# 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

Android Candy: Google Keep | Dec 19, 2014 |

Handling the workloads of the Future | Dec 18, 2014 |

Raspi-Sump | Dec 16, 2014 |

diff -u: What's New in Kernel Development | Dec 12, 2014 |

Non-Linux FOSS: Don't Type All Those Words! | Dec 10, 2014 |

Computing without a Computer | Dec 08, 2014 |

- Android Candy: Google Keep
- Readers' Choice Awards 2014
- Raspi-Sump
- Handling the workloads of the Future
- How Can We Get Business to Care about Freedom, Openness and Interoperability?
- Days Between Dates?
- Synchronize Your Life with ownCloud
- diff -u: What's New in Kernel Development
- Computing without a Computer
- December 2014 Issue of Linux Journal: Readers' Choice

## Editorial Advisory Panel

Thank you to our 2014 Editorial Advisors!

- Jeff Parent
- Brad Baillio
- Nick Baronian
- Steve Case
- Chadalavada Kalyana
- Caleb Cullen
- Keir Davis
- Michael Eager
- Nick Faltys
- Dennis Frey
- Philip Jacob
- Jay Kruizenga
- Steve Marquez
- Dave McAllister
- Craig Oda

- Mike Roberts
- Chris Stark
- Patrick Swartz
- David Lynch
- Alicia Gibb
- Thomas Quinlan
- Carson McDonald
- Kristen Shoemaker
- Charnell Luchich
- James Walker
- Victor Gregorio
- Hari Boukis
- Brian Conner
- David Lane