# 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

As Linux continues to play an ever increasing role in corporate data centers and institutions, ensuring the integrity and protection of these systems must be a priority. With 60% of the world's websites and an increasing share of organization's mission-critical workloads running on Linux, failing to stop malware and other advanced threats on Linux can increasingly impact an organization's reputation and bottom line.

Sponsored by Bit9

Most companies incorporate backup procedures for critical data, which can be restored quickly if a loss occurs. However, fewer companies are prepared for catastrophic system failures, in which they lose all data, the entire operating system, applications, settings, patches and more, reducing their system(s) to “bare metal.” After all, before data can be restored to a system, there must be a system to restore it to.

In this one hour webinar, learn how to enhance your existing backup strategies for better disaster recovery preparedness using Storix System Backup Administrator (SBAdmin), a highly flexible bare-metal recovery solution for UNIX and Linux systems.

Sponsored by Storix

## Trending Topics

## Free Webinar

### One Click, Universal Protection: Implementing Centralized Security Policies on Linux Systems

Join editor Bill Childers and Bit9's Paul Riegle on April 27 at 12pm Central to learn how to keep your Linux systems secure.

Free to *Linux Journal* readers.

Getting Good Vibrations with Linux | Aug 26, 2014 |

Cluetrain at Fifteen | Aug 25, 2014 |

Cluetrain at Fifteen | Aug 25, 2014 |

New Products | Aug 22, 2014 |

diff -u: What's New in Kernel Development | Aug 20, 2014 |

Security Hardening with Ansible | Aug 18, 2014 |

- Cluetrain at Fifteen
- Getting Good Vibrations with Linux
- Embedding Python in Your C Programs
- [<Megashare>] Watch Mrs Brown's Boys Movie Online Full Movie HD 2014
- New Products
- Readers' Choice Awards 2013
- Customizing Vim
- Security Hardening with Ansible
- Memory Ordering in Modern Microprocessors, Part I
- Tech Tip: Really Simple HTTP Server with Python

## Featured Jobs

Linux Systems Administrator | Houston and Austin, Texas | Host Gator |

Senior Perl Developer | Austin, Texas | Host Gator |

Technical Support Rep | Houston and Austin, Texas | Host Gator |

UX Designer | Austin, Texas | Host Gator |

Web & UI Developer (JavaScript & j Query) | Austin, Texas | Host Gator |