PARIusers - User's Guide to PARI GP C Batut K Belabas D...

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User’s Guide to PARI / GP C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier Laboratoire A2X, U.M.R. 9936 du C.N.R.S. Universit´ e Bordeaux I, 351 Cours de la Lib´ eration 33405 TALENCE Cedex, FRANCE e-mail: [email protected] Home Page: http://pari.math.u-bordeaux.fr/ Primary ftp site: ftp://pari.math.u-bordeaux.fr/pub/pari/ last updated 11 December 2003 for version 2.2.7
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Copyright c circlecopyrt 2000–2003 The PARI Group Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy and distribute modified versions, or translations, of this manual under the conditions for verbatim copying, provided also that the entire resulting derived work is distributed under the terms of a permission notice identical to this one. PARI/GP is Copyright c circlecopyrt 2000–2003 The PARI Group PARI/GP is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation. It is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY WHATSOEVER.
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Table of Contents Chapter 1: Overview of the PARI system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The PARI types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Operations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2: Specific Use of the GP Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Defaults and output formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Simple metacommands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Input formats for the PARI types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 GP operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 The general GP input line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.6 The GP/PARI programming language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7 Errors and error recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.8 Interfacing GP with other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.9 The preferences file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.10 Using GP under GNU Emacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.11 Using GP with readline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 3: Functions and Operations Available in PARI and GP . . . . . . . . . . . . . . . . 53 3.1 Standard monadic or dyadic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Conversions and similar elementary functions or commands . . . . . . . . . . . . . . . . . . . . . . . 58 3.3 Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5 Functions related to elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.6 Functions related to general number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.7 Polynomials and power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.8 Vectors, matrices, linear algebra and sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.9 Sums, products, integrals and similar functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 3.10 Plotting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.11 Programming under GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Chapter 4: Programming PARI in Library Mode . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Introduction: initializations, universal objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 Important technical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3 Creation of PARI objects, assignments, conversions . . . . . . . . . . . . . . . . . . . . . . . . . . 175 4.4 Garbage collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.5 Implementation of the PARI types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 4.6 PARI variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.7 Input and output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.8 A complete program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.9 Adding functions to PARI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter 5: Technical Reference Guide for Low-Level Functions . . . . . . . . . . . . . . . . . 207 5.1 Level 0 kernel (operations on unsigned longs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 5.2 Level 1 kernel (operations on longs, integers and reals) . . . . . . . . . . . . . . . . . . . . . . . . 208 5.3 Level 2 kernel (operations on general PARI objects) . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Appendix A: Installation Guide for the UNIX Versions . . . . . . . . . . . . . . . . . . . . . 221 Appendix B: A Sample program and Makefile . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Appendix C: Summary of Available Constants . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
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Chapter 1: Overview of the PARI system 1.1 Introduction .
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