User`s manual
Contents
1. alphaCertified gt Polys x y 2 2 y 9 gt Vare I 2 gt Points 0 0 1 1 1 gt PointsData alphaCertified alphaPath Polys Vars Points 6 2 alphaCertifiedExp The Maple procedure alphaCertified constructs the input files for a polynomial exponential system runs alphaCertified and loads output data PointsData alphaCertifiedExp alphaPath Funcs Vars Points Settings e alphaPath A string which is the path to an alphaCertified executable file e Funcs A list containing the polynomial exponential system e Vars A list containing the variables of the polynomial exponential system e Points A two dimensional list containing the points to test e Settings optional A Maple record containing the configuration settings See Section 6 4 for more details con structing this record e PointsData A Maple record containing output data from alphaCertified The fields in this record are alpha A list containing an upper bound of a at each point beta A list containing an approximation of at each point gamma A list containing an upper bound of y at each point refinedPts A two dimensional list containing the refined points isApproxSoln A list that describes if each point is a certifiable approximate solution Each value is either Unknown Yes or No isDistinctSoln A list that describes if each point corresponds to a dis2. exponential equations Preprint 2011 Available at www math tamu edu jhauenst 15 Appendix A Configurations The configurations for alphaCertified are presented below along with a brief description Table A 1 Configurations for alphaCertified NAME ACCEPTABLE VALUES DEFAULT VALUE ALGORITHM 0 1 2 2 ARITHMETICTYPE 0 or 1 0 PRECISION gt 64 96 REFINEDIGITS gt 0 0 NUMRANDOMSYSTEMS gt 2 2 RANDOMDIGITS gt 0 10 RANDOMSEED gt 0 random NEWTONONLY 0 or 1 0 NUMITERATIONS gt 0 2 REALITYCHECK 1 0 or 1 1 REALITYTEST 0 or 1 0 Configurations e ALGORITHM If ALGORITHM is 0 alphaCertified only determines which points are certifiably approxi mate solutions for the given polynomial system If ALGORITHM is 1 alphaCertified also determines which certifiable approximate solutions correspond to distinct solutions If AL GORITHM is 2 and the polynomial system is real i e has only real coefficients alphaCer tified also determines which certifiable approximate solutions correspond to real solutions e ARITHMETICTYPE If ARITHMETICTYPE is 0 alphaCertified performs all computations using rational cer tifiable arithmetic If ARITHMETICTYPE is 1 alphaCertified performs all computations using floating point arithmetic In this case the results of alphaCertified are soft certified since the floating point errors are not fully controlled One way to control local errors is
3. A string which is the name of the file to create e Points A two dimensional list containing the points to print 12 e arithmeticType The value either 0 or 1 of the configuration setting ARITHMETICTYPE If 0 the coor dinates of the points are printed using a rational representation otherwise the points are printed using a floating point representation Example gt pointsName testPoints gt Points 0 0 1 1 1 I gt arithmeticType 1 gt printPoints pointsName Points arithmeticType 6 7 printPolyExpSystem The Maple procedure printPolyExpSystem constructs a polynomial exponential system file printPolyExpSystem funcName Funcs Vars e funcName A string which is the name of the file to create e Funcs A list containing the polynomial exponential system e Vars A list containing the variables of the polynomial exponential system Example gt funcName polyExpSystem gt Funcs x y z y exp 3 x gt Vars ey gt printPolyExpSystem funcName Funcs Vars 6 8 printPolynomialSystem The Maple procedure printPolynomialSystem constructs a polynomial system file printPolynomialSystem polyName Polys Vars e polyName A string which is the name of the file to create 13 e Polys A list containing the polynomial system e Vars A list containing the variables of the polynomial system Example gt polyName polynomial
4. The Maple procedure alphaCertified constructs the input files for a polynomial system runs al phaCertified and loads output data PointsData alphaCertified alphaPath Polys Vars Points Settings e alphaPath A string which is the path to an alphaCertified executable file e Polys A list containing the polynomial system e Vars A list containing the variables of the polynomial system e Points A two dimensional list containing the points to test e Settings optional A Maple record containing the configuration settings See Section 6 4 for more details con structing this record e PointsData A Maple record containing output data from alphaCertified The fields in this record are alpha A list containing an upper bound of a at each point beta A list containing an approximation of 8 at each point gamma A list containing an upper bound of y at each point refinedPts A two dimensional list containing the refined points isApproxSoln A list that describes if each point is a certifiable approximate solution Each value is either Unknown Yes or No isDistinctSoln A list that describes if each point corresponds to a distinct solution Each value is either Unknown Yes or No isRealSoln A list that describes if each point corresponds to a real solution Each value is either Unknown Yes or No Example gt alphaPath
5. 0 004 1 0 4 1 2 1 12 3 001 9 0 100 2 0 000 70 2 00101 0 000 1 Table 1 Example of a polynomial system file for f 3 2 Polynomial exponential systems Starting with version 1 2 alphaCertified implements the algorithms of 5 for certifying solutions to square systems of polynomial exponential functions The polynomial exponential system must be of the form P Bis sci Dias CS eight Yi Gi Bike 7 1 m where each P is a polynomial with complex rational coefficients 3 is a complex rational number and g z is either exp z sin z cos z sinh z or cosh z Moreover polynomial exponential certification must use floating point arithmetic see Appendix A for more details The first line of the file lists both the number of variables e g n m and the number of polynomials in the polynomial exponential system e g n Then the file contains a block for each polynomial following the structure described in Section 3 1 The final block contains a line for each additional function which lists the integer o a string describing the function g and the real and imaginary parts of The following table lists the strings for the possible functions function string exp z X sin z S cos z C sinh z SH cosh z CH Table 2 Functions and strings r 3xry z4 For a complete example consider f x y z 3 3i xy z 9z 2x 7 The polynomial z cos 3 4i y exponentia
6. configuration settings The fields in this record are algorithm arithmeticType precision refineDigits numRandomSystems randomDigits randomSeed newtonOnly numIterations realityCheck realityTest deleteFiles Example The following constructs the Maple record for the configurations settings described in Table 6 gt Settings defaultSettings gt Settings algorithm 0 gt Settings arithmeticType 1 gt Settings precision 192 11 6 5 loadOutput The Maple procedure loadOutput loads output data PointsData loadOutput newtonOnly algorithm isReal numVars e newtonOnly The value either 0 or 1 of the configuration setting NEWTONONLY e algorithm The value either 0 1 or 2 of the configuration setting ALGORITHM e isReal A boolean value i e either true or false that describes if the input polynomial system is a real polynomial system e numVars The number of variables for the polynomial system e PointsData A Maple record containing output data from alphaCertified See Section 6 1 for the structure of this record Example gt newtonOnly 0 gt algorithm 1 gt isReal true gt numVars 2 gt PointsData loadOutput newtonOnly algorithm isReal numVars 6 6 printPoints The Maple procedure printPoints constructs an input point file printPoints pointsName Points arithmeticType e pointsName
7. to increase PRECISION 16 PRECISION If ARITHMETICTYPE is 1 PRECISION indicates the starting level of precision in bits That is all computations for each point start with this precision but the internal working precision can be increased as needed Standard settings include 64 bits roughly 19 decimal digits 96 28 128 38 160 48 192 57 224 67 and 256 77 In general N bits is equivalent to Nlog 9 2 decimal digits REFINEDIGITS If REFINEDIGITS is positive say 7 all of the certifiable approximate solutions will be refined using Newton s method to be within 1077 of the corresponding solution If ARITHMETIC TYPE is 1 the precision will automatically be increased internally so that the refined point is computed using a precision that has at least decimal digits NUMRANDOMSYSTEMS When the polynomial system is overdetermined alphaCertified will analyze NUMRANDOM SYSTEMS number of randomized square systems RANDOMDIGITS When the polynomial system is overdetermined each randomized square system must have a solution within 1077 to be considered an approximate solution for the overdetermined system where T is RANDOMDIGITS RANDOMSEED RANDOMSEED is the seed for the random number generator NEWTONONLY If NEWTONONLY is 1 alphaCertified performs NUMITERATIONS number of Newton iterations on the points If ARITHMETICTYPE is 1 the precision will automatically be increased internally following each iter
8. Section 3 3 lists the points which are certifiably approximate solutions that correspond to distinct real solutions This file is created when ALGORITHM is 2 and the polynomial system is real redundantSolns This file in the format of the input points files described in Section lists the points which are certifiably approximate solutions and correspond to the same solution as another certifiably approximate solution that precedes it in the list of points This file is created when ALGORITHM is at least 1 refinedPoints This file in the format of the input points files described in Section 3 lists the most accurate internally computed approximation of the corresponding solution If REFINEDIGITS is positive say T then for each certifiable approximate solution the point listed in this file is within 1077 of the corresponding solution This file is always created e summary This file is a human readable summary for each point The first part of this file reprints the onscreen summary of the results This is followed by a block for each point which lists the point the results for that point and the computed values of a and y for both the original point and its corresponding point printed in refinedSolns see 4 for details on how these values are computed The last part of this file contains configuration settings and version information for alphaCertified This file is always created e unknownPoints This file in the for
9. System gt Polys z y 2 2 y y gt Vars eyl gt printPolynomialSystem polyName Polys Vars 6 9 printSettings The Maple procedure printSettings constructs a configuration settings file printSettings settingsName Settings e settingsName A string which is the name of the file to create e Settings A Maple record containing the configuration settings See Section 6 4 for more details con structing this record Example The following constructs the configurations settings file described in Table 6 gt settingsName settings gt Settings defaultSettings gt Settings algorithm 0 gt Settings arithmeticType 1 gt Settings precision 192 gt printSettings settingsName Settings 14 Bibliography 1 D J Bates J D Hauenstein A J Sommese and C W Wampler Bertini Software for numerical algebraic geometry Available at www nd edu sommese bertini 2 L Fousse G Hanrot V Lefevre P Pelissier and P Zimmermann MPFR A Multiple Precision Binary Floating Point Library with Correct Rounding ACM Trans Math Softw 33 2 Art 13 2007 3 T Granlund GNU MP the GNU multiple precision arithmetic library Available at 4 J D Hauenstein and F Sottile alphaCertified certifying solutions to polynomial systems To appear in ACM Trans Math Softw 5 J D Hauenstein and V Levandovskyy Certifying solutions to square systems of polynomial
10. alphaCertified Jonathan D Hauenstein September 16 2011 alphaCertified 1 Introduction to alphaCertified The program alphaCertified by Jonathan D Hauenstein and Frank Sottile implements algorithms based on Smale s a theory to certify solutions to polynomial and polynomial exponential systems This manual provides detailed instructions on how to use alphaCertified while 4 5 provides more information regarding the mathematical theory underlying alphaCertified 2 Compiling alphaCertified The program alphaCertified is written in C and uses the GMP 3 and MPFR libraries to perform rational and arbitrary floating point arithmetic To compile alphaCertified the user needs to verify the settings in Makefile associated with the C compiler and the location of these libraries The following is an example of the first three lines in Makefile which specifies using gcc along with the locations of the GMP and MPFR installation directories COMP gcc GMP home GMP_432 MPFR home MPFR _2_4 2 The following is the next two lines in Makefile which specifies to the C compiler which libraries to link to and location of the header files LIB lm L MPFR lib Impfr L GMP lib lgmp INC I MPFR include I GMP include When using gcc for example if the proper static libraries were created adding the static option on the LIB line displayed below will create a statically linked executable LIB static Im L MPFR li
11. ation NUMITERATIONS The number of Newton iterations to perform when NEWTONONLY is 1 that is when alphaCertified is only setup to only perform Newton iterations on the points REALITYCHECK If REALITYCHECK is 1 alphaCertified assumes that the polynomial system defines a real map Otherwise the value of REALITYCHECK instructs alphaCertified on which tests to perform to determine if the polynomial system defines a real map If REALITYCHECK is 0 alphaCertified only checks to see if all of the coefficients are real If REALITYCHECK is 1 alphaCertified checks the coefficients as well as checking to see if the polynomial system as a set is invariant under conjugation REALITYTEST If REALITYTEST is 0 the local approach for determining reality of associated solutions presented in 4 is used If REALITYTEST is 1 the global approach presented in 4 is used 17
12. b Impfr L GMP lib lgmp Once Makefile is setup simply run make to compile alphaCertified 3 Using alphaCertified To use alphaCertified the user needs to create at least two files which specify the polynomial or polynomial exponential system and the points to test An optional third file can be used to adjust the configuration settings See Appendix A for a detailed description of each configuration setting 3 1 Polynomial systems A polynomial system is entered into a file by listing the monomials and the coefficients appearing in each polynomial The system must be square or overdetermined and each coefficient must be a complex rational number The first line of the file lists both the number of variables and the number of polynomials for the polynomial system Then the file contains a block for each polynomial which contains the number of terms followed by the degrees of each variable in the monomial and the real and imaginary parts of its coefficient For example if the polynomial system depends upon three variables say x y and z the term Ean a i z 2 a 1 2 1 1 2 3 would be written as since the degrees of x y and z in xyz are 1 2 and 1 respectively and the coefficient has real 1 part 5 and imaginary part 3 x 3zy 24 For a complete example consider f x y z 3 3i ry z 9z 2x 7 The polyno 10 _ z i mial system file for f is as follows 3 3 3 200 1 0 110 3
13. ed was not able to certify that it was an approximate solution This file is always created isDistinctSoln This file in the same format as isApproxSoln lists values describing whether each point corresponds to a distinct solution A value of 2 means that the point was not able to be certified as an approximate solution A value of 1 means that the point corresponds to a solution that is distinct from the certifiably approximate solutions that precede it in the list of points A nonnegative value say j means that this point and the jt point correspond to the same solution where the points are numbered staring with 0 This file is created when ALGORITHM is at least 1 isRealSoln This file in the same format as isApproxSoln lists values describing whether each point corresponds to a real solution A value of 2 means that the point was not able to be certified as an approximate solution Otherwise the value is a boolean value 0 or 1 describing if the point corresponds to a real solution This file is created when ALGORITHM is 2 and the polynomial system is real nonrealDistinctSolns This file in the format of the input points files described in Section 3 3 lists the points which are certifiably approximate solutions that correspond to distinct nonreal solutions This file is created when ALGORITHM is 2 and the polynomial system is real realDistinctSolns This file in the format of the input points files described in
14. figuration settings and the polynomial system as well as an onscreen summary The following lists the pos sible files created by alphaCertified their format and the settings needed to create them See Appendix A for more details regarding the configurations settings e approxSolns This file in the format of the input points files described in Section 3 3 lists the points which are certifiably approximate solutions This file is always created e constantValues This file lists an upper bound of a an approximation of 3 and an upper bound of y for each point see 4 for details on how these values are computed These values are always printed using a 16 digit floating point representation The format is similar to that presented in Table 5 In particular the first line lists the number of points and then a block for each point which lists the computed values for a 8 and y on separate lines This file is always created distinctSolns This file in the format of the input points files described in Section 3 3 lists the points which are certifiably approximate solutions that correspond to distinct solutions This file is created when ALGORITHM is at least 1 isApproxSoln This file lists boolean values 0 or 1 describing whether each point has been certified to be an approximate solution The first line lists the number of points and then the boolean value is listed for each point Note that a value of 0 means that alphaCertifi
15. l system file for f is as follows 3 2 3 2001 0 1 1 0 3 0 0041 0 4 1 2 1 1 2 3 00 1 9 0 1002 0 0007 0 2 C 3 4 Table 3 Example of a polynomial exponential system file for f 3 3 Input points The points are read into alphaCertified using the requested arithmetic type see Appendix A for more details In particular if alphaCertified is configured to use rational arithmetic the real and imaginary parts of the coordinates for each point must be a rational number Likewise if alphaC ertified is configured to use P bit floating point arithmetic the the real and imaginary parts of the coordinates for each point must be a floating point number and are read into alphaCertified using P bit precision When using floating point arithmetic many of the output files generated by Bertini can be used for input into alphaCertified The first line of the file lists the number of points in the file This is followed by a block for each point which contains the real and imaginary parts of each coordinate For example the following presents 2 i 1 5 20 3 S 7 74 2i 3i 1 8i using rational and 10 digit floating point representation 2 2 3 2 7 0 0 3 1 8 0 7 2 1 2 5 Table 4 Example using rational numbers for 2 0 6666666667 2 0 0 3 0 125 0 7 2 1 0 4 Table 5 Example using floating point numbers for 3 4 Configuration settings To adjust the configuration se
16. mat of the input points files described in Section 3 3 lists the points which can not be certified as approximate solutions This file is always created 5 Performing Newton iterations using alphaCertified When using alphaCertified for certifying solutions the certified approximate solutions can be refined using Newton s method to any given accuracy using the REFINEDIGITS configuration setting In stead of only refining the certified approximate solutions alphaCertified can also be used to perform Newton iterations on all of the input points using the NEWTONONLY and NUMITERATIONS configuration settings See Appendix A for more details regarding these configuration settings Note that if floating point arithmetic is begin used i e ARITHMETICTYPE is 1 the internal working precision is automatically increased during each iteration 6 Maple interface for alphaCertified The Maple interface for alphaCertified can be used to construct the input files run alphaCertified and read in output files After updating libname to include the folder where the alphaCerti fied Maple interface is located it can be loaded with the following command gt with alphaCertifiedMaple yielding the following output alphaCerti fied alphaCerti fiedExp de faultExpSettings de faultSettings loadOutput printPoints printPolyExpSystem print PolynomialSystem printSettings The following describes how to use these nine procedures 6 1 alphaCertified
17. tinct solution Each value is either Unknown Yes or No isRealSoln A list that describes if each point corresponds to a real solution Each value is either Unknown Yes or No Example gt alphaPath alphaCertified gt Funes y 2 y exp 3 x gt Vers 2 2 gt Points 0 1 2 1 gt PointsData alphaCertifiedExp alphaPath Funcs Vars Points 6 3 defaultExpSettings The Maple procedure defaultExpSettings constructs the Maple record containing the default con figuration settings for a polynomial exponential system Settings defaultExpSettings e Settings A Maple record containing the default configuration settings The fields in this record are algorithm arithmeticType precision refineDigits numRandomSystems randomDigits randomSeed newtonOnly numIterations realityCheck realityTest deleteFiles 10 Example The following constructs the Maple record for the configurations settings described in Table 6 gt Settings defaultExpSettings gt Settings algorithm 0 gt Settings arithmeticType 1 gt Settings precision 192 6 4 defaultSettings The Maple procedure defaultSettings constructs the Maple record containing the default configu ration settings Settings defaultSettings e Settings A Maple record containing the default
18. ttings the user needs to create a file listing the configurations along with the requested values See Appendix A for each configurable setting and its acceptable values 4 For example the following presents settings that would only certify approximate solutions using 192 bit floating point precision ALGORITHM 0 ARITHMETICTYPE 1 PRECISION 192 Table 6 Example of configuration settings 3 5 Running alphaCertified The command line arguments for alphaCertified specify the names of the files to use The default names for the corresponding files are polynomialSystem testPoints and settings The command line arguments correspond to the names of these files in order For example if there are two arguments the first is the name of the system file and the second is the name of the file containing the points to test Default names will be used when less than three command line arguments are used Errors will be returned if either the system or the test points file do not exist If the configuration settings file does not exist the default settings will be used For example if there is no file named settings in the current folder the following Linux command runs alphaCertified using the default settings on the system file named polySys and the test points file named points gt alphaCertified polySys points 4 Output of alphaCertified The output of alphaCertified is a collection of files that are dependent upon the con
Download Pdf Manuals
Related Search