete Under X11 the physical window opened by plotdraw or any of the ploth functions is completely separated from gp technically a fork is done and the non graphical memory is imme diately 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 3 10 3 Functions for PostScript output in the same way that printtex allows you to have a TEX output corresponding 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
204. ence 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 13 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 other 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 z y gand z y and gnot x 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 y where x and y must be in R and gequal 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 Comparisons to special constants are implemented and should be used instead of gequal gempO x x 0 gemp1 x x 1 and gemp_1 z Note that gcmp0 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 ise
205. ependently 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 b respectively see below 2 11 17 parisize default 4M resp 8M on a 32 bit resp 64 bit machine 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 11 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 11 19 prettyprinter default tex2mail TeX noindent ragged by_par the name of an external prettyprinter to use when output is 3 alternate prettyprinter Note that the default tex2mail
206. er 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 24 centerlift x v lifts an element x a mod n of Z nZ to a in Z and similarly lifts a polmod to a polynomial This is the same as 1ift 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 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 25 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 26 components of a PARI object There are essentially three ways to extract the component
207. eric 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 across a set of routines it would break backward compatibility The only advantage of explicit numeric values is that they are fast to type so their use is 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 lt as in fun x flag 0 If flag is equal to 1 AGM use an agm formula dots which means that one can use indifferently fun x 1 or fun z 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
208. eries evaluation and plotting are included See the last sections of Chapter 3 for details And now you should really have a look at the tutorial before proceeding 25 26 EMACS Chapter 2 Specific Use of the gp Calculator 2 1 Introduction Originally gp was designed as a debugging device for the PARI system library and not much thought had been given to making it user friendly The situation has changed and gp is very useful as a stand alone tool The operations and functions available in PARI and gp are described in the next chapter In the present one we describe the specific use of the gp programmable calculator If you have GNU Emacs you can work in a special Emacs shell described in Section 2 14 Specific features of this Emacs shell are indicated by an EMACS sign in the left margin 2 1 1 Startup To start the 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 11 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 If a
209. ermine whether L K is Galois The library syntax is rnfisnorminit pol polrel flag 3 6 145 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 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 6 146 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 159 3 6 147 rnfnormgroup bnr pol bnr being a big ray cl
210. es 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 36 norml2 square of the L norm of x More precisely if x is a scalar norm12 1 is defined to be x conj a If x is a row or column vector or a matrix norm12 x is defined recursively as norm12 x where x run through the components of x In particular this yields the usual YN xi resp Y x z 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 eE norml2 1 2 3 4 5 6 recursively defined 4 91 The library syntax is gnorml2 z 3 2 37 numerator x numerator of x The meaning of this is clear when x is a rational number or function If x is an integer or a polynomial it is treated as a rational number of function respectively and the result is x itself For polynomials you probably want to use numerator content x instead In other cases numerator x is defined to be denominator x x This is the case when z is a vector or a matrix but also for t_COMPLEX or t_QUAD In particular since a t_PADIC or t_INTMOD has denominator 1 its numerator is itself Warning multivariate objects are created according to variable priorities with possibly surprising side effects
211. es 108 Buchmann 122 123 124 141 163 Buchmann McCurley 108 buchnarrow 26 e 0 246 we wae 127 buchreal a aos ew eee ee 2 108 C Cantor Zassenhaus 100 Garact 44 44 484 as i ak 170 CATA sss ee eee A tee ee das 170 Carhe SS e omiso week de SRS ew 170 Geil fog 4 Be Bon ee 2 a he Qo bee a 80 c terlift y i erosa g whe e 80 221 COnterllELO scrisori bees a E es 80 certifybuchall is s sisie recia 123 changevar sos sos a Ye we a A 39 80 character string ooo 34 character iaa we E ES 120 character s s srona ag o say 128 130 131 characteristic polynomial 170 charpoly miseria was 170 CharpolyO asias aE ek ee 170 Chebyshev 2 167 ChineSe so es so ee A ee nawa 96 Ghinesed sis saiat og a 96 classgrouponly 124 ClaSSNO 0 0 10 oe ior a boine be 106 ClASSNO 4 eos FR ROR Re ee 106 CUB os Sh qe Bet te st cet abe ied 121 CLISP is momo e we e e a 52 cmdtool soria a A 58 Code words ss 208 bee E oe ao a ea e 80 COdIff soo e e Se ee oe bee Cas 121 CO ama Aaa A ae A ee eS 75 COLES oa a atch ld ae ata RPS 54 column vector 20 34 comparison operators 74 compatibles io unica ook td Ge BS a 54 completion 4 66 complex number 20 21 31 COMPO speke rane E do 80 COMPONENt a a s a poa aa ee A 80 components si hin o me se a aa 80 COMPOSICI N so g e o oE o ae e 106 compositum 0 150 COMPLA
212. euristic 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 20 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 21 mathnfmodid z d outputs the upper triangular Hermite normal form of x concate nated with d times the identity matrix Assumes that x has integer entries The library syntax is hnfmodid z d 3 8 22 matid n creates the n x n identity matrix The library syntax is matid 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
213. 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 will not protest if you skip this Nothing prevents you from modifying the help of built in PARI functions But if you do we would 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 correspond 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 3 11 2 3 allocatemem x 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 16x x 16 bytes is allocated all the PAR
214. f a row vector containing the information about ideals of a given norm in no specific order depending on the value of flag The possible values of flag are 0 give the bid associated to the ideals without generators 1 as 0 but include the generators in the bid 2 in this case nf must be a bnf with units Each component is of the form bid U where bid is as case O and U is a vector of discrete logarithms of the units More precisely it gives the ideallogs with respect to bid of bnf tufu This structure is technical and only meant to be used in conjunction with bnrclassnolist or bnrdisclist 3 as 2 but include the generators in the bid 4 give only the HNF of the ideal nf nfinit x 2 1 L ideallist nf 100 L 1 3 1 0 0 11 A single ideal of norm 1 L 65 74 4 There are 4 ideals of norm 4 in Zii If one wants more information one could do instead nf nfinit x 2 1 138 L ideallist nf 100 0 1 L 25 vector 1 i 1 i clgp 43 20 20 16 4 4 20 20 1 1 mod 4 25 18 0 1 1 1 2 mod 5 5 0 0 5 O 1 3 mod 46 25 7 0 11 El where we ask for the structures of the Z i I for all three ideals of norm 25 In fact for all moduli with finite part of norm 25 and trivial archimedean part as the last 3 commands show See ideallistarch to treat general moduli The library syntax is ideallistO nf bound fl
215. f poly nomials 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 z y xa Zyl 2 but how do we know 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 obtained via library mode would produce an invalid 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 bef
216. 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 154 3 6 123 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 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 6 124 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 6 125 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 discrimin
217. function tends to 0 slowly like z Here it is essential to give the correct a if possible but on the other hand a lt 2 is equivalent to 0 in other words to no a at all The last two codes are reserved for oscillating functions Let k gt 0 real and g x a nonoscil lating function tending to 0 then e a kl assumes that the function behaves like cos kx g x e a kI assumes that the function behaves like sin kx g x Here it is critical to give the exact value of k If the oscillating part is not a pure sine or cosine one must expand it into a Fourier series use the above codings and sum the resulting contribu tions Otherwise you will get nonsense Note that cos ka and similarly sin kx means that very function and not a translated version such as cos kx a If for instance f x cos kx g x where g x tends to zero exponentially fast as exp az it is up to the user to choose between 1 a and 1 kJ but a good rule of thumb is that if the oscillations are much weaker than the exponential decrease choose 1 a otherwise choose 1 kI although the latter can reasonably be used in all cases while the former cannot To take a specific example in the inverse Mellin transform the function to be integrated is almost always exponentially decreasing times oscillating If we choose the oscillating type of integral we perhaps obtain the best results at the expense of having to recompute our functions
218. h would be used to enter a series see Section 2 3 10 secondly the prime p and finally the number of significant p adic digits k Note that it is not checked whether p is indeed prime but results are undefined if this is not the case you can work on 10 adics if you want but disasters will happen as soon as you do something non trivial like taking a square root Note that 0 25 is not the same as 0 572 you want the latter For example you can type in the 7 adic number 2x7 21 3 4x7 2 7 2 0 773 exactly as shown or equivalently as 905 7 0 773 Note that this type is available for convenience not for speed internally t_PADICs are stored as p adic units modulo some p Each elementary operation involves updating p multiplying or dividing by powers of p and a reduction mod p In particular additions are slow n 1 0 2720 for i 1 1075 n time 86 ms n Mod 1 2 20 for i 1 1075 n time 48 ms n 1 for i 1 1075 n time 38 ms 31 2 3 7 Quadratic numbers type t_QUAD first you must define the default quadratic order or field in which 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
219. he 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 158 If rnfisnorminit has determined or was told that L K is Galois and flag 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 y 2 2xy 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 3 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 6 144 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 Nx a for x in L and a in 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 det
220. he instructions in the next section Look especially for the gmp readline and X11 libraries and the perl and gunzip or zcat binaries 2 2 Compilation To compile the GP binary and build the documentation type make all To only compile the GP binary type make gp in the toplevel directory If your make program supports parallel make you can speed up the process by going to the Oxxzx directory that Configure created and doing a parallel make here for instance make j4 with GNU make It should even work from the toplevel directory 212 2 3 Basic tests 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 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 went wrong The testing utility directs you to files containing the differences between the test output and the expected results Have a look and decide for yourself if something is amiss If it looks like a bug in the Pari system we would appreciate a repor
221. he library syntax is ideallllred nf x vdir prec where an omitted vdir is coded as NULL 3 6 68 idealstar nf flag 1 outputs a bid structure necessary for computing in the finite abelian group G ZK I Here nf is a number field and J is a modulus either an 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 This bid is used in ideallog to compute discrete logarithms It also contains useful information which can be conveniently retrieved as bid mod the modulus bid clgp G as a finite abelian group bid no the cardinality of G bid cyc elementary divisors and bid gen generators If flag 1 default the result is a bid structure without generators If flag 2 as flag 1 but including generators which wastes some time 141 If flag 0 deprecated Only outputs Zx 1 as an abelian group i e as a 3 component vector h d g h is the order d is the vector of SNF cyclic components and g the corresponding generators This flag is deprecated it is in fact slightly faster to compute a true bid structure which contains much more information The library syntax is idealstar0 nf J flag 3 6 69 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 a
222. herwise The correspond respectively to e v 1 the class number e v2 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 v 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 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 cz and buchreal D flag ci c2 3 4 56 quaddisc 1 discriminant of the quadratic field Q zx where x Q The library syntax is quaddisc z 108 3 4 57 quadhilbert D pq relative equation defining the Hilbert class field of the quadratic field of discriminant D If D lt 0 uses complex multiplication Schertz s variant The technical component pq if supplied is a vector p q where p q are the prime numbers needed for the Schertz s method More precisely prime ideals above p and q should be non principal and coprime to all reduced representatives of the class group In addition if one of these ideals has order 2 in the class group they should have the same class Finally for efficiency ged 24 p 1 q 1 should be as large as possible The routine returns 0 if p q is not suitable If D gt
223. hese assumptions is not correct the behaviour of the routine is undefined e an dele is a 2 component vector the first being an ideal as above the second being a Ri Ra component row vector giving Archimedean information as complex numbers 119 3 6 3 Finite abelian groups A finite abelian group G in user readable format is given by its Smith Normal Form as a pair h d or triple h d g Here h is the cardinality of G d is the vector of elementary divisors and gi is a vector of generators In short G i lt n Z diZ gi with dn da d and d R This information can also be retrieved as G no G cyc and G gen e a character on the abelian group 9 Z d Z g is given by a row vector x a1 such that x g exp 2ia Y ain di e given such a structure a subgroup H is input as a square matrix whose column express generators of H on the given generators g Note that the absolute value of the determinant of that matrix is equal to the index G H 3 6 4 Relative extensions When defining a relative extension the base field nf must be defined by a variable having a lower priority see Section 2 5 4 than the variable defining the extension For example you may use the variable name y to define the base field and x to define the relative extension e rnf denotes a relative number field i e a data structure output by rnfinit e A relative matriz is a matrix whose entries are elements of a fixed
224. his 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 10 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 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 algorithms using scripts before actually coding them in C for the sake of speed The GP2C comp
225. hisms 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 Zk 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 96 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 4 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 restriction 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
226. htly different polynomials Sometimes faster The library syntax is sumpos void E GEN eval GEN void GEN a long prec Also available is sumpos2 with the same arguments flag 1 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 suggested ideas or submitted 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 easy to port or expand There are three types of graphic functions 3 10 1 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 3 10 2 Low level plotting functions called rectplot functions sharing the prefix plot where every drawing primitive point line box etc is specified by the user These low level functions 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 an
227. iable and the derivative is taken with respect to the main variable used in the base ring R The library syntax is deriv z v where v is a long and an omitted v is coded as 1 When x is a t_POL derivpol 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 If x is a character string eval a executes x as a GP command as if directly input from the keyboard and returns its output For convenience x is evaluated as if strictmatch was off In particular unused characters at the end of x do not prevent its evaluation eval 1a Vol 1 The library syntax is geval z 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 04 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 algo
228. ibrary syntax is hess z 3 8 18 mathilbert x x being a long creates the Hilbert matrixof order zx i e the matrix whose coefficient i j is 1 i j 1 The library syntax is mathilbert z 3 8 19 mathnf z flag 0 if x is a not necessarily square matrix with integer entries 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 x 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 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 173 If flag 4 as in case 1 above but uses a h
229. ic 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 PARI GP 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 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 executa bles 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 independent and can be done simultaneously Technical note Configure accepts many other flags besides prefix 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 you probably want to enable it 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 report genuine problems If anything should have been found and was not consider that Configure failed and follow t
230. ic 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 A number of factoring methods are used by a rather sophisticated factoring engine to name a few Shanks s SQUFOF Pollard s rho Lenstra s ECM the MPQS quadratic sieve These routines output strong pseudoprimes which may be certified by the APRCL test There is also a large package to work with algebraic number fields All the usual operations on elements ideals prime ideals etc are available More sophisticated functions are also implemented like solving Thue equations finding integral bases and discriminants of number fields computing class groups and fundamental units computing in relative number field extensions class field theory and also many functions dealing with elliptic curves over Q or over local fields 1 5 5 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 addi tion standard numerical analysis routines like univariate integration using the double exponential method real root finding when the root is bracketed polynomial interpolation infinite s
231. ic index entry A Abelian extension 154 159 ADS h hh Groh ots ey GOERS Gee a 88 ACCUTACY 2a ke wR ewe 23 ACOs cups a Be es 88 ACOSH Bud ke woe RE dee Be ee 88 addello 2565464644 6534 4 445 gA H addhelpP ose soe o sra a e a 47 205 addprimes ua do Es 49 95 153 AGG ek er a SS SG oS 12 adjoint matrix 172 OGM ce bw es Pee ele a Pace a eye 88 aketi nog fig ok we de ek Me h 112 algdep 2454468444 644 4 64 169 170 algdepO v s os web eee a ee ee 170 algebraic dependence 169 algebraic number 2 o ee 119 algtobasiS o 143 alias fon ace eon o as 48 205 allocatemem 57 205 allocatemoremem 205 alternating series 192 And as 24 eee See hin aed ee ee 74 and mest Geeta ats Ge oy Ge GS os Ree Ss 79 anell voceros mar a 112 pell tae ra oe E S lig Apell2 usa do cee BE Se 112 area oso t aue PRA woo SSE ESS SA a AE e e E e A ie 88 Artin L function 131 Artin root number 131 ASIN escri ae ee 88 e oS ase cn PA bie we ke Ble a 88 assmat ku kw nw Be deo Ke be 172 ata p uin 444654464 SAS A 88 atanh 2 fete eee ewe ee ee 88 automatic simplification 58 available commands 61 B backslash character 29 base s o soa a Be ee BO ee ae 143 DASE j s b e eo ee bee EG OR Ee ee 143 baSistoale wick ey ae o ee Ged 143 220 BerlekaMp o 101 bernfraG aoa a be
232. ic 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 39 matsolve x y x being an invertible matrix and y a column vector finds the solution u of r u y using Gaussian elimination This has the same effect as but is a bit faster than 27 x y The library syntax is gauss z y 176 3 8 40 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 Xa Mi jtj 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 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 ro
233. ich the integral transform is to be applied which will multiply the weights of integration and m is as in intnuminit If flag is nonzero assumes that s X s X which makes the computation twice as fast See intmellininvshort for examples of the use of this function which is particularly useful when the function s X is lengthy to compute such as a gamma product The library syntax is intfuncinit void E GEN eval GEN void GEN a GEN b long m long flag long prec Note that the order of m and flag are reversed compared to the GP syntax 3 9 6 intlaplaceinv X sig z expr tab numerical integration of expr X e with respect to X on the line R X sig divided by 2ir in other words inverse Laplace transform of the function corresponding to expr at the value z sig is coded as follows Either it is a real number o equal to the abcissa of integration and then the function to be integrated is assumed to be slowly decreasing when the imaginary part of the variable tends to 00 Or it is a two component vector o a where is as before and either a 0 for slowly decreasing functions or a gt 0 for functions decreasing like exp at Note that it is not necessary to choose the exact value of a tab is as in intnun It is often a good idea to use this function with a value of m one or two higher than the one chosen by default which can be viewed thanks to the function intnumstep or to increase the abcissa of integ
234. icit divisions and many variables This also affects functions like numerator denominator or content denominator x y 1 1 denominator y x 12 x content x y 43 1 y content y x 44 y content 2 x 15 2 Can you see why Hint 2 y 1 y x x is in Q y x and denominator is taken with respect to Q y x y x y x x 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 dependence on implicit orderings This will hopefully be corrected in future versions 2 5 5 Multivariate power series Just like multivariate polynomials power series are funda mentally single variable objects It is awkward to handle many variables at once since PARI s implementation cannot handle multivariate error terms like O x y It can handle the polyno mial O y x x which is a very different thing see below The basic assumption in our model is that if variable x has higher priority than y then y does not depend on zx
235. 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 Vo b Yo gt Volx for all prime ideals in x and v b gt 0 for all other p The library syntax is idealchinese nf x y 3 6 52 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 3 6 53 idealdiv nf x y flag 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 54 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 55 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
236. ie the quotient closest to 00 is chosen hence the remainder would belong to y 2 y 2 When x is a vector or matrix the operator is applied componentwise The library syntax is gdivround z y for x y 71 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 integer 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 modulo y When y is a non integral real number x y is defined as x a2 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 70 5 3 KK 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 x is a vector or matrix the operator is applied componentwise The library syntax is gmod z y for x y 3 1 8 divrem z 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 zx 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
237. ield by increasing modulus norm up to bound bound The ramified Archimedean places are given by arch all possible values are taken if arch is omitted The alternative syntax bnrdisclist bnf list is supported where list is as output by ideal list or ideallistarch with units in which case arch is disregarded The output v is a vector of vectors where v i 5 is understood to be in fact V 2 i 1 j of a unique big vector V This akward scheme allows for larger vectors than could be otherwise represented V k is itself a vector W whose length is the number of ideals of norm k We consider first the case where arch was specified Each component of W corresponds to an ideal m of norm k and gives invariants associated to the ray class field L of bnf of conductor m arch Namely each contains a 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 its absolute discriminant We set d r D 0 if m is not the finite part of a conductor 130 If arch was omitted all t 2 possible values are taken and a component of W has the form im d1 71 D1 d 7 D where mis the finite part of the conductor as above and d ri Di are the invariants of the ray class field of conductor m v where v is the i th archimedean component ordered by inverse lexicographic
238. ies 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 77 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 fac torization of the discriminant 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 flag fa An extended version is nfbasis z amp d flag fa where d receives
239. iler often eases this part Note also 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 efficiently import foreign functions for use under gp see Section 3 11 2 13 We are aware of four PARI related public domain packages to embed PARI in other languages 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 The first is the Math Pari Perl module see any CPAN 52 mirror written by Ilya Zakharevich The second is PariPython by St fane Fermigier which is no more maintained Starting from Fermigier s work Wiliam Stein has embedded PARI into his Python based SAGE system Finally Michael Stoll has integrated PARI into CLISP which is a Common Lisp implementation by Bruno Haible Marcus Daniels and others this interface has been updated for pari 2 by Sam Steingold These provide interfaces to gp functions for use in perl python or Lisp programs respectively 2 11 Defaults 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 13 but some of them are useful interactively set timer on
240. in T X installation After that the doc directory contains various dvi files libpari dvi manual for the PARI library users dvi manual for the gp calculator tutorial dvi a 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 equivalent it is in Landscape orientation and assumes A4 paper size If the pdftex package is part of your TX 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 see the last section 217 5 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 extgcd c program which is commented in the manual This should produce a statically linked binary extgcd sta standalone a dynamically linked binary extgcd dyn loads libpari at runtime and a shared library libextgcd which can be used from gp to install your new extgcd command The standalone binary should be bulletproof but the other two may fail for various reasons If whe
241. 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 3 3 this pointer argument is optional for all documented functions hence the amp will always appear between brackets as in Z_issquare z amp 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 70 3 1 Standard monadic or dya
242. in w is drawn The virtual cursor does not move 197 3 10 7 plotclip 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 3 10 8 plotcolor w c set default color to c in rectwindow w In present version 2 3 3 this is only implemented for the X11 window system and you only have the following palette to choose from 1 black 2 blue 3 sienna 4 red 5 green 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 9 plotcopy wl w2 dz dy copy the contents of rectwindow wl to rectwindow w2 with offset dx dy 3 10 10 plotcursor w give as a 2 component vector the current scaled position of the virtual cursor corresponding to the rectwindow w 3 10 11 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 1 yl w2 12 y2 the windows wl w2 etc are physically placed with their upper left corner at physical position 1 y1 12 y2 respectively and are then drawn together Overlapping regions will thus be drawn twice and the windows are considered transparent
243. index less than 3 2 2 1 O 1 1 0O O 2 2 O O 1 3 O O 1 1 0O O 1 subgrouplist 6 2 1 31 index 3 73 3 0 0 1 bnr bnrinit bnfinit x 120 1 1 L subgrouplist bnr 8 In the last example L corresponds to the 24 subfields of Q C 120 of degree 8 and conductor 12000 by setting flag we see there are a total of 43 subgroups of degree 8 vector L i galoissubcyclo bnr L i will produce their equations For a general base field you would have to rely on bnrstark or rnfkummer The library syntax is subgrouplistO bnr bound flag where flag is a long integer and an omitted bound is coded by NULL 3 6 153 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 161 CAVEAT This implementation is not satisfactory and must be rewritten In particular e 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 fewer significant digits than expected Computing with 28 eight digits of relative
244. ining polynomial for the number field which must be in Z X irreducible and monic In fact if you supply a non monic polynomial at this point gp issues a warning then transforms your polynomial so that it becomes monic The nfinit routine will return a different result in this case instead of res you get a vector res Mod a Q where Mod a Q Mod X P gives the change of variables In all other routines the variable change is simply lost The numbers c lt cz are positive real numbers which control the execution time and the stack size For a given c set co c to get maximum speed To get a rigorous result under GRH you must take c2 gt 12 or c2 gt 6 in P is quadratic Reasonable values for c are between 0 1 and 2 The default is c co 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 Warning Make sure you understand the above By default most of the bnf routines depend on the correctness of a heuristic assumption which is stronger than the GRH In particular any of the class number class group structure class group generators regulator and fundamental units may be wrong independently of each other Any result computed from such a bnf may be wrong The only guarantee is that the units given generate a subgroup of finite index in the full unit group
245. initO F flag prec Also available are initell E prec flag 0 and smallinitell E prec flag 1 3 5 17 ellisoncurve E 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 It is allowed for z to be a vector of points in which case a vector of the same type is returned The library syntax is ellisoncurve E z Also available is oncurve E z which returns a long but does not accept vector of points 3 5 18 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 19 elllocalred E p calculates the Kodaira type of the local fiber of the elliptic curve E at the prime p E must be an sell as output by ellinit and is assumed to have all its coefficients 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 Ig 2 3 and 4 mean types II III and IV respectively 4 v with y gt 0 means type l finally the opposite values 1 2 etc refer to the starred types Ig II etc The third component v is itself a vector u r s t giving the coordinate changes done during
246. invertible square matrix and in that case the result is zx 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 rational function in general To obtain the approximate real value of the quotient of two integers add O 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 34 The library syntax is gdiv x y for x y 3 1 5 The expression x X y is the Euclidean quotient of x and y If y is a real scalar this is defined as floor a y if y gt 0 and ceil x y if y lt 0 and the division is not exact Hence the remainder x 2Wy 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 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 t
247. ion 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 24 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 xy For the value at s 0 the function returns in fact for each character x a 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 cy sx O s xt1 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 y 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 Example bnf bnfinit x 2 229 bnr bnrinit bnf 1 1 bnrL1
248. ional 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 29 2 12 1 command gp on line help interface 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 being obtained by simply typing 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 If the line before the copyright message indicates that extended help is available this means perl is present on your system and the PARI distribution was correctly installed you can add more signs for extended functionalities 59 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 e
249. ions from unrelated libraries For instance it makes the function system obsolete install system vs sys libc so sys 1s gp gp c gp h gp_rl c 206 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 Caution This function may not work on all systems especially when 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 x 0 OK unused characters f x 0 a kill
250. iority 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 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 32 writetex filename str as write in TeX format 210 Appendix A Installation Guide for the UNIX Versions 1 Required tools Compiling PARI requires an ANSI C or a C compiler If you do not have one we suggest that you obtain the gcc g compiler As for all GNU software mentioned afterwards you can find the most convenient site to fetch gcc at the address http www gnu org order ftp html On Mac OS X this is also provided in the Xcode tool suite You can certainly compile PARI with a different compiler but the PARI kernel takes advantage of optimizations provided by gcc This results in at least 20 speedup on most architectures Optional packages The following programs and libraries are useful in conjunction with gp but not mandatory In any case get them before pro
251. is restored to its preceding value which is 0 local variables are initialized to 0 by 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 is a corrected version FirstPrimeDiv x forprime p 2 x if x p 0 return p Implementation note For the curious reader here is how values of variables are handled a hashing function is computed from the variable name and used as an index in hashtable a table of linked list of structures type entree The linked list is searched linearly for the identifier each list typically has less than 10 components 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 restored to the state preceding the last call of the main evaluator 2 7 Member functions Member functions use the dot notation to retrieve information from complicated structures by default bid ell galois 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 ell or outputs an error message if ell doesn t have the correct type To define your
252. is allowed to be undefined but continuous at a or b for example the function sin x x at x 0 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 190 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 1e1000 However beware of real underflow when dealing with rapidly decreasing functions For example if one wants to compute the ie e dx to 28 decimal digits then one should set infinity equal to 10 for example and certainly not to 1e1000 The library syntax is intnumromb void E GEN eval GEN void GEN a GEN b long flag long prec where eval 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 12 intnumstep give the value of m used in al
253. is as before and either 0 for slowly decreasing functions or a gt 0 for functions decreasing like exp at such as gamma products Note that it is not necessary to choose the exact value of a and that a 1 equivalent to sig alone is usually sufficient tab is as in intnun As all similar functions this function is provided for the convenience of the user who could use intnum directly However it is in general better to use intmellininvshort Ap 105 intmellininv s 2 4 gamma s 3 time 1 190 ms reasonable Xp 308 intmellininv s 2 4 gamma s 3 time 51 300 ms slow because of T s The library syntax is intmellininv void E GEN eval GEN void GEN sig GEN z GEN tab long prec 185 3 9 8 intmellininvshort sig z tab numerical integration of s X z with respect to X on the line R X sig divided by 237 in other words inverse Mellin transform of s X at the value z Here s X is implicitly contained in tab in intfuncinit format typically tab intfuncinit T 1 1 s sig I T or similar commands Take the example of the inverse Mellin transform of T s given in int mellininv Ap 105 oo 1 for clarity A intmellininv s 2 4 gamma s 3 time 2 500 ms not too fast because of T s function of real type decreasing as exp 37 2 t tab intfuncinit t 00 3 Pi 2 o0 3 Pi 2 gamma 2 I t 73 1 time 1 370 ms intmellinin
254. is ie e tt l dt The library syntax is incgamc s x prec 3 3 37 log x principal branch of the natural logarithm of x i e such that Im log x 7 7 The result is complex with imaginary part equal to 7 if x R and x lt 0 In general the algorithm uses the formula T x _ _ mlog 2 2agm 1 4 s ETR log x if s 12 is large enough The result is exact to B bits provided s gt 28 2 At low accuracies the series expansion near 1 is used p adic arguments are also accepted for x with the convention that log p 0 Hence in particular exp log 1 x is not in general equal to 1 but to a p 1 th root of unity or 1 if p 2 times a power of p The library syntax is glog z prec 3 3 38 Ingamma z principal branch of the logarithm of the gamma function of x This function is analytic on the complex plane with non positive integers removed Can have much larger arguments than gamma itself The p adic lngamma function is not implemented The library syntax is gngamma z prec 91 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 0 5 2 n a lt 1 Uses the functional equation linking the values at x and 1 x to restrict to the case x lt 1 then the power series when x lt 1 2 and the power series expansion in log x otherwise
255. is the real period and wa2 w1 belongs to Poincar s half plane eta quasi periods 71 72 such that mw now ir roots roots of the associated Weierstrass equation tate u u v in the notation of Tate wW Mestre s w this is technical 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 Q The use of member functions is best described by an example E ellinit 0 0 0 0 11 The curve y z 1 7 E a6 12 1 7 E c6 3 864 E disc 4 432 Some functions in particular those relative to height computations see ellheight require also that the curve be in minimal Weierstrass form which is duly stressed in their description below This is achieved by the function ellminimalmodel Using a non minimal model in such a routine will yield a wrong result 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 The requirements are given as the minimal ones any richer structure may replace the ones requested For instance in functions which have no use for the extra information given by an ell structure the curve can be given either as a five component vector as an sell or as an ell if an sell is requested an ell may equally be given 111 3 5 1 el
256. ive recursive plot If this flag is set only one curve can be drawn at a 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 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 print the y axis e 32 no_Frame do not print frame e 64 no_Lines only plot reference points do not join them e
257. iven and an exact square root had to be computed in the checking process puts that square root in n This is the case when x is a t_INT t_FRAC t_POL or t_RFRAC or a vector of such objects issquare 4 amp n 41 1 n 42 2 issquare 4 x72 amp n 13 1 1 both are squares n 4 2 x the square roots This will not work for t_INTMOD use quadratic reciprocity or t_SER only check the leading coefficient The library syntax is gissquarerem z n Also available is gissquare z 3 4 34 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 3 4 35 kronecker z y Kronecker symbol x y 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 arguments with the following special rules for y 0 1 or 2 e x 0 1 if z 1 and 0 otherwise e x 1 1if x gt 0 and 1 otherwise e 1 2 0 if x is even and 1 if x 1 1mod8 and 1 if x 3 3 mod 8 The library syntax is kronecker z y the result 0 or 1 is a long 3 4 36 lem z y least common multiple of
258. l the intnum and sumnum programs hence such that the integration step is equal to 1 2 The library syntax is intnumstep long prec 3 9 13 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 equal 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 this has degree 5050 time 3 335 ms prod i 1 100 1 Xi 1 0 X 101 time 43 ms 12 1 X X72 X75 X77 X 12 X715 X722 X 26 X 35 X 40 X 51 X 57 X 70 X 77 X792 X7100 O X 101 The library syntax is produit entree ep GEN a GEN b char expr GEN x 3 9 14 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 void E GEN eval GEN void GEN a GEN b long prec 3 9 15 prodinf X a expr flag 0
259. l 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 installs all such packages 4 2 The GPRC file Copy the file misc gprc dft or gprc dos if you are using GP EXE to HOME gprc Modify it to your liking For instance if you are 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 5 Getting Started 5 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 pla
260. l to know the output Let x t be the change of variable which is used tab 1 contains the integer m as above either given by the user or computed from the default precision and can be recomputed directly using the function intnumstep tab 2 and tab 3 contain respectively the abcissa and weight corresponding to t 0 0 and 0 tab 4 and tab 5 contain the abcissas and weights corresponding to positive t nh for 1 lt n lt N and h 1 2 nh and nh Finally tab 6 and tab 7 contain either the abcissas and weights corresponding to negative t nh for N lt n lt 1 or may be empty but not always if p t is an odd function implicitly we would have tab 6 tab 4 and tab 7 tab 5 The library syntax is intnuminit GEN a GEN b long m long prec 3 9 11 intnumromb X a b expr flag 0 numerical integration of expr smooth in Ja bl with respect to X This function is deprecated use intnum instead 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
261. laced 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 9 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 5l 2 9 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 gdiver division by 0 invmoder impossible inverse modulo archer not available on t
262. ladd z1 22 sum of the points z1 and 22 on the elliptic curve corresponding to 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 an sell as output 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 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 0 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 is slower than the previous method as soon as p is greater tha
263. lds 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 above 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 mis 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 135 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 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
264. lean 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 14 READL true if gp is compiled with readline support see Section 2 15 1 VERSION op number where op is in the set gt lt lt gt and number is a PARI version number 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 13 1 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 11 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 Fo
265. lh fe oak te oh ape aoe O92 Dro ALEIDA st sec xe Rats RT O dae pd Wiech wily SCP se SRR ey A AA AAA SO eA Won Ge oh ee 2 DO e Bw eal gn ar athe a a a ar E Gye eae e ao eee Se Se o Ok of gl dak 6 2 ie ae Bl ae 6 Sh be A A Ob oe dS 93 REET 5 ae ace Se ae EE ey CSA AD Re Cel he ae Nee BE A HG he eB ES Ge OP Os Re He BO SS AG tans MA E Ge Se A A A BE pt BO ee ADA we a Oe ea a Gee ces oe Ml se a DA S A T tans ee ay ae Wh ae Bm ea oO ob Se a ke gan b Boe ar 2 OR toe aol ng Ba a GS ee 4 2 3 39 48 te1chmuller e it aie RI a A Be ee Eee RR eel e Bat me ye OA 3 32 49 theta ni A AA A A A A A et A e ee DA S23 50 thetanullk ce aa der ar RA AA Be ek A AA a O oe DA DS L WEDE e A A Woe A A A e tte O A A A A A O ELA ca A NR AAA A IN AN A A a e 3 4 Arithmetic functions oaoa O SAA primes Y s Be ss air Bee Mig dp oe a i AG ae OR ee ay ee EE AA SR ASE CA AO 3 4 2 bestappr a taa Be a A E Ave Toate dee do 06 E A A hh ee late SAA DEZOULTES acc us Ai A Is a DA A DE Be ee A a a a O da ter da DO 3 4 5 DigOMEsa A AA A AAA oN AE Ge ele A amp 496 3 40 binomial s Aa ias A A a a a a O 3 dar chimeser ae RI a adds da AA ads a sb a 0 SS CODEN e ao e ee Be pe a Oe Se opi e a A o a AA e ee E 3 4 0 COM TAC 00d a a e rs A Oe BY oe e oie me a A a Bt i a OF 3 4 10 eontiracpngi S a E AAA AN DR e ah AAA AO a A E es Boo OT DARLES Atak AA te A SON E AAA A AA AN A Y S41 2Coredise carr AA a ek a ad a a eee a A AS SAT ordirdive g td AA O BA A OM
266. 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 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 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 2 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 e
267. lowing approach inversemod a b local m m polsylvestermatrix polrecip a polrecip b m matinverseimage m matid m 1 Polrev vecextract m Str poldegree b variable b inversemod a b 12 Mod 2 4 y 5 Mod 3 4 y7 3 Mod 1 4 y 2 Mod 3 4 y Mod 2 4 This is not guaranteed to work either since it must invert pivots See Section 3 8 The library syntax is gpow x n prec for x7n 73 3 1 10 bittest x n outputs the n bit of x starting from the right i e the coefficient of 2 in the binary expansion of x The result is 0 or 1 To extract several bits at once as a vector pass a vector for n See Section 3 2 17 for the behaviour at negative arguments The library syntax is bittest x n where n and the result are longs 3 1 11 shift x n or x lt lt n x gt gt n shifts z 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 217 with a remainder of the same sign as x hence is not the same in general as 112 The library syntax is gshift x n where n is a long 3 1 12 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 shifts Euclidean division takes place when n lt 0 h
268. ls If flag 1 and the precision happens to be insufficient for obtaining the fundamental units the internal precision is doubled and the computation redone until the exact results are obtained 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 large real quadratic fields For this case there are alternate compact representations for algebraic numbers implemented in PARI but currently not available in GP 123 If flag 2 the fundamental units and roots of unity are not computed Hence the result has only 7 components the first seven ones The library syntax is bnfclassunitO P flag tech prec 3 6 10 bnfclgp P tech as bnfinit but only outputs bnf clgp i e the class group The library syntax is classgrouponly P tech prec where tech is as described under bnfinit 3 6 11 bnfdecodemodule nf m if m is a module as output in the first component of an exten sion given by bnrdisclist outputs the true module The library syntax is decodemodule nf m 3 6 12 bnfinit P flag 0 tech initializes a bnf structure Used in programs such as bnfisprincipal bnfisunit or bnfnarrow By default the results are conditional on a heuristic strengthening of the GRH see 3 6 7 The result is a 10 component vector bnf This implements Buchmann s sub exponential algorithm for computing the class group the regulator and a s
269. lves 2 12 20 s prints the state of the PARI stack and heap This is used primarily as a debugging device for PARI 2 12 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 12 22 u prints the definitions of all user defined functions 2 12 23 um prints the definitions of all user defined member functions 2 12 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 12 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 2 12 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
270. mat The library syntax is ellsearch N Also available is ellsearchcurve N that only accept complete curve names 3 5 29 ellsub 21 22 difference of the points z1 and 22 on the elliptic curve corresponding to E The library syntax is subell E z1 22 3 5 30 elltaniyama E computes the modular parametrization of the elliptic curve E where E is an sell as 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 2v ayu az is equal to f z dz a differential form on H Ty N where N is the conductor of the curve The variable used in the power series for u and v is x which is implicitly understood to be equal to exp 2i7 2 It is assumed that the curve is a strong Weil curve and that 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 elltaniyama F prec and the precision of the result is determined by prec 117 3 5 31 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
271. meaningful gp will provide it together with e an ending comma if we are completing a default e a pair of parentheses if we are 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 periment with this mechanism as often as possible you will 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 _ 67 68 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 3 3 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 un
272. 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 intmod a prime or real numbers but is usually slower in other base rings The library syntax is charpoly0 A v flag where v is the variable number Also available are the functions caract A v flag 1 carhess 4 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 z 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 use Mat instead see the example there 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 and bottom row otherwise The empty matrix is considered to have a number of rows compatible with any operation in partic
273. mods and variables of lower priority for external operations typically between a poly nomial and a polmod See Section 2 5 4 for a definition of priority and a discussion of PARI s idea of multivariate polynomial arithmetic For instance Mod x x72 1 Mod x x72 1 1 Mod 2 x x72 1 24 or 2i with i 1 x Mod y y 2 1 12 x Mod y y 2 1 Al in Q i z y Mod x x72 1 3 Mod x y x72 1 Ain Q y i The first two are straightforward but the last one may not be what you want y is treated here as a numerical parameter not as a polynomial variable If the main variables are the same it is allowed to mix t_POL and t_POLMODs The result is the expected t_POLMOD For instance x Mod x x72 1 1 Mod 2 x x72 1 32 2 3 9 Polynomials type t_POL type the polynomial in a natural way not forgetting to put a 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 1 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 machine
274. n 100 say No checking is done that p is indeed prime E must be an sell as output by ellinit defined over Q F or Q E must be given by a Weierstrass equation minimal at p The library syntax is ellapO p flag Also available are apell E p corresponding to flag 0 and apell2 E p flag 1 3 5 5 ellbil 21 22 if z1 and 22 are points on the elliptic curve E assumed to be integral given by a minimal model this function computes the value of the canonical bilinear form on z1 22 h E 21422 h E z1 h E 22 2 where denotes of course addition on E In addition z1 or 22 but not both can be vectors or matrices The library syntax is bilhell F 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 x u22 r y u y su x t E must be an sell as output by ellinit The library syntax is coordch E v 3 5 7 ellchangepoint x v changes the coordinates of the point or vector of points x using the vector v u r s t ie if x and y are the new coordinates then x u x r y uy su72 t see also ellchangecurve The library syntax is pointch z v 112 3 5 8 ellconvertname name converts an elliptic curve name as found in the elldata database from a string to a triplet conductor isogeny class index It will also convert a triplet back to a
275. n Bernstein and most feedback should be directed to those They are e pari announce to announce major version changes You cannot 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 The BTS forwards the mail it receives to this list e pari users for everything else You may send an email to the last two without being subscribed You will have to confirm that your message is not unsollicited bulk email aka Spam To subscribe send empty messages respectively to pari announce subscribe list cr yp to pari users subscribe list cr yp to 218 pari dev subscribe list cr yp to You can also write to us at the address pari math u bordeaux fr 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 3 3 year 2006 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 219 Index Some Word refers to PARI GP concepts SomeWord is a PARI GP keyword SomeWord is a gener
276. n be a regular factorization as produced with any factor command A few examples factorback 2 2 3 1 99 1 12 factorback 2 2 3 1 12 12 factorback 5 2 3 13 30 factorback 2 2 3 1 nfinit x 3 2 14 16 0 0 o 16 0 o O 16 nf nfinit x 2 1 fa idealfactor nf 10 75 2 1 1 2 1 1 1 2 LES 2 1 1 1 1 2 1 1 LES 2 1 1 1 2 11 1 factorback fa kkk forbidden multiplication t_VEC t_VEC factorback fa nf 46 10 0 0 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 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 z p factors the polynomial z 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 factorc
277. n 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 11 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 2 insert right left pairs where appropriate 4 insert discretionary breaks in polynomials to enhance the probability of a good line break 2 11 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 12 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 opt
278. n 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 realprecision see Section 1 2 6 3 30 2 3 3 Intmods type t_INTMOD to enter n mod m type Mod n m not n m Internally all oper ations are done on integer representatives belonging to 0 m 1 Note that this type is available for convenience not for speed each elementary operation involves a reduction modulo m 2 3 4 Rational numbers types t_FRAC all fractions are automatically reduced to lowest terms so it is impossible to work with reducible fractions To enter n m just type it as written As explained in Section 3 1 4 division is not performed only reduction to lowest terms Note that this type is available for convenience not for speed each elementary operation involves computing a gcd 2 3 5 Complex numbers type t_COMPLEX to enter x iy type x I y not x ix 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 or t_PADIC 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 0 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 whic
279. n 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 37 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 x 3 8 38 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 outputs U V D where U and V are two unimodular matrices such that UX V is the diagonal matrix D Otherwise output only the diagonal of D 2 gener
280. n running extgcd dyn you get a message of the form DLL not found then stick to statically linked binaries or look at your system documentation to see how to indicate at linking time where the required DLLs may be found E g on Windows you will need to move 1libpari d11 somewhere in your PATH 5 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 See the file examples EXPLAIN for some explanations 5 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 5 5 The PARI Community PARI s home page at the address http pari math u bordeaux fr maintains an archive of mailing lists dedicated to PARI documentation including Frequently Asked Questions a download area and our Bug Tracking System BTS Bug reports should be submitted online to the BTS which may be accessed from the navigation bar on the home page or directly at http pari math u bordeaux fr Bugs Further information can be found at that address but to report a configuration problem make sure to include the relevant dif files in the Oxxx directory and the file pari cfg There are three mailing lists devoted to PARI GP run courtesy of Da
281. n the file filename subject to string expansion If filename 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 31 the file is loaded and the last object in the file is returned 3 11 2 22 readvec str reads in the file filename subject to string expansion If filename is omitted re reads the last file that was fed into gp The return value is a vector whose components are the evaluation of all sequences of instructions contained in the file For instance if file contains 1 3 then we will get NXra 1 1 12 2 43 3 read a 74 3 readvec a 45 1 2 3 In general a sequence is just a single line but as usual braces and may be used to enter multiline sequences 3 11 2 23 reorder x 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 decreasing 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 24 setrand n reseed
282. nal 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 The library syntax is externO str where str is a char 205 UNIX 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 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 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 situa
283. nce 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 2 2 or call the function gvar x which returns the variable number of x 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 In 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 86 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 Xp 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
284. nce 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 2 8 Strings and Keywords 2 8 1 Strings GP variables can hold values of type character string internal type t_STR This section describes how they are actually used as well as some convenient tricks automatic concate nation 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 x 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
285. nd the result is a row vector a a with two components such that x aZk 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_eltO nf x a where an omitted a is entered as NULL 3 6 70 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 71 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 syntax is principalidele nf x 3 6 72 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
286. nderlying expression to speed up the computation In the present version 2 3 3 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 3 9 20 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 for 3 consecutive evaluations The expressions must always evaluate to a complex number If the series converges slowly make sure realprecision is low even 28 digits may be too much In this case if the series is alternating or the terms have a constant sign sumalt and sumpos should be used instead p28 suminf i 1 1 7i i suminf user interrupt after 10min 20 100 ms sumalt i 1 1 i i log 2 time 0 ms 1 2 524354897 E 29 The library syntax is suminf void E GEN eval GEN void GEN a long prec 3 9 21 sumnum X a sig expr tab flag 0 numerical summation of expr the variable X taking integer values from ceiling of a to 00 where expr is assumed to be a holomorphic function f X for R X gt 0 The parameter o R is coded in the argument sig as follows it is either e a real number Then the function f is assumed to decrease at least as 1 X at infinity but not exponentially e a two component vector o a
287. nexact t_PADIC coefficients this is not always well defined in this case the equation is first made integral then lifted to Z Hence the roots given are approximations of the roots of a polynomial which is p adically close to the input 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 C 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 166 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
288. 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 157 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 i e such that d pol Z kx f70 L K nf rnf 6 is currently unused 3 5 is currently unused rnf 7 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 rnf rnf 10 is nf is the inverse matrix of the integral basis matrix with coefficients polmods in nf 9 is currently unused 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 3 is an abstract root of pol and a the generator of nf the generator whose root is vabs will be G k
289. ng 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 15 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 string The library syntax is gtovecsmall z 78 3 2 16 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 z 3 2 17 bitand z y bitwise and of two integers x and y that is the integer ye and y 2 i Negative numbers behave 2 adically i e the result is the 2 adic limit of bitand n Yn where Zn and y are non negative integers tending to x and y respectively The result is an ordinary integer possibly negative bitand 5 3 1 1 bitand 5 3 12
290. ng 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 zx The library syntax is polredO z flag fa where an omitted fa is coded by NULL Also available are polred x and factoredpolred z fa both corresponding to flag 0 3 6 115 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 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 z 4 gives all polynomials of minimal T norm of the two poly
291. nical note 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 as 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 specific thing to look for is to check the readline h include file wherever your readline include files are for the string RL PROMPT_START_IGNORE If it is there you are safe Another 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 54 2 11 2 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 compatible 0 no backward compatibility In this mode a very handy function to be described in Section 3 11 2 28 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
292. nly the significant coefficients are taken into account but this time by increasing order of degree The library syntax is gtocol x 3 2 2 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 3 Mat x transforms the object x into a matrix If x is already a matrix a copy of x is created 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 unless all elements are column resp row vectors of the same length in which case the vectors are concatenated sideways and the associated big matrix is returned Mat x 1 i x 1 Vec matid 3 712 1 0 O 0 1 O O O 1 Mat 3 1 0 0 o 1 0 0 0 1 Col 1 2 3 4 4 1 21 3 4 Mat 5 1 2 3 4 The library syntax is gtomat z 3 2 4 Mod z y flag 0 creates the PARI object x mod y i e an intmod 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 flag is obsolete and should n
293. nomials P x and P z 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 6 116 polredord x finds polynomials with reasonably small coefficients and of the same degree as that of x defining suborders of the order defined by x One 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 153 3 6 117 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 6 118 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 6 119 rnfbasis bnf M let K the field represented by bnf as o
294. 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 the O p The library syntax is gevtoi x amp e where e is a long integer Also available is gtrunc z 3 2 49 valuation x p computes the highest exponent of p dividing x If p is of type integer x must be an integer an intmod 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 x 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 50 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 si
295. nt 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 6 110 nfsnf nf x given a torsion module x as a 3 component row vector 4 I 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 In other words x is isomorphic to Zx d 9 9 Zk d and di divides d _1 for gt 2 The link between x and A I J is as follows if e is the canonical basis of K I b bn and J a an then x is isomorphic to biei Pep bnen a1 Az Der An An gt 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 6 111 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 150 3 6 112 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 err
296. nt 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 62 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 58 The library syntax is fundunit z 109 3 4 63 removeprimes x removes the primes listed in x from the prime number table In particular removeprimes 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 64 sigma z k 1 sum of the kt powers of the positive divisors of x x and k 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 65 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 sqrti x Also available is sqrtremi x amp r which returns s such that s g4 r withO lt r lt 2s 3 4 66 zncoppersmith P N X B N finds all integers xo with ao lt X such that gcd N P x0 gt B If N is prime or
297. nt x y in the number field nf The library syntax is element_div nf x y 3 6 82 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 83 nfeltdivmodpr nf zx y pr given two elements x and y in nf and pr a prime ideal in modpr format see nfmodprinit computes their quotient x y modulo the prime ideal pr The library syntax is element_divmodpr nf x y pr 3 6 84 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 nfdivrem nf x y 3 6 85 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 y The library syntax is nfmod nf x y 3 6 86 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 87 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 y modulo the prim
298. ntreltoabs rnf x 3 6 129 rnfeltup rnf x rnf being a relative number field extension L K as output by rnfinit and 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 155 3 6 130 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 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 6 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 5 4 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
299. ntries like multivariate polynomials The library syntax is det x flag 0 and det2 x flag 1 172 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 z 3 8 14 matdiagonal 1 x being a vector creates the diagonal matrix whose diagonal entries are those of x The library syntax is diagonal x 3 8 15 mateigen x gives the eigenvectors of x as columns of a matrix The library syntax is eigen 3 8 16 matfrobenius M flag 0 v x returns the Frobenius form of the square matrix M If flag 1 returns only the elementary divisors as a vectr of polynomials in the variable v If flag 2 returns a two components vector F B where F is the Frobenius form and B is the basis change so that M B FB The library syntax is matfrobenius M flag v where v is the variable number 3 8 17 mathess x Hessenberg form of the square matrix x The l
300. o not need to install it You can also use this feature together with a symbolic link named readline in the PARI toplevel 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 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 Oxxx directory e g readline linux i686 Known problems e on 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 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 e on OS X 4 Tiger comes equipped with a fake readline which is not sufficient for our purpose As a result gp is built without readline support Since readline is not trivial to install in this environment a step by step solution can be f
301. o special set of commands becoming available during a break loop as they would in most debuggers 50 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 9 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 inspection Of course break loops are no longer available the new handler has rep
302. o your query e g elliptic curve or quadratic field 27 If gp was properly installed see Appendix A a line editor is available to correct the command line get automatic completions and so on See Section 2 15 1 or readline for a short summary of the line editor s commands 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 you 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 12 2 1 3 Input Just type in an instruction e g 1 1 or Pi No action is undertaken until you hit the lt Return gt key Then computation starts and a result is eventually printed To suppress printing of the result end the expression with a sign Note that many systems use to indicate end of input Not so in gp this will hide the result from you Which is certainly useful if it occupies
303. 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 3 8 32 matmultodiagonal x y product of the matrices x and y assuming that the result is a diagonal matrix Much faster than zxy in that case The result is undefined if x y is not diagonal The library syntax is matmultodiagonal z y 3 8 33 matpascal x q creates as a matrix the lower triangular Pascal triangle of order x 1 ie 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 x q where x is a long and q NULL is used to omit q Also available is matpascal z 3 8 34 matrank zx rank of the matrix z The library syntax is rank x and the result is a long 175 3 8 35 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 36 matrixqz x p x being an m x n matrix with m gt n with rational or integer entries this function has varying behaviour depending on the sig
304. ogram is usually pessimistic when it comes to choosing the integration step 2 It is often possible to improve the speed by trial and error Continuing the above example test M tab intnuminit 1 1 M intnum x 1 1 intnum y sqrt 1 x 2 sqrt 1 x 2 x 2 y72 tab tab Pi 2 m intnumstep what value of m did it take 11 7 test m 1 189 time 1 790 ms 2 2 05 E 104 4 2 times faster and still OK test m 2 time 430 ms 3 1 11 E 104 16 24 times faster and still OK test m 3 time 120 ms 3 7 23 E 60 64 2 times faster lost 45 decimals The library syntax is intnum void E GEN eval GEN void GEN a GEN b GEN tab long prec where an omitted tab is coded as NULL 3 9 10 intnuminit a b m 0 initialize tables for integration from a to b where a and b are coded as in intnum Only the compactness the possible existence of singularities the speed of decrease or the oscillations at infinity are taken into account and not the values For instance intnuminit 1 1 is equivalent to intnuminit 0 Pi and intnuminit 0 1 2 1 is equiv alent to intnuminit 1 1 1 2 If mis not given it is computed according to the current precision Otherwise the integration step is 1 2 Reasonable values of m are m 6 or m 7 for 100 decimal digits and m 9 for 1000 decimal digits The result is technical but in some cases it is usefu
305. oid 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 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 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 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 polynomi
306. olcoeff as a polynomial of degree zero The library syntax is truecoeff z n The third method is specific to vectors or matrices in 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 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 m row of x and x n represents the nt column of zx Finally note that in library mode the macros gcoeff and gmael are available as direct accessors to a GEN component See Chapter 4 for details 3 2 27 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 intmods fractions or p adics The only forbidden type is polmod see conjvec for this The library syntax is gconj x 3 2 28 conjvec 1 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 intmod 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
307. olynomials one also has modulargcd x 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 O meaning the place at infinity Otherwise p needs not be given and x and y can be of compatible types integer fraction real intmod a prime result is undefined if the modulus is not prime or p adic The library syntax is hil x 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 x but the simpler function isfundamental x which returns a long should be used if x is known to be of type integer 3 4 30 ispower z k amp n if k is given returns true 1 if x is a k th power false 0 if not In this case x may be an integer or polynomial a rational number or function or an intmod a prime or p adic If k is omitted only integers and fractions are allowed and the function returns the maximal k gt 2 such that x n is a perfect power or 0 if no such k exist in particular ispower 1 ispower 0 and ispower 1 all return 0 If a third argument amp n is given and a k th root was computed in the process then n is set to that root The library syntax is ispower z k amp n the result is a long Omitted k or n are coded as NULL 102 3 4 31 isprime z flag 0 true 1 if x is a proven prime number fal
308. on see Section 2 12 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 do not need anymore 2 11 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 11 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 LaTeX output to the logfile can be further customized using TeXstyle 2 11 14 logfile default pari log name of the log file to be used when the log toggle is on Environment and time expansion are performed 56 UNIX 2 11 15 new_galois_format default 0 if this is set the polgalois command will use
309. 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 zx such that r lt arg x lt r The library syntax is garg z prec 3 3 10 asin x principal branch of sin x i e such that Re asin w 7 2 7 2 Ife R and x gt 1 then asin x is complex The library syntax is gasin z prec 3 3 11 asinh z principal branch of sinh x i e such that Im asinh x 1 2 7 2 The library syntax is gash z prec 88 3 3 12 atan x principal branch of tan x i e such that Re atan 7 2 7 2l The library syntax is gatan z prec 3 3 13 atanh x principal branch of tanh 2 i e such that Im atanh x 7 2 7 2 If x Rand z gt 1 then atanh x is complex The library syntax is gath x prec 3 3 14 bernfrac x Bernoulli number Bz where Bo 1 By 1 2 By 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 By is returned as a real number with the current precision The library syntax is bernreal z prec 3 3 16 bernvec x creates a vector containing as rational numbers the Bernoulli numbers Bo Ba Boz This routine is obsolete Use bernfrac instead each time you need a Bernoulli number in exact
310. onverted to a power series using the current precision held in the variable precdl 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 series 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 3 3 will have a 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 log 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 y 0 57721 Note that Euler is one of the few special reserved names which cannot be used for variables the others are 1 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 87 3 3 3 I
311. or 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 follows 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 C5 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 x715 260 x714 355 x713 95 x712 1460 x711 3279 x 10 3660 x 9 2005 x78 705 x77 9210 x 6 13506 x75 7145 x 4 2740 x73 1040 x72 320 x 256 a z 2 a5 5 a is a fifth root of 5 13 0
312. order so v 0 0 ve 1 0 0 etc Again we set dj ri D 0 if m ui is not a conductor Finally each prime ideal pr p a e f 8 in the prime factorization m is coded as the integer 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 Note that to compute such data for a single field either bnrclassno or bnrdisc is more efficient The library syntax is bnrdisclistO bnf bound arch 3 6 32 bnrinit bnf f flag 0 bnf is as output by bnfinit f is a modulus initializes data linked to the ray class group structure corresponding to this module a so called bnr structure The following member functions are available on the result bnf is the underlying bnf mod the modulus bid the bid structure associated to the modulus finally clgp no cyc clgp refer to the ray class group as a finite abelian group its cardinality its elementary divisors its generators The last group of functions are different from the members of the underlying bnf which refer to the class group use bnr bnf xxx to access these e g bnr bnf cyc to get the cyclic decomposition of the class group They are also different from the members of the underlying bid which refer to Vx f use bnr bid xxx to access these e g bnr bid no to get f If flag 0 default the generators of the ray class group are not computed
313. ore starting any real computations You could also define variables from your GPRC to have a consistent 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 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 Al 0 Py hx heey these two take place in Q y x x Mod 1 y 13 Mod 1 y x A in Q y yQ y z Qiz Mod x y 4 0 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 39 In the last example the division by y takes place in Q y x 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 impl
314. ormalizations E is assumed to be integral given by a minimal model The library syntax is mathell E x prec 3 5 15 ellidentify E look up the elliptic curve E over Z in the elldata database and return CIN M G C where N is the name of the curve in J E Cremona database M the minimal model G a Z basis of the free part of the Mordell Weil group of E and C the coordinates change see ellchangecurve The library syntax is ellidentify 3 5 16 ellinit E flag 0 initialize an e11 structure associated to the elliptic curve E E isa 5 component vector a1 az a3 a4 ag defining the elliptic curve with Weierstrass equation Y a XY a3Y X a2X a4X ag or a string in this case the coefficients of the curve with matching name are looked in the elldata database if available For the time being only curves over a prime field F and over the p adic or real numbers including rational numbers are fully supported Other domains are only supported for very basic operations such as point addition The result of ellinit is a an ell structure by default and a shorted sell if flag 1 Both contain the following information in their components 01 42 43 44 06 ba ba be bg C4 C6 J All are accessible via member functions In particular the discriminant is F disc and the j invariant is F j The other six components are only present if flag is O or omitted Their content depends on whether the curve is
315. ors 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 integers contained in the vector x or the single integer x to a special table of user defined primes and returns that table Whenever factor is subsequently called it will trial divise by the elements 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 and in fact they need only be positive integers The algorithm makes sure that all elements in the table are pairwise coprime so it may end up containing divisors of the input integers It is a useful trick to add known composite numbers which 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 95 3 4 2 bestappr z A B if B is omitted finds the best rational approximation to
316. ot be used The library syntax is gmodulo z y 76 3 2 5 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 Warning this is not a substitution function It will not transform an object containing variables of higher priority than v Pol x y y Pol variable must have higher priority in gtopoly The library syntax is gtopoly x uv where v is a variable number 3 2 6 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 7 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 Negative definite forms are not implemented use their positive definite counterpa
317. ound in the PARI FAQ see http pari math u bordeaux fr 3 6 Testing 3 6 1 Known problems if BUG shows up in make bench 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 xxIX 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 like an Illegal Instruction on startup It does not affect gp sta e Some implementations of the diff utility on HPUX for instance output No differences encountered or some similar message instead of the expected empty input Thus producing a spurious BUG message 215 3 6 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 argumen
318. 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 It could be implemented as x j if type x t_VEC length x lt 14 error this is not a proper elliptic curve x x 13 Typing um will output the list of user defined member functions 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 myj 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 45 Warning it is advised not to apply a member whose name starts with e or E to an integer constant where it would be confused with the usual floating point exponent E g compare 7 x e2 x 1 7 1 e2 1 100 000000000 taken to mean 1 052 1 e2 12 2 7 1 0 e2 73 2 00000000000 Note Member functions were not meant to be too complicated or to depend on any data that wouldn t be global He
319. put in the columns of the matrix which is the natural direction for solving equations afterwards The use of this matrix can be essential when 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 T of the first kind in variable v The library syntax is tchebi n v where n and v are long integers v is a variable number 3 7 27 polzagier n m creates Zagier s polynomial Pe 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 ap X and y bp X then serconvol z y az by X The library syntax is convol z y 3 7 29 serlaplace x x must be a power series with non negative exponents If x
320. qual to the ring of integers of nf and outputs the four component row vector A 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 Zk module Zz its image in SKo Zx The library syntax is rnfsteinitz nf x 160 3 6 152 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 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 Tf flag 0 default and bnr is as output by bnrinit 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 12 0 0 2 6 3 0O 1 2 1 O 1 3 0 0O 2 1 0 0 21 6 0 O 1 2 0 O 1 3 0 O 1 1 0 O 1 subgrouplist 6 2 3
321. r 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 functionalities 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 TREX 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 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 12 2 comment Everything between the stars is ignored by gp These comments can span any number of lines 2 12 3 one line comment The rest of the line is ignored by gp 60 2 12 4 Na n prints the object number n n in
322. r 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 72 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 Ta 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 nf nfinit x 3 12 initialize number field Q X X73 12 nf pol defining polynomial 12 x 3 12 nf disc field discriminant 3 972 nf index index of power basis order in maximal order 14 2 nf zk integer basis lifted to Q X 5 1 x 1 2 x72 nf sign signature 46 1 1 factor abs nf disc determines ramified primes ht 2 2 3 5 idealfacto
323. r input evaluates it then prints the result and assigns it to an history array if it is not void see next section More precisely you input either a metacommand or a sequence of expressions Metacommands described in Section 2 12 are not part of the GP language and are simple shortcuts designed to alter gp s internal state such as the working precision or general verbosity level or speed up input output An expression is formed by combining constants variables operator symbols functions in cluding user defined functions and control statements It always has a value which can be any PARI object There is a distinction between lowercase and uppercase Also outside of character strings blanks are completely ignored in the input to gp An expression is evaluated using the conventions about operator priorities and left to right associativity are Several expressions are combined on a single line by separating them with semicolons Such an expression sequence will be called simply a seg A seg also has a value which is the value of the last expression in the sequence Under gp the value of the seg and only this last value becomes an history entry The values of the other expressions in the seq are discarded after the execution of the seg is complete except of course if they were assigned into variables In addition the value of the seq is printed if the line does not end with a semicolon 2 2 2 The gp history
324. r 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 O anda 1 differ The second principle 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 with 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 3 3 this is not implemented 1 5 Operations and functions The available operations and functions in PARI are described in detail in Chapter 3 Here is a brief summary 1 5 1 Standard arithmetic operations Of course
325. r 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 Bgp7A e m B 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 63 prompt H M gp gt since the preprocessor directive applies to the 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 Hif VERSION gt 2 2 3 ah 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 13 2 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 e gp checks whether the environment variable GPRC is set Under DOS you can set it in AU
326. r nf 2 8 Clas 0 0 1 3 1 0 1 0 31 BBS In case pol has a huge discriminant which is difficult to factor the special input format pol B is also accepted where pol is a polynomial as above and B is the integer basis as would be computed by nfbasis This is useful if the integer basis is known in advance or was computed conditionnally pol polcompositum x 5 101 polcyclo 7 1 B nfbasis pol 1 faster than nfbasis pol but conditional 148 nf nfinit pol B factor abs nf disc 5 18 7 25 101 24 B is conditional when its discriminant which is nf disc can t be factored In this example the above factorization proves the correctness of the computation 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 z 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
327. r respectively They take as first argument a number field of that 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 On the other hand if the function requires a bnf it will not launch bnfinit for you which is a costly operation Instead it will give you a specific error message In short the types nf lt bnf lt bnr are ordered each function requires a minimal type to work properly but you may always substitute a larger type The data types corresponding to the structures described above are rather complicated Thus as we already have seen it with elliptic curves GP provides member functions to retrieve data from these structures once they have been initialized of course The relevant types of number fields are indicated between parentheses bid bnr bid ideal structure bnf bnr bnf Buchmann s 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 index bnr bnf nf
328. r we are computing a power x 1 00274 x x 3 1 0 275 1 0 274 74 44 1 0 276 note the difference between 2 and 3 above The library syntax is gsqr x 3 3 44 sqrt x principal branch of the square root of x i e such that Arg sqrt x 7 2 7 2 or in other words such that R sqrt x 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 Intmod 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 intmods is in the interval 0 p 2 When the argument is an intmod a non prime or a non prime adic the result is undefined The library syntax is gsqrt x prec 3 3 45 sqrtn x n amp z principal branch of the nth root of x i e such that Arg sqrt x r n Tt n Intmod 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 In the case this argument is present and no square root exist 0 is returned instead or raising an error sqrtn Mod 2 7 2 1 Mod 4 7 sqrtn Mod 2 7 2 amp z Z 92 Mod 6 7 sqrtn Mod 2 7 3 sqrtn nth root does not exist in gsqrtn sqrtn Mod 2 7 3 amp z 2 0 Z 3 0 The following script comp
329. rary libpari dvi Important note A tutorial for gp is provided in the standard distribution A tutorial for PARI GP tutorial dvi and you should read this first You can then start over and read the more boring stuff which lies ahead 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 refer to the complete function description in Chapter 3 How to get the latest version Everything can be found on PARI s home page http pari math u bordeaux fr From that point you may access all sources some binaries version informations the complete mailing list archives frequently asked questions and various tips All threaded and fully searchable How to report bugs Bugs are submitted online to our Bug Tracking System available from PARI s home page or directly from the URL http pari math u bordeaux fr Bugs Further instructions can be found on that page 19 1 2 The PARI types The GP language is not typed in the traditional sense it is dynamically typed in particular variables have no type In library mode the type of all PARI objects is GEN a generic type On the other hand each object has a specific internal type depending on the mathematical object it represents The crucial word is recursiveness most of the types PARI knows about are recursive For example the basic internal type t_COMPLEX exists However the components i e the
330. ration o For example p 105 intlaplaceinv x 2 1 1 x 1 time 350 ms 1 7 37 E 55 1 72 E 54x I not so good 184 m intnumstep 42 7 intlaplaceinv x 2 1 1 x m 1 1 time 700 ms 43 3 95 E 97 4 76 E 98 I better intlaplaceinv x 2 1 1 x m 2 1 time 1400 ms 74 0 E 105 O E 106x1I perfect but slow intlaplaceinv x 5 1 1 x 1 time 340 ms 5 5 98 E 85 8 08 E 85 I better than 1 intlaplaceinv x 5 1 1 x m 1 1 time 680 ms 76 1 09 E 106 O E 104 1 perfect fast intlaplaceinv x 10 1 1 x 1 time 340 ms 7 4 36 E 106 0 E 102 I perfect fastest but why sig 10 intlaplaceinv x 100 1 1 x 1 time 330 ms 47 1 07 E 72 3 2 E 72xI XX too far now The library syntax is intlaplaceinv void E GEN eval GEN void GEN sig GEN z GEN tab long prec 3 9 7 intmellininv X sig z expr tab numerical integration of expr X 27 with respect to X on the line R X sig divided by 27 in other words inverse Mellin transform of the function corresponding to expr at the value z sig is coded as follows Either it is a real number equal to the abcissa of integration and then the function to be integrated is assumed to decrease exponentially fast of the order of exp t when the imaginary part of the variable tends to too Or it is a two component vector c a where g
331. raw format If the number n is omitted print the latest computed object 2 12 5 b n Same as Na in prettyprint i e beautified format 2 12 6 Ac 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 command see above For user defined functions member functions see Nu and Xun 2 12 7 Md prints the defaults as described in the previous section shortcut for default see Section 3 11 2 4 2 12 8 le n switches the echo mode on 1 or off 0 If n is explicitly given set echo to n 2 12 9 g n sets the debugging level debug to the non negative integer n 2 12 10 gf n sets the file usage debugging level debugfiles to the non negative integer n 2 12 11 gm n sets the memory debugging level debugmem to the non negative integer n 2 12 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 2 12 13 1 logfile switches log mode on and off If a logfile argument is given change the default logfile nam
332. re X is the main variable of x 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 precision0 x n 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 42 random N 2 returns a random integer between 0 and N 1 N is an integer which can be arbitrary large This is an internal PARI function and does not depend on the system s random number generator The resulting integer is obtained by means of linear congruences and will not be well dis tributed in arithmetic progressions The random seed may be obtained via getrand and reset using setrand Note that random 2731 is not equivalent to random although both return an integer be tween 0 and 2 1 In fact calling random with an argument generates a number of random words 32bit or 64bit depending on the architecture rescaled to the desired interval The default uses directly a 31 bit generator The library syntax is genrand N Also available are pari_rand which returns a random unsigned long 32bit or 64bit depending on the architecture and pari_rand31 which returns a 31bit long integer 3 2 43 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 i
333. 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 1 rational numbers Q i real numbers R i C or even elements of Z nZ in Z nZ t t 1 or p adic numbers when p 3 mod 4 Q i This feature must not be used too rashly in library mode for example you are in principle allowed to create objects which are complex numbers of complex numbers This is not possible under gp But do not expect PARI to make sensible use of such objects you will mainly get nonsense On the other hand one thing which s 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 intmod 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 t_xxx 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 Intmods integers modulo n type t_FRAC Q Rational numbers in irreducible form type t_COMP
334. recision kernel comes in three non exclusive flavours See Appendix A for how to set up these on your system various compilers are supported but the GNU gcc compiler is the definite favourite A first version is written entirely in ANSI C with a C compatible syntax and should be portable without trouble to any 32 or 64 bit computer having no drastic memory constraints We do not know any example of a computer where a port was attempted and failed In a second version time critical parts of the kernel are written in inlined assembler At present this includes e the whole ix86 family Intel AMD Cyrix starting at the 386 up to the Xbox gaming console including the Opteron 64 bit processor e three versions for the Sparc architecture version 7 version 8 with SuperSparc processors and version 8 with MicroSparc I or II processors UltraSparcs use the MicroSparc II version e the DEC Alpha 64 bit processor e the Intel Itanium 64 bit processor e the PowerPC equipping modern macintoshs G3 G4 etc e the HPPA processors both 32 and 64 bit A third version uses the GNU MP library to implement most of its multiprecision kernel It improves significantly on the native one for large operands say 100 decimal digits of accuracy or more You should enable it if GMP is present on your system Parts of the first version are still in use within the GMP kernel but are scheduled to disappear An historical version of the PARI GP kernel
335. reg a 1 used to contain an obsolete check number the number of roots of unity and a generator bnf tu the fundamental units bnf fu 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 GDa 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 bnrisprincipal or those corresponding to the relations in W and B when the bnf internal precision needs to be increased An sbnf 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 r1 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
336. rithm due to Buchmann and Lenstra which is usually less efficient The library syntax is factorpadic4 pol p r where r is a long integer 163 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 z is a rational function it is assumed that the base ring is an integral domain of characteristic zero The library syntax is integ z 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 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 Zp or can be an integral element of a finite extension of Qp given as a t_POLMOD at least one of whose coefficients is a t_PADIC In this case the result is the vector of roots belonging to the same extension of Q as a The library syntax is padicappr pol a 3 7 7 polcoeff x s 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 polynomial of degree 0 and to rational functions provided the denominator is a monomial The librar
337. rns 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 z There are no known composite numbers passing this test in particular all composites lt 10 are correctly 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 long should be used if x is known to be of type integer 3 4 33 issquare x amp n true 1 if x is a square false 0 if not What being a square means depends on the type of x all t_COMPLEX are squares as well as all non negative t_REAL for exact types such as t_INT t_FRAC and t_INTMOD squares are numbers of the form s with s in Z Q and Z NZ respectively issquare 3 as an integer hi 0 issquare 3 as a real number 42 1 issquare Mod 7 8 in Z 8Z 43 0 issquare 5 0 1374 Ml in Q_13 14 0 103 If n is g
338. rovides a solution in heuristic subexponential time in D assuming the GRH The library syntax is qfbsolve Q n 3 4 55 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 101 or when the structure is wanted It is a special case of bnfinit which is slower but more robust 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 3 3 this does not work at all use the general function bnfnarrow Optional parameter tech is a row vector of the form c1 c2 where c lt cz are positive real numbers which control the execution time and the stack size For a given c1 set ca C to get maximum speed To get a rigorous result under GRH you must take cz gt 6 Reasonable values for c are between 0 1 and 2 More precisely the algorithm will assume that prime ideals of norm less than ca log D generate the class group but the bulk of the work is done with prime ideals of norm less than c log D A larger c means that relations are easier to find but more relations are needed and the linear algebra will be harder The default is cy co 0 2 so the result is not rigorously proven The result is a vector v with 3 components if D lt 0 and 4 ot
339. rt instead 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 8 Ser z 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 precd1 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 warning given for Pol also 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 9 Set x converts x into a set i e into a row vector of character strings with strictly increasing entries with respect to lexicographic ordering 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 TT 3 2 10 Str x converts its argument list into a single character string type t_STR the empty string if x is omitted To recover an ordinary GEN from a string apply eval to it The arguments of Str are evalua
340. runs from 1 to n In particular vector n i expr is not equivalent to v vector n for i 1 n vlil expr as the following example shows 3 v vector n vector n i i gt 2 3 4 v vector n for i 1 n vlil i gt 2 0 4 The library syntax is vecteur GEN nmax entree ep char expr 3 8 61 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 62 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 In the descriptions the letter X will always denote any simple variable name and represents the formal parameter used in the function The expression to be summed integrated etc is any legal
341. s in 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 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 is very small you may try setting flag 1 which uses a naive search 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 given to p adic precision r Multiple roots are not repeated p is assumed to be a prime and pol to be non zero modulo p Note that this is not the same as the roots in Z p Z rather it gives approximations in Z p Z of the true roots living in Qp If pol has i
342. s 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 n column if x is a polynomial the n 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 which can be recalled by typing t under gp The library syntax is compo z n where n is a long 80 The two other methods are more natural but more restricted The function polcoeff z 7 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 23 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 p
343. s given rec is omitted a break loop will be started see Section 2 9 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 Note The interface is currently not adequate for trapping individual exceptions In the current version 2 3 3 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 archer not available on this architecture or operating system errpile the PARI stack overflows gdiver division by 0 invmoder impossible inverse modulo siginter SIGINT received usually from Control C talker miscellaneous error typeer wrong type user user error from the error function 209 3 11 2 27 type x this is useful only under gp Returns the internal type name of the PARI object x as a string Check out existing type names with the metacommand Xt For example type 1 will return t_INT The library syntax is type0 7 though the macro typ is usually simpler to use since it return an integer that can easily be matched with the symbols t_ The name type was avoided due to the fact that type is a reserved identifier for some C compilers 3
344. s 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 44 round z amp e If x is in R rounds z to the nearest integer and sets e to the number of error bits that is the binary 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 84 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 24x X 1 7 truncate C 24x whereas q 24 X2 17Y_ 2 X 2 roun xX xX 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 x e where e is a long integer Also available is ground z 3 2 45 simplify x this function simplifies x as much as it can Specifically a complex or quadratic number whose imaginary part is an exact 0 i e not an approximate one as a 0 3 or 0 E 28 is converted to its real part and a polynomial of degree 0 is converted to its constant term Simplifications occur recursively This function is especially useful before using arithmetic functions which expect integer argu ments T X T A O hi 1
345. s in reasonable time as long as the integrand is regular It is also reasonable to compute numerically integrals in several variables although more than two becomes lengthy The integration domain may be non compact and the integrand may have reasonable singularities at endpoints To use intnum the user must split the integral into a sum of subintegrals where the function has possible singularities only at the endpoints Polynomials in logarithms are not considered singular and neglecting these logs singularities are assumed to be algebraic in other words asymptotic to C x a for some a such that a gt 1 when z is close to a or to correspond to simple discontinuities of some higher derivative of the function For instance the point 0 is a singularity of abs x See also the discrete summation methods below sharing the prefix sum 3 9 1 intcirc X a R expr tab numerical integration of expr with respect to X on the circle X al R divided by 2iz In other words when expr is a meromorphic function sum of the residues in the corresponding disk tab is as in intnum except that if computed with intnuminit it should be with the endpoints 1 1 p105 intcirc s 1 0 5 zeta s 1 time 3 460 ms 1 2 40 E 104 2 7 E 106 I The library syntax is intcirc void E GEN eval GEN void GEN a GEN R GEN tab long prec 183 3 9 2 intfouriercos X a b z expr tab numerical integration
346. s may differ slightly from their C counterparts 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 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 This is not a reference to the variable x i e an lvalue thus x 2 3 35 is invalid 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 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
347. s must be in R or a vector of steps s1 Sn 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 7 forsubgroup H G B seq evaluates 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 2Zgs G 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 92 This routine is intended to treat huge groups when subgrouplist is not an option due to the sheer size of the output 203 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 O 1 1 0 O 2 2 0 O 1 1 0 O 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 th
348. s respectively which should be enough If you ever need hundreds of variables you should probably be using vectors instead See Section 2 5 4 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 23 Caveat Power series with inexact coefficients sometimes have a non intuitive behaviour if k significant terms are requested an inexact zero is counted as significant even if it is the coefficient of lowest degree This means that useful higher order terms may be disregarded If the series precision is insufficient errors may occur mostly division by 0 which could have been avoided by a better global understanding of the computation A 1 y 0 B 1 O y 7 B denominator A 12 0 E 28 OCy 7 A B eK division by zero A B k Warning normalizing a series with O leading term EK division by zero A 1 B k Warning normalizing a series with O leading term 13 1 0000000000000000000
349. s 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 x k Corresponding to flag 1 is idealpowred nf vp k prec where prec is a long 140 3 6 65 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 p is given the result is undefined The result is a vector of pr structures each representing one of the prime ideals above p in the number field nf The representation P p a e f b of a prime ideal means the following The prime ideal is equal to pZk aZx where Zg is the ring of integers of the field and a 5 aw where the w form the integral basis nf zk e is the ramification index f is the residual index and b represents a 3 Zg such that P Zg B pZ kx 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
350. s the random number generator to the value n The initial seed is n 1 The library syntax is setrand n where n is a long Returns n 208 UNIX 3 11 2 25 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 26 trap e rec seq tries to evaluate seg 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 9 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 72 INFINITY If seg is omitted defines rec as a default action when catching exception 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 s print reorder writebin crash If no recovery code i
351. se 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 See also factor If flag 0 use a combination of Baillie PSW pseudo primality test see ispseudoprime Selfridge p 1 test if x 1 is smooth enough and Adleman Pomerance Rumely Cohen Lenstra APRCL for general z 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 passing 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 1is 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 32 ispseudoprime z flag true 1 if x is a strong pseudo prime see below false 0 otherwise If this function retu
352. se 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 11 1 s 2 1 plotscale 0 0 1000 0 1000 3 10 16 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 17 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 y1 to 2 y2 where 1 yl is not necessarily the position of the virtual cursor use plotmove w x1 y1 before using this function 3 1
353. setting y to a function of x after some computations with bivariate power series does not make sense a priori This is because implicit constants in expressions like O x depend on y whereas in O y they can not depend on x For instance 7 O x x y 1 O x 0 y x 12 OCy x Here is a more involved example A 1 x72 1 0 x B 1 x 1 0 x73 subst z A z B 42 x 3 x 2 x 1 1 0 x 7 Be A 13 x 3 x7 2 x 1 0 1 zak A 14 z x 2 z 0 x 40 The discrepancy between 2 and 3 is surprising Why does 42 contain a spurious constant term which cannot be deduced from the input Well we ignored the rule that forbids to substitute an expression involving high priority variables to a low priority variable The result 4 is correct according to our rules since the implicit constant in O x may depend on z It is obviously wrong if z is allowed to have negative valuation in x Of course the correct error term should be O xz but this is not possible in PARI 2 6 User defined functions 2 6 1 Definition It is easy to define a new function in 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 21 local to t1 3 local the first newline is disregarded due to the preceding sign and the others because
354. sons 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 gqfbclassno 400000028 time 3 140 ms hi 1 quadclassunit 400000028 no time 20 ms much faster 12 1 gfbclassno 400000028 time O ms 13 7253 correct and fast enough quadclassunit 400000028 no time O ms 4 7253 The library syntax is qfbclassno0 D flag Also available classno D qfbclassno D classno2 D qfbclassno D 1 and finally we have the function hclassno D which computes the class number of an imaginary quadratic field by counting reduced forms an O D algorithm See also qfbhclassno 3 4 47 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 3 4 48 qfbhclassno x Hurwitz class number of x where x is non negative and congruent to 0 or 3 modulo 4 For x gt 5 10 we assume the GRH and use quadclassunit with default parameters The library syntax is hclassno z 106 3 4 49 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 a la Atkin l is any positive constant but for optimal speed one should take 1
355. ss than b In this variant an explicit b must be provided In both these cases x is assumed to have integral entries The implementation uses low precision floating point computations for maximal speed which gives incorrect result when x has large entries The condition is checked in the code and the routine will raise an error if large rounding errors occur A more robust but much slower implementation is chosen if the following flag is used If flag 2 x can have non integral real entries In this case if b is omitted the minimal vectors only have approximately the same norm If bis omitted mis an upper bound for the number of vectors that will be stored and returned but all minimal vectors are nevertheless enumerated If m is omitted all vectors found are stored and returned note that this may be a huge vector The library syntax is qfminim0O z b m flag prec also available are minim x b m flag 0 minim2 x b m flag 1 In all cases an omitted b or m is coded as NULL 3 8 49 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 minimal vectors As a side note to old timers this used to fail bluntly when x had more than 5000 minimal vectors Bewar
356. st 3 2 3 Mat 3 2 4 Mod 3 2 5 Pol 3 2 6 Polrev 3 2 7 Qfb 3 2 8 Ser 3 2 9 Set 3 2 10 Str 3 2 11 Strchr 3 2 12 Strexpand 3 2 13 Strtex 3 2 14 Vec 3 2 15 Vecsmall 3 2 16 binary 3 2 17 bitand 3 2 18 bitneg 3 2 19 bitnegimply 3 2 20 bitor 3 2 21 bittest 3 2 22 bitxor 3 2 23 ceil 3 2 24 centerlift 3 2 25 changevar 3 2 26 components of a PARI object 3 2 27 conj 67 69 71 71 71 71 71 71 71 72 72 72 74 74 74 74 75 75 75 75 75 75 76 76 76 76 STI 77 77 KIT 77 78 78 78 78 78 78 79 79 79 79 79 80 80 80 80 80 80 81 3 2 28 conjvec 3 2 29 denominator 3 2 30 floor 3 2 31 frac 3 2 32 imag 3 2 33 length 3 2 34 lift 3 2 35 norm 3 2 36 norml2 3 2 37 numerator 3 2 38 numtoperm 3 2 39 padicprec 3 2 40 permtonum 3 2 41 precision 3 2 42 random 3 2 43 real 3 2 44 round 3 2 45 simplify 3 2 46 sizebyte 3 2 47 sizedigit 3 2 48 truncate 3 2 49 valuation 3 2 50 variable 3 3 Transcendental functions 331 3 3 2 Euler 3 3 3 I 3 3 4 Pi 3 3 5 abs 3 3 6 acos 3 3 7 acosh 3 3 8 agm 3 3 9 arg 3 3 10 asin 3 3 11 asinh 3 3 12 atan 3 3 13 atanh 3 3 14 bernfrac 3 3 15 bernreal 3 3 16 bernvec 3 3 17 besselh1 3 3 18 besselh2 3 3 19 besseli 3 3 20 besselj 3 3 21 besseljh 3 3 22 besselk 3 3 23 besseln 3 3 24 cos 3 3 25 cosh 81
357. st be in absolute value less than 2288135454 i e roughly 80807123 digits The precision of real numbers is also at most 80807123 significant decimal digits and the binary exponent must be in absolute value less than 27 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 the native PARI kernel does not contain asymptotically fast DFT based techniques Hence although it is possible to use PARI to do computations with 107 decimal digits better programs can be written for such huge numbers At the very least the GMP kernel should be used at this point For reasons of backward compatibility we cannot enable GMP by default but you probably should enable it Integers and real numbers are non recursive types and are sometimes called leaves 1 2 2 Intmods rational numbers p adic numbers polmods and rational functions these are recursive but in a restricted way For intmods 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 ap proximation to the p adic number Here Z is considered as the projective
358. 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 91 ex2 a 2 b ex concat b a 2 ex2 46 concat a b 13 2ex 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 used 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 a string is expected 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 t
359. t see the last section 3 Troubleshooting and fine tuning In case the default Configure run fails miserably try Configure a interactive mode and answer all the questions there are not that many Of course Configure still provides 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 see the last section 3 1 Installation directories 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 Architecture independent files 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 elldata or 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 3 2 Environment variables Configure lets the following environment variable override the defaults if set AS Assembler
360. t 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 37 2 5 Variables and symbolic expressions 2 5 1 Variable names In GP you can use up to 16383 variable names up to 65535 on 64 bit machines A valid identifier name starts with a letter and contain only valid keyword characters or alphanumeric characters A Za z0 9 You may not use built in function names see the list with c including the constants Pi Euler and I 1 Note that GP names are case sensitive This means for instance that the symbol i is perfectly safe to use and will not be mistaken for y 1 and that o is not synonymous to 0 We will see in Section 2 6 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 2 5 2 Vectors and matrices If the variable x contains a vector or list x m refers to its m th entry You can assign a result to x m i e write something like z k expr If x is a matrix x m n referes to its m n entry you can assign a result to z m n but not to x m If you want to assign an expression to the m th column of a matrix x use x m expr instead Similarly use z m expr
361. t egwi g grits de Be 116 power series 20 21 33 PONTE ss eon ek Swe ee e 72 87 powraw 2 A a ie e o 107 DT oe ee ee ee a A 140 DESCAL asss poa egansa a ea we 87 PTECISION s 08 a a aa oe Bw e A 86 precision a se ta al wee A 83 precisionO 0 84 precprime oo 105 preferences file 2f Ba bez prettymatriz format 57 prettyprint format 4 57 prettyprinter 57 PLINE he eon a a a es 105 primedec 000 140 primeform lt s eo sor eac es krema 107 primelimit 57 153 PEIMOPI aa oe o a a 105 Primes sa coa a we ws 105 principal ideal 141 principalideal 141 principalidele 142 PEINE y tes he hes 46 48 207 PHANG ue ra ae a Seine Gow woe amp 207 PYINtp 2 464 ae eas Gah we es 207 DEIDTPD L yay a Bow ee ed et es 207 PRINttex fa sak Pe wa ESE Ss 207 PLIOLIGY e os ds oe be A 35 e s eee Be eke BS ee eee Bs 191 prodeuler a soas kaa ota axe da os 191 prodinf sc coso e e torace ew ees 191 prodinfi 2 6 ei 8b bee a 191 prodct s sa Kd keels Ee ae 71 Produit casae eda aa ee 191 programming 201 projective module o oo o 120 230 PLOMPL dice a ee we ee a es 57 Prompt_cont ss es sest Hak s mai 58 PSdraw aes 62 aaa E aie x 201 pseudo basis sire ee 120 pseudo matrig 2 ee 120 Psiile local dl 58 197 A 92 PEPLO DO so we ew A ae aa 201
362. t will print the list of available extra tests among all available targets The make bench and make test compat runs produce a Postscript file pari ps in Oxxx which you can send to a Postscript printer The output should bear some similarity to the screen images 3 6 3 Heavy duty testing Optional There are a few extra tests which should be useful only for developpers make test kernel checks whether the low level kernel seems to work and provides simple diagnostics if it does not Only useful if make bench fails horribly e g things like 1 1 do not work make test all runs all available test suites Slow 4 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 are an avid gp fan do not forget to delete the old pari libraries once in a while This installs in the directories chosen at Configure time the default GP executable
363. tandalone foo would be enough though we do not 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 Finally an optional argument between braces followed by a star like 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 p 4 p2 and so on using the available consecutive powers of 2 69 Mnemonics for flags Num
364. tclassunit0 Vox rado aa 1 23 DOTFCLEP ea ee ek 124 bnfdecodemodule 124 131 bnfinit los 4 2 2e4548 108 119 124 DOTIDITO di ebb bk Be Si doe EGG 125 bnfisintnorm 125 126 bnfisnorm les cards a0 a4 125 126 bnfisprincipal 108 125 126 DATISSUIDIE ecc iaa ee ac 126 DATISUNDIC as goes a e E 126 DRATMAKe os o wok ee ae Ka Be 126 bnfnarrow 108 127 DATOE eras bead a 127 bnfsignunit issos s sosi a egt a 127 DE SUDATE s s sise sofa E agert 127 128 DATUNTE id tao Ge uo a Be 128 DN Gees a Gd RS RSS 45 119 DONTC ASS a s toea i ea eS 129 bnrclassO 129 bDnrelassno e cio ome 4 3 24 129 131 bnrclassnolist 129 138 bnrconductor 2 66 Ske ee ee OS 129 bnrconductorofchar 130 PHYdISC y io Rw ew ae 130 131 DOATATSCO 6 so i misg Fa ek ke as 130 b rdisclist e da rada aoa E 130 138 bHrdpsclistO a 64 4 44h ay ass 131 D TINIG ma Greta ee Gy Ge e Es 129 131 DOTINITO ea s coem id aa MW He al bHYISCONdUCTOY 3 2 aie oP kk we 1 31 bnrisprincipal 125 131 e al on os Yee dae pe 128 129 bnrrootnumber 131 132 bnrstark 109 132 161 boolean operators 74 brace characters 30 break OOD cco unu See a wh aoe eS 50 209 Break e esma aca BE By eS Gah ees 50 202 Breil oe aa ale amp aa me eS ee 112 DUCATI oe eines Bia ah eae hye Bo i a 128 b chimag ee soe soa oe Be ee ee
365. ted discrete logarithm problems e bnr denotes a ray number field i e a data structure output by bnrinit corresponding to the ray class group structure of the field for some modulus f It contains a bnf the modulus f the ray class group Cly A and data associated to the discrete logarithm problem therein 3 6 2 Algebraic numbers and ideals An algebraic number belonging to K Q X T is given as e a t_INT t_FRAC or t_POL implicitly modulo T or e a t_POLMOD modulo T or e a t_COL v of dimension N K Q representing the element in terms of the computed integral basis as sum i 1 N v i nf zk i Note that a t_VEC will not be recognized An ideal is given in any of the following ways e an algebraic number in one of the above forms defining a principal ideal e a prime ideal i e a 5 component vector in the format output by idealprimedec e a t_MAT square and in Hermite Normal Form or at least upper triangular with non negative coefficients whose columns represent a basis of the ideal One may use idealhnf to convert an ideal to the last preferred format Note Some routines accept non square matrices but using this format is strongly discouraged Nevertheless their behaviour is as follows If strictly less than N K Q generators are given it is assumed they form a Zx basis If N or more are given a Z basis is assumed If exactly N are given it is further assumed the matrix is in HNF If any of t
366. ted in string context see Section 2 8 x2 0 i 2 Str x i 91 xo eval 12 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 3 2 11 Strchr x converts x to a string translating each integer into a character Strchr 97 1 a Vecsmall hello world 72 Vecsmal1 104 101 108 108 111 32 119 111 114 108 100 Strchr 3 hello world 3 2 12 Strexpand 1 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 11 This feature can be used to read environment variable values Strexpand HOME doc 1 home pari doc The individual arguments are read in string context see Section 2 8 3 2 13 Strtex 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 8 3 2 14 Vec x transforms the object x into a row vector The vector will be with one component only except when x is a vector or a quadratic form in which case the resulting vector is simply the initial object considered as a row vector a matrix the vector of columns comprising the matrix is return a character stri
367. ted to a power series with positive valuation the result is MEA z 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 21 The library syntax is eta x prec 3 3 31 exp 1 exponential of x p adic arguments with positive valuation are accepted The library syntax is gexp z prec 3 3 32 gammah z gamma function evaluated at the argument x 1 2 The library syntax is ggamd z prec 90 3 3 33 gamma 2 gamma function of z The library syntax is ggamma zx prec 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 The argument x and s are complex numbers x must be a positive real number if s 0 The result returned is jis e tts 1 dt When y is given assume of course without checking that y s For small x this will 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 3 3 36 incgamc s x complementary incomplete gamma function The arguments x and s are complex numbers such that s is not a pole of T and z s 1 is not much larger than 1 otherwise the convergence is very slow The result returned
368. th precision varying as needed Lehmer s trick as generalized by Schnorr If flag 1 G has integer entries still positive but not necessarily definite i e x needs not have maximal rank The computations are all done in integers and should be slower than the default unless the latter triggers accuracy problems flag 4 G has integer entries gives the kernel and reduced image of x flag 5 same as case 4 but G may have polynomial coefficients The library syntax is qflllgramo0 G flag prec Also available are Mgram G prec flag 0 lllgramint G flag 1 and Wlgramkerim G flag 4 178 3 8 48 qfminim x b m flag 0 x being a square and symmetric matrix representing a positive definite quadratic form this function deals with the vectors of x whose norm is less than or equal to b enumerated using the Fincke Pohst algorithm The function searches for the minimal non zero vectors if b is omitted The precise behaviour depends on flag If flag 0 default seeks at most 2m vectors The result is a three component vector the first component being the number of vectors found 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 The vectors are returned in no particular order In this variant an explicit m must be provided If flag 1 ignores m and returns the first vector whose norm is le
369. 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 A4 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 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 0 146 H concat H N i K nfinit y 2 7 polL x 4 y x 3 3 x72
370. the function type 2 4 GP operators Loosely speaking an operator 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 is performed first Operations at the same priority level are performed in the order they were written i e from left to right Anything enclosed between parenthesis is considered a complete subexpression and is 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 OPa Z OPy Y Ops Y is equivalent to op y opz 2 op x ops y GP contains 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 operator
371. 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 elllocalred F p 115 3 5 20 elllseries E s 4 1 E being an sell as output by ellinit this computes the value of the L series of E at s It is assumed that E is defined over Q not necessarily minimal 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 NV algorithm The library syntax is elllseries E s A prec where prec is a long and an omitted A is coded as NULL 3 5 21 ellminimalmodel amp v return the standard 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 code
372. the rectangle which is such that the points x1 y1 and x1 dx yl dy are opposite corners where x1 y1 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 24 plotrecth w X a b ezpr flag 0 n 0 writes to rectwindow w the curve output of ploth w X a b expr flag n 3 10 25 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 26 plotrline w dz dy draw in the rectwindow w the part of the segment x1 yl x1 dx yl dy which is inside w where x1 y1 is the current position of the virtual cursor and move the virtual cursor to 11 dx yl dy even if it is outside the window 3 10 27 plotrmove w dx dy move the virtual cursor of the rectwindow w to position x1 dx yl dy where x1 yl is the initial position of the cursor i e to position dx dy relative to the initial cursor 3 10 28 plotrpoint w dx dy draw the point x1 dx yl dy on the rectwindow w if it is inside w where x1
373. 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 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 betw
374. theses 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 do not want a toggle you can use M 0 M 1 to specifically switch it on or off Note In some 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 do not 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 15 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
375. thm for computing 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 which for ease of presentation is in fact output as a one column matrix It is not a bnf you need bnfinit for that First we describe the default behaviour flag 0 v 1 is equal to the polynomial P 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 product of cyclic groups of order n v clgp cyc and the corresponding generators of the class group of respective orders n v clgp gen 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 polynomia
376. ting 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 You can type a anywhere It is interpreted as above only if apart from ignored whitespace characters it is immediately followed by a newline For example you can type 73 4 29 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 since an equal sign can never end a valid expression gp disregards 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 An opening brace at the beginning of a line modulo spaces as usual signals that you are typing a multi line command and newlines are ignored until you type a closing brace There is an important but easily obeyed restriction inside an open brace close brace pair all your input lines are 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 this is interpreted as a bb c which assigns the value of c to both bb and a 2 3 The PARI
377. tion is dangerous p 3 fix 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 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 14 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 the Libpari Manual 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 funct
378. to Hastad Lagarias and Schnorr STACS 1986 If the precision is too low the routine may enter an infinite loop If flag 2 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 element is defined over the field of coefficients of x and is in general not integral If flag 3 uses the PSLQ algorithm This may return a real number B indicating that the input accuracy was exhausted and that no relation exist whose sup norm is less than B If flag 4 uses an experimental 2 level PSLQ which does not work at all Should be rewritten The library syntax is lindepO z flag prec Also available is lindep z prec flag 0 171 3 8 5 listcreate n creates an empty list of maximal length n This function is useless in library mode 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
379. to assign an expression to the m th row of x This process is recursive so if x is a matrix of matrices of an expression such as x 1 1 3 4 1 is perfectly valid and actually identical to x 1 1 4 3 1 assuming that all matrices along the way have compatible dimensions 2 5 3 Variables and polynomials The main thing to understand is that PARI GP is not a symbolic manipulation package One of the main consequences of this fact is that all expressions are evaluated as soon as they are written they never stay in an abstract form As an important example consider what happens when you use a variable name before assigning a value into it x say This is perfectly acceptable it is considered as a monomial of degree 1 in the variable x p x 2 1 11 x 2 1 x 2 x 24 1 43 5 p 14 x72 1 eval p 45 5 As is shown above assigning a value to a variable does not affect polynomials that used it to take into account the new variable s value one must use the function eval see Section 3 7 3 It is in general preferable to use subst rather than assigning values to polynomial variables 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 38 2 5 4 Variable priorities multivariate objects PARI has no sparse representation o
380. ts 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 5 4 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 nf x 2 y OK nffactor nf x72 Mod y y 2 1 OK nffactor nf x 2 Mod z z72 1 WRONG NN OY Y The library syntax is nffactor nf x 3 6 94 nffactormod nf x pr factorization of the univariate polynomial z 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 intmod modulo the rational prime under pr polmod or polynomial with intmod coefficients column vector of intmod 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 5 4 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 nffactormod nf x pr 145 3 6 95 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 automorp
381. ts of accuracy to an integer or division by 0 e g inverting 0E1000 when 49 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 9 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 break return or C d EOF any of which will let gp perform its usual cleanup
382. types We 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 A note on efficiency The following types are provided for convenience not for speed t_INTMOD t_FRAC t_PADIC t_QUAD t_POLMOD t_RFRAC Indeed they always perform a reduction of some kind after each basic operation even though it is usually more efficient to perform a single reduction at the end of some complex computation For instance in a convolution product gt jan Ciyj in Z NZ common when multiplying polynomials it is wasteful to perform n reductions modulo N In short basic individual operations on these types are fast but recursive objects with such components could be handled more efficiently programming with libpari will save large constant factors here compared to GP 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 is the supremum of the input precision and the default precision For example if the default precision is 28 digits typing 2 gives a number with internal precision 28 but typing a 45 significant digit real number gives 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 no
383. ular concatenation Note that this is definitely not the case for empty vectors or 170 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 42 1 3 2 4 concat 1 2 3 4 5 6 43 1 2 5 3 4 6 concat 7 8 1 2 3 411 75 1 25 7 3 4 6 8 1 2 3 4 The library syntax is concat z y 3 8 4 lindep z flag 0 a being a vector with p adic or real complex coefficients finds a small integral linear combination among these coefficients If x is p adic flag is meaningless and the algorithm LLL reduces a suitable dual lattice Otherwise the value of flag determines the algorithm used in the current version of PARI we suggest to use non negative values since it is by far the fastest and most robust implementation See the detailed example in Section 3 8 1 algdep If flag gt 0 uses a floating point variable precision LLL algorithm This is in general much faster than the other variants If flag 0 the accuracy is chosen internally using a crude heuristic If flag gt 0 the computation is done with an accuracy of flag decimal digits In that case the parameter flag should be between 0 6 and 0 9 times the number of correct decimal digits in the input If flag 1 uses a variant of the LLL algorithm due
384. umber 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 e Here y must be an integer but x can be any PARI object The library syntax is binomial z y where y must be a long 3 4 7 chinese z y if x and y are both intmods or both polmods creates with the same type a z in the same residue class as 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 chinese is applied recursively to the components of z yielding a residue belonging to the same class as all components of x Finally chinese z 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 chinese x y Also available is chinese1 x corresponding to an om mitted y 96 3 4 8 content 1 computes the gcd of all the coefficients of x when this gcd makes sense This is the natural definition if x is a polynomial and by extension a power series or a vector matrix This is in general a weaker notion than the ideal generated by the coefficients content 2 x y 41 1 ged 2 y over Qly If x is a
385. urrently q must not have large prime factors Beware that Mod 7 19 7 1 2 1 Mod 11 19 is any square root sqrt Mod 7 19 12 Mod 8 19 is the smallest square root Mod 7 19 3 5 3 Mod 1 19 1 37 5 3 14 Mod 1 19 Mod 7 19 is just another cubic root If the exponent is a negative integer an inverse must be computed For non invertible t_INTMOD this will fail and implicitly exhibit a non trivial factor of the modulus Mod 4 6 1 kk impossible inverse modulo Mod 2 6 Here a factor 2 is obtained directly In general take the gcd of the representative and the modulus This is most useful when performing complicated operations modulo an integer N whose factorization is unknown Either the computation succeeds and all is well or a factor d is discovered and the computation may be restarted modulo d or N d For non invertible t_POLMOD this will fail without exhibiting a factor Mod x 2 x73 x 1 non invertible polynomial in RgXQ_inv a Mod 3 4 ry73 Mod 1 4 b y 6ty 5 y 4 y 3 y 2 y 1 Mod a b 1 non invertible polynomial in RgXQ_inv In fact the latter polynomial is invertible but the algorithm used subresultant assumes the base ring is a domain If it is not the case as here for Z 4Z a result will be correct but chances are an error will occur first In this specific case one should work with 2 adics In general one can try the fol
386. usly prove that the result is prime use isprime The library syntax is nextprime z 3 4 39 numdiv x number of divisors of x x must be of type integer The library syntax is numbdiv z 3 4 40 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 8c n n must be of type integer and 1 lt n lt 10 The algorithm uses the Hardy Ramanujan Rademacher formula The library syntax is numbpart n 3 4 41 omega x number of distinct prime divisors of x x must be of type integer The library syntax is omega z the result is a long 3 4 42 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 2 3 4 43 prime z the x prime number which must be among the precalculated primes The library syntax is prime x x must be a long 3 4 44 primepi x the prime counting function Returns the number of primes p p lt x Uses a naive algorithm so that x must be less than primelimit The library syntax is primepi z 3 4 45 primes z creates a row vector whose components are the first x prime numbers which must be among the precalculate
387. ut 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 arow vector with 9 components nf 1 contains the polynomial pol nf pol nf 2 contains r1 r2 nf sign nf r1 nf r2 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 nf index 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 r2 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 Ta 2 Y o x o running over the embeddings of K into C e The 72 component is deprecated and currently unused 147 e T is the n x n matrix whose coefficients are Tr w w where the w are the elements of the integral basis 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 T71 generates the codifferent ideal e Finally MDI is a two element representation fo
388. utes all roots in all possible cases sqrtnall x n 93 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 r z n 1 while r2 r r2x z n V vector 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 x hyperbolic tangent of z The library syntax is gth z prec 3 3 48 teichmuller x Teichm ller character of the p adic number 2 i e the unique p 1 th root of unity congruent to 2 p modulo p The library syntax is teich x 3 3 49 theta q z Jacobi sine theta function The library syntax is theta q z prec 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 x flag 0 one of Weber s three f functions If flag 0 returns f x exp im 24 n 2 1 2 m x such that j f 16 f where j is the elliptic j invariant see the function e11j If flag 1 returns fi 2 2 m x such that j ff 16 fi Finally if flag 2 returns felz V2n 2x n x such that j f2 16 f Note the identities f8 i ES and f fife V2 The library syntax is weber0 x flag prec Asso
389. ution 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 name 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 A which use special symbols This help system can also be menu driven by using the command M Xc which opens a help menu window which enables you to choose the categor
390. utput by bnfinit Misa projective Zg module given by a pseudo basis as output by rnfhnfbasis The routine returns either a true Z x basis of M if it exists or an n l element generating set of M if not where n is the rank of M over K Note that n is the size of the pseudo basis It is allowed to use a polynomial P with coefficients in K instead of M in which case M is defined as the ring of integers of K X P P is assumed irreducible over K viewed as a Z k module The library syntax is rnfbasis bnf x 3 6 120 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 L expressed on the relative integral basis The library syntax is rnfbasistoalg rnf x 3 6 121 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 T a v where v is a variable number 3 6 122 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
391. vshort 2 4 tab A time 50 ms 74 1 26 3 25 E 109 I 50 times faster than A and perfect tab2 intfuncinit t 00 oo gamma 2 I t 73 1 intmellininvshort 2 4 tab2 6 1 2 E 42 3 2 E 109 I AM 63 digits lost In the computation of tab it was not essential to include the exact exponential decrease of T 2 4t But as the last example shows a rough indication must be given otherwise slow decrease is assumed resulting in catastrophic loss of accuracy The library syntax is intmellininvshort GEN sig GEN z GEN tab long prec 3 9 9 intnum X a b expr tab numerical integration of expr on a b possibly infinite interval with respect to X where a and b are coded as explained below The integrand may have values belonging to a vector space over the real numbers in particular it can be complex valued or vector valued If tab is omitted necessary integration tables are computed using intnuminit according to the current precision It may be a positive integer m and tables are computed assuming the integration step is 1 2 Finally tab can be a table output by intnuminit in which case it is used directly This is important if several integrations of the same type are performed on the same kind of interval and functions and the same accuracy since it saves expensive precomputations If tab is omitted the algorithm guesses a reasonable value for m depending on the current precision That value may b
392. w 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 matsolvemod0 m d y flag Also available are gaussmodulo m d y flag 0 and gaussmodulo2 m d y flag 1 3 8 41 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 42 mattranspose x or x transpose of x This has an effect only on vectors and matrices The library syntax is gtrans z 3 8 43 minpoly A v x flag 0 minimal polynomial of A with respect to the variable v i e the monic polynomial P of minimal degree in the variable v such that P A 0 The library syntax is minpoly A v where v is the variable number 3 8 44 qfgaussred q decomposition into squares of the quadratic form represented by the sym metric matrix q 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 q x X aja ajej j gt i The library
393. wed as a Zg module The library syntax is rnfidealhermite rnf x 3 6 135 rnfidealmul rnf x y rnf being a relative number field extension L K as output by rnfinit and x 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 156 3 6 136 rnfidealnormabs rnf x rnf being a relative number field extension L K as output by rnfinit and zv 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 idealnorm rnfidealnormrel rnf x but faster The library syntax is rnfidealnormabs rnf x 3 6 137 rnfidealnormrel 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 relative norm of x as a ideal of K in HNF The library syntax is rnfidealnormrel rnf x 3 6 138 rnfidealreltoabs rnf x rnf being a relative number field extension L K as output by rnfinit and z being a relative ideal 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 pol The following routine might be useful return y rnfidealreltoabs rnf as an id
394. wise The library syntax is pollead x v where v is a long and an omitted v is coded as 1 Also available is leading_term z 3 7 16 pollegendre n v x creates the n 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 assumes the base ring is a domain If flag 0 uses the subresultant algorithm 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 polresultantO 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 z y flag 1 165 3 7 19 polroots pol flag 0 complex roots of the polynomial pol given as a column vector where each root is repeated according to its multiplicity The precision is given as for transcendental function
395. with respect to the sorting criterion is not changed 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 x 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 vecsortO x k flag To omit k use NULL instead You can also use the simpler functions sort 1 vecsort0 z NULL 0 indexsort x vecsort0 x NULL 1 lexsort x vecsort0 x NULL 2 Also available are sindexsort 1 and sindexlexsort x which return a t_VECSMALL v where v 1 v n contain the indices 181 3 8 60 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 Avoid modifying X within expr if you do the formal variable still
396. written in 1985 was specific to 680x0 based computers and was entirely written in MC68020 assembly language It ran on SUN 3 xx Sony News NeXT cubes and on 680x0 based Macs It is no longer part of the PARI distribution to run PARI with a 68k assembler micro kernel one should now use the GMP kernel 1 4 The PARI philosophy The basic principle 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 Specifically an exact operation between exact objects will yield an exact object For example dividing 1 by 3 does not give 0 33333 but simply the rational number 1 3 To get the result as a floating point real number evaluate 1 3 or add 0 to 1 3 Conversely the result of operations between imprecise objects will be as precise as possible Consider for example 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 and also on the size of x y From this data PARI works out the right precision for the result Even if it is working in calculator mode gp where there is a notion of default precision which is only used to convert exact types to inexact ones 23 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 fo
397. 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 2 A B Also available is bestappr x A corresponding to an omitted B 3 4 3 bezout x y finds u and v minimal in a natural sense such that x x u y v gcd x 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 algorithm uses subresultant assumes the base ring is a domain The library syntax is vecbezoutres z y to get the vector or subresext z y Zu 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 n
398. xactzero z Also note that the gcmp and gequal 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 74 3 1 14 lex z y gives the result 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 Aid lex 1 3 1 3 1 2 1 lex 1 1 13 1 lex 1 1 714 0 The library syntax is lexcmp z y 3 1 15 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 16 max z y and min z y
399. xtended 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 priority To get the default help use 7 default log 7 default simplify 727 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 e g t_INT or integer as well as various miscellaneous keywords such as edit short summary of line edito
400. y integral element of nf which must have valuation equal to 0 at all prime ideals dividing I bid 1 This function 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 x lt di and x or mod I Note that when is a module this implies also sign conditions on the embeddings The library syntax is zideallog nf x bid 3 6 61 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 62 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 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 0 where as usual prec is a C long integer representing the precision 3 6 63 idealnorm nf x computes the norm of the ideal x in the number field nf The library syntax is idealnorm nf x 3 6 64 idealpow nf x k flag 0 compute
401. y of commands for which you want an explanation Nevertheless if extended help is available on your system see Section 2 12 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 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 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 65 You also have at your disposal a few other commands and many possible customizations colours prompt
402. y 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 The degree of 0 is a fixed negative number whose exact value should not be used The degree of a non zero scalar is 0 Finally when x is a non zero polynomial or rational function returns the ordinary degree of x Raise an error otherwise The library syntax is poldegree z 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 elementary 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 discrimin
403. y where a suitable version of PARI can be built The GP binary built above with make all or make gp 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 3 4 Multiprecision kernel The kernel can be fully specified via the kernel fqkn 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 hppa64 ia64 ix86 x86_64 m68k ppc sparcv7 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 gt e A name not containing a dash is an alias An alias stands for name none but gmp stand for auto gmp e The default kernel is auto none 214 3 5 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 will not 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 d
404. y x 1 rnfgaloisconj K polL K automorphisms of L The library syntax is galoisconj0 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 97 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 98 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 99 nfhnfmod nf x deta given a pseudo matrix A I and an ideal detx which is contained in read integral multiple of the determinant of 4 1 finds a pseudo basis in Hermite normal form of the module generated by 4 1 This avoids coefficient explosion detz can be computed using the function nfdetint The library syntax is nfhermitemod nf x detz 3 6 100 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 nfzrzx functions b
405. yl is the current position of the cursor and in any case move the virtual cursor to position x1 dx yl dy 3 10 29 plotscale w x1 x2 yl y2 scale the local coordinates of the rectwindow w so that x goes from z1 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 30 plotstring w x flag 0 draw on the rectwindow w the String x see Section 2 8 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 31 psdraw list same as plotdraw except that the output is a PostScript program appended to the psfile 3 10 32 psploth X a b expr same as ploth except that the output is a PostScript program appended to the psfile 3 10 33 psplothraw listz listy same as plothraw except that the output is a PostScript pro gram appended to the psfile 20
406. 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 is the canonical integral basis of the quadratic order i e w Vd 2 if d 0mod4 and w 1 vd 2 if d 1 mod4 where d is the discriminant and to enter yw you just type x yx w 2 3 8 Polmods type t_POLMOD exactly as for intmods to enter mod y where x and y are polynomials type Mod x y not x y 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 Note that this type is available for convenience not for speed each elementary operation involves a reduction modulo y Important remark Mathematically the variables occurring in a polmod are not free variables But internally a congruence class in R t y is represented by its representative of lowest degree which is a t_POL in R t and computations occur with polynomials in the variable t PARI will not recognize that Mod y y 2 1 is the same as Mod x x72 1 since x and y are different variables To avoid inconsistencies polmods must use the same variable in internal operations i e be tween pol
407. yped 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 involving 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 2 8 2 Keywords Since there are cases where expansion is not desirable we now distinguish between Keywords and Strings String is what has been described so far Keywords are special relatives of
408. ystem of fundamental units of the general algebraic number field K defined by the irreducible polynomial P with integer coefficients 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 we insist on finding the fundamental units exactly Be warned that this can take a very long time when the coefficients of the fundamental units on the integral basis are very large If the fundamental units are simply too large to be represented in this form an error message is issued They could be obtained using the so called compact representation of algebraic numbers as a formal product of algebraic integers The latter is implemented internally but not publicly accessible yet When flag 2 on the contrary it is initially agreed that units are not computed Note that the resulting bnf will not be suitable for bnrinit and that this flag provides negligible time savings compared to the default In short it is deprecated When flag 3 computes a very small version of bnfinit a small Buchmann s number field or sbnf for short which contains enough information to recover the full bnf vector very rapidly but which is much smaller and hence easy to store and print It is supposed to be used in conjunction with bnfmake tech is a technical vector empty by default see 3 6 7 Careful use of this parameter may speed up your
409. zeta E z 3 5 34 ellztopoint F z E being an ell as output by ellinit computes the coordinates x y on the curve E corresponding to the complex number z Hence this is the inverse function of el1 pointtoz In other words if the curve is put in 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 z prec 118 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 are found in section Section 3 4 Arithmetic functions 3 6 1 Number field structures Let K Q X T a number field Zx its ring of integers T Z X is monic Three basic number field structures can be associated to K in GP e nf denotes a number field i e a data structure output by nfinit This contains the basic arithmetic data associated to the number field signature maximal order given by a basis nf zk discriminant defining polynomial T etc e bnf denotes a Buchmann s number field i e a data structure output by bnfinit This contains nf and the deeper invariants of the field units U K class group CI K as well as technical data required to solve the two associa

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