The Pari Package On Linux

Fast math is Pari's claim to fame. Klaus-Peter Nischke introduces us to a small, fast, flexible calculator with symbolic and numerical theoretic abilities.

The pari package (named after the French capital Paris, where the idea for this package originated) is a computer algebra system designed to work under several Unix derivatives, and of course Linux is one of them. It is well-known to a small group of mathematicians, and most probably useful for anyone who wants to perform symbolic or numerical computations or who just likes to have a powerful calculator. Its features include arbitrary-precision numerical computation, symbolic calculations, matrix/vector operations, plotting facilities (text mode or X11), and tons of number theoretic functions. Pari provides an interactive interface (the GP calculator) as well as its own programming facilities and a library for using the kernel within its own C/C++ programs. An emacs lisp file (pari.el) for using the GP calculator within an emacs buffer is included in the package. Pari is not so extensive as the commercial packages Maple, Mathematica, or Axiom are, but its major advantage is its speed. Pari claims to be 5 to 100 times faster than the commercial counterparts. I personally like its very economical use of memory. It performs really well on my “low end” 386/40 with 8 meg RAM.

Pari is available by anonymous ftp from megrez.math.u-bordeaux.fr as pari-1.39a.tar.gz, together with examples, benchmarks, and a manual (160 pgs.) which includes a function reference and a tutorial. The authors are C. Batut, D. Bernardi, H. Cohen, and M. Olivier, who are well-known number-theorists. You can contact them at pari@math.u-bordeaux.fr.

Unpacking and Compiling

On megrez.math.u-bordeaux.fr, you will find precompiled Linux binaries (gplinux.tar.gz) as well as the source package pari-1.39a.tar.gz. Because it contains the documentation and the examples, I recommend getting the source package even if you get the binaries. pari-1.39a.tar.gz unpacks into three subdirectories: doc, examples, and src. If you have gcc installed, recompiling is quite straightforward. After running configure i386 and performing a minor hack in the Makefile (read the src/INSTALL file), you are prepared to run make. You can optionally compile the gp calculator with readline support, meaning you have a command history, programmable keystrokes, and other features as within GNU bash. The source to bash also contains the necessary readline library.

It's easy to install the pari library and the gp calculator by issuing make install as root. Installing emacs support is a little bit tricky and requires you to edit some pathnames and constants defined in pari.el to match your configuration. Once pari.el is installed, you can start gp by issuing M-x gp and get an overview via M-x describe-mode, like most emacs modes.

A First Session

After compiling and installing it successfully, let's start gp and try a few expressions at the “?” prompt:


? 2*3
%1 = 6
? 4/3*5/14
%2 = 10/21
? 4.0/3*5/14
%3 = 0.4761904761904761904761904761

As you can see, pari tries to use exact integer and rational numbers as long as possible. As soon as you introduce one real (floating point) number, the result will be real. You may request (almost) arbitrary precision:


? \precision=50
    precision = 50 significant digits
? pi
%4 = 3.1415926535897932384626433832795028841971693993751

You may enter expressions with indeterminates


? (x+2)*(x^2+1)
%5 = x^3 + 2*x^2 + x + 2

assign variables:


? x=2
%6 = 2

and evaluate, e.g., our (x+2)*(x2+1)


? eval(%5)
%7 = 20

or compute some factorial


? 1000!
%8 =
4023872600770937735437024339230039857193748642107146
3254379991042993851239862902059204420848696940480047
998861019719605863166687299480855890132382966994459...

within milliseconds.

Number Types

Of pari's basic types, until now you have seen integer, rational, and real numbers and rational expressions with indeterminates (polynomials/rational functions). Integers can store values with up to 315,623 decimal digits. The precision of reals is controlled by the

\precision=n setting, where n is restricted to be not greater than 315623. Further, you can work with complex numbers, power series, row or column vectors, matrices, and more. You can combine these types (i.e. vectors of matrices); pari handles these using a recursive technique.

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