# The Pari Package On Linux

In addition to the standard mathematical operations +, -, *, and /, you find transcendental and number theoretical functions, functions dealing with elliptic curves, number fields, polynomials, power series, linear algebra, sums, and products, as well as functions for plotting.

For example, you can factor numbers and polynomials:

? factor(249458089531) %9 = [7 2] [48611 1] [104729 1]

meaning 249458089531=72*48611*104729, or

? factor(t^3+t^2-2*t-2) %10 = [t + 1 1] [t^2 - 2 1]

meaning *t3+t2-2*t-2=(t+1)*(t2-2)*, where
*t2-2* cannot be factored further using rational
coefficients. It is only possible to factor polynomials in one
indeterminate.

To solve a linear equation *x=3*y,
y=2*x-1* (using the gauss method), you rewrite it as
*x-3*y=0*, *-2*x+y=-1*, take
the coefficient matrix A, the right side b and compute

? A=[1,-3;-2,1] %11 = [1 -3] [-2 1] ? b=[0;-1] %12 = [0] [-1] ? gauss(A,b) %13 = [3/5] [1/5]

giving you the result x=3/5, y=1/5.

To determine the roots of a polynomial you may just enter roots:

? \precision=4 precision = 4 significant digits :? roots(t^3+t^2-2*t-2) %14 = [-1.414 + 0.0000*i, -1.000 + 0.0000*i, 1.414 + 0.0000*i]~

Plotting gives you a quick overview of a function even in text-mode; see Figure 1. For plotting to a separate X11-window, enter:

? ploth(x=-pi,pi,sin(x))

**Figure 1. A
Function Plot in Text Mode**

Instead, to get the graph in Figure 2, enter:

? plot(x=-pi,pi,sin(x))

The gp commands may be classified into expressions (which are
evaluated immediately), function definitions, meta-commands, and
help. Via the ? key, you obtain help for the meta-commands
controlling gp as well as for each of the built-in functions. The
meta-commands allow you to control the way of printing pari results
as well as reading and writing from or to a file. **\w
<filename>** saves your complete session (from
starting gp up to issuing this command) to a file, **\r
<filename>** does the reverse job, reading the
session, bringing you to (or returning you to) the exact state that
you previously saved. Other useful features include the writing of
expressions in TeX/LaTeX format (via **texprint**)
and switching the printing of timing information by the # command.
You may also of course run gp as a batch job using standard I/O
redirection. You span input over several lines by using the \
continuation character.

Defining your own functions in gp is quite simple. As an example, cube returns the third power of its argument:

? cube(x)=x*x*x ? cube(3) %15 = 27 ? cube(t+1) %16 = t^3 + 3*t^2 + 3*t + 1

You can use control structures as **if, while,
until**, for (there are some special variants),
**goto** and **label** as well as
functions for printing or clearing variables. Though pari already
provides a function **fibo**, let us try to program
a function for the Fibonacci sequence. This sequence is defined by
*f0=1, f1=1, fn=fn-1+fn-2* for
*n>=2*, yielding *f2=1+1=2,
f3=2+1=3, f4=5,..*. The (probably) shortest such function
uses recursion. Here you need the **if** expression
to test for the special cases *f0=1* and
*f1=1*. **if(a,seq1,seq2)**
evaluates **seq1** if a is nonzero and
**seq2** otherwise:

?fib(n)=if(n==0,1,\ if(n==1,1,fib(n-1)+fib(n-2))) ? fib(5) %17 = 8

For small *n* this is okay. A faster way
is to compute the Fibonacci numbers by iteration. In each step the
new value *h=fn* is computed as the sum of the
last two values *g=fn-1* and
*f=fn-2*, and afterwards these values are
exchanged. For this you need variables **f, g, h**,
and **m** (counter). To avoid conflicts with
variables defined outside the function, these four are declared as
local by writing them at the end of the parameter list. The
**for(x=a,b,seq)** expression evaluates
**seq** for each value of *x*
running from **a** to **b**.
Expressions separated by a semicolon ; form a sequence, and a
sequence's value is always that of its last expression:

? fib2(n, m,f,g,h)= f=1; g=1; \ for(m=2, n, h=f+g; f=g; g=h); g ? fib2(5) %18 = 8

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