PARI/GP
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PARI/GP is a computer algebra system with the main aim of facilitating number theory computations. As of October 5, 2006, the stable version is 2.3.1 and the development version is 2.4.0. It is free software since Version 2.1.0 and is distributed under the terms of the GNU General Public License as published by the Free Software Foundation. It runs on most common operating systems.
PARI/GP was originally developed in 1985 by a team led by Henri Cohen at Laboratoire A2X and is now maintained by Karim Belabas at the University of Bordeaux with the help of many volunteer contributors. PARI/GP's progenitor was a program named ISABELLE, an interpretor for higher arithmetic written in 1979 by Henri Cohen and François Dress (of the Université de Bordeaux I).
The PARI system is a package that is capable of doing formal computations on recursive types at high speed. It is primarily aimed at number theorists, but can be used by anybody whose primary need is speed. It can perform arbitrary precision calculations. It can be used to compute factorizations, to perform elliptic curve computations and to perform algebraic number theory. It also allows computations with matrices, polynomials, power series, algebraic numbers and implements many special functions.
Although quite an amount of symbolic manipulation is possible in PARI, the system does very badly compared to more sophisticated systems like AXIOM, Macsyma, Maple, Mathematica or Reduce on such manipulations (e.g. multivariate polynomials, formal integration, and so on). On the other hand, the three main advantages of the system are its speed (which can be between 5 and 100 times better on many computations), the possibility of directly using data types that are familiar to mathematicians, and its extensive algebraic number theory module, which has no equivalent in the above-mentioned systems.
It is possible to use PARI in two different ways:
- As a library, which can be called from an upper-level language application (for instance, written in C, C++, Pascal, Fortran, Perl, or Python).
- As a sophisticated programmable calculator, named gp, which contains most of the control instructions of a standard language like C. It can be programmed in a language called GP.
A compiler called gp2c can be used to translate GP scripts to the C language.
Below are some samples of the calculator usage:
? \p 200 realprecision = 202 significant digits (200 digits displayed) ? 123456! +0.0 time = 1,650 ms. %1 = 2.604069904929137872951393056092656881827327040950301958461018557 9952057379676834157935607166171279087355200170616660008572612714566985893 730865282934317244121152865814030204645985573419251305342231136 E574964
? sin(x) time = 0 ms. %2 = x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 + O(x^17)
? for(z=25,30, print (factor(2^z-1))) [31, 1; 601, 1; 1801, 1] [3, 1; 2731, 1; 8191, 1] [7, 1; 73, 1; 262657, 1] [3, 1; 5, 1; 29, 1; 43, 1; 113, 1; 127, 1] [233, 1; 1103, 1; 2089, 1] [3, 2; 7, 1; 11, 1; 31, 1; 151, 1; 331, 1] time = 5 ms.
? e = ellinit([0,0,1,-7,6]); ellidentify(e) time = 28 ms. %3 = [["5077a1", [0, 0, 1, -7, 6], [[-2, 3], [-1, 3], [0, 2]]], [1, 0, 0, 0]]
(a famous rank 3 elliptic curve of conductor 5077)
? K = bnfinit(x^2 + 23); K.cyc time = 1ms. %4 = [3]
(this number field has class number 3)
[edit] External links
- PARI/GP Development Headquarter[s]
- Resources for PARI/GP at Karim Belabas's website
- "Le langage et l'interpréteur 'Isabelle', spécialement conçus pour utilisations arithmétiques," François Dress, Seminaire de Théorie des Nombres de Bordeaux, Vol. 9 (1979-1980); an article on ISABELLE, PARI/GP's ancestor progam created in 1979.
