# 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

Linux has become a key foundation for supporting today's rapidly growing IT environments. Linux is being used to deploy business applications and databases, trading on its reputation as a low-cost operating environment. For many IT organizations, Linux is a mainstay for deploying Web servers and has evolved from handling basic file, print, and utility workloads to running mission-critical applications and databases, physically, virtually, and in the cloud. As Linux grows in importance in terms of value to the business, managing Linux environments to high standards of service quality — availability, security, and performance — becomes an essential requirement for business success.

Sponsored by Red Hat

If you already use virtualized infrastructure, you are well on your way to leveraging the power of the cloud. Virtualization offers the promise of limitless resources, but how do you manage that scalability when your DevOps team doesn’t scale? In today’s hypercompetitive markets, fast results can make a difference between leading the pack vs. obsolescence. Organizations need more benefits from cloud computing than just raw resources. They need agility, flexibility, convenience, ROI, and control.

Stackato private Platform-as-a-Service technology from ActiveState extends your private cloud infrastructure by creating a private PaaS to provide on-demand availability, flexibility, control, and ultimately, faster time-to-market for your enterprise.

Sponsored by ActiveState

## Trending Topics

diff -u: What's New in Kernel Development | Jul 23, 2014 |

Great Scott! It's Version 13! | Jul 21, 2014 |

Adminer—Better Than Awesome! | Jul 17, 2014 |

It Actually Is Rocket Science | Jul 16, 2014 |

Android Candy: Repix, Not Just Another Photo App | Jul 14, 2014 |

Wanted: Your Embedded Linux Projects | Jul 10, 2014 |

- diff -u: What's New in Kernel Development
- Divx# Watch The Other Woman Full HD Online Streaming Viooz
- Numerical Python
- Use Linux as a SAN Provider
- Why Python?
- NSA: Linux Journal is an "extremist forum" and its readers get flagged for extra surveillance
- Download "Linux Management with Red Hat Satellite: Measuring Business Impact and ROI"
- Great Scott! It's Version 13!
- RSS Feeds
- 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 |