# MuPAD

We have loosely tossed about the term “domain”. We shall now look at this in a bit (but not much) more detail. Domains are fundamental to the way in which MuPAD works, and we need to have a basic understanding of them in order to use MuPAD effectively.

A domain in MuPAD is either an algebraic structure (such as finite field or permutation group) or a data type (such as Matrix, Polynomial or Fraction), for which overloaded operators or functions defined on that domain always return results in the domain (or the result FAIL if no result exists).

To give some examples, suppose we investigate matrices over the integers modulo 29. Since 29 is prime, these integers form a Galois field, and so the matrices should respond to all standard arithmetic operations.

First the definition:

>> M29:=Dom::Matrix(Dom::IntegerMod(29)); Dom::Matrix(Dom::IntegerMod(29))

We have two domains being used here:
**Dom::Matrix**, which creates a matrix domain, and
**Dom::IntegerMod(29)**, which creates the field of
integers modulo 29.

>> A:=M29([[100,200,-30],[47,-97,130],[13,33,-1001]]); +- -+ | 13 mod 29, 26 mod 29, 28 mod 29 | | | | 18 mod 29, 19 mod 29, 14 mod 29 | | | | 13 mod 29, 4 mod 29, 14 mod 29 | +- -+Notice that the result returned by MuPAD is automatically normalized so that the matrix elements are in the required field. If we enter values which can't be normalized (say, decimal fractions), MuPAD will return an error message.

>> 1/A; +- -+ | 3 mod 29, 8 mod 29, 15 mod 29 | | | | 28 mod 29, 9 mod 29, 22 mod 29 | | | | 12 mod 29, 19 mod 29, 13 mod 29 | +- -+Here the inverse operator returns a suitable result. Let's check this.

>> %*A; +- -+ | 1 mod 29, 0 mod 29, 0 mod 29 | | | | 0 mod 29, 1 mod 29, 0 mod 29 | | | | 0 mod 29, 0 mod 29, 1 mod 29 | +- -+This is the identity for our particular matrix ring. Now we can try a few other matrix operations.

>> linalg::det(A); 12 mod 29 >> linalg::gaussElim(A); +- -+ | 13 mod 29, 26 mod 29, 28 mod 29 | | | | 0 mod 29, 12 mod 29, 2 mod 29 | | | | 0 mod 29, 0 mod 29, 9 mod 29 | +- -+ >> linalg::gaussJordan(A); +- -+ | 1 mod 29, 0 mod 29, 0 mod 29 | | | | 0 mod 29, 1 mod 29, 0 mod 29 | | | | 0 mod 29, 0 mod 29, 1 mod 29 | +- -+ >> A^10; +- -+ | 22 mod 29, 10 mod 29, 3 mod 29 | | | | 3 mod 29, 21 mod 29, 16 mod 29 | | | | 4 mod 29, 18 mod 29, 5 mod 29 | +- -+ >> exp(A); FAILThe matrix exponential exp(X) is defined as 1 + X + (X^2)/2 + (X^3)/6 + (X^4)/24 + . . . + (X^n)/n! + . . . As you might expect, this is not defined for matrices over our field. For another example, consider polynomials over the integers modulo 2. The definition is similar to the matrix definition above.

>> PK:=Dom::Polynomial(Dom::IntegerMod(2)); Dom::Polynomial(Dom::IntegerMod(2))Now we'll create a polynomial in this domain.

>> p1:=PK(x^17+1); 17 x + 1For good measure, we'll create a second polynomial which looks the same, but is not in our domain.

>> p2:=x^17+1; 17 x + 1Even though they look the same on the screen, MuPAD knows all about them; the type command will tell us.

>> type(p1); Dom::Polynomial(Dom::IntegerMod(2)) >> type(p2); "_plus"(The result of this last command is that

**p2**is an object formed by adding things together.)

>> Factor(p1); 3 4 5 8 2 4 6 7 8 1 (x + 1) (x + x + x + x + 1) (x + x + x + x + x + x + 1) >> Factor(p2); 2 3 4 5 6 7 8 9 10 11 12 13 (x + 1) (x - x - x + x - x + x - x + x - x + x - x + x - x 14 15 16 + x - x + x + 1)The

**domains**package is part of MuPAD which is very much in a state of constant revision and enhancement. For example, at present, it is not possible to perform polynomial division in a polynomial domain.

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.

Considering Legacy UNIX/Linux Issues | Aug 28, 2014 |

Getting Good Vibrations with Linux | Aug 26, 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
- Python Scripts as a Replacement for Bash Utility Scripts
- Considering Legacy UNIX/Linux Issues
- [<Megashare>] Watch Mrs Brown's Boys Movie Online Full Movie HD 2014
- Memory Ordering in Modern Microprocessors, Part I
- Getting Good Vibrations with Linux
- New Products
- Security Hardening with Ansible
- RSS Feeds
- Putlocker!! Watch Begin Again Online 2014 Streaming Full Movie

## 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 |

## Comments

## Sellout

Mupad has been bought out by mathworks and all code is now under matlab (junk) licence.

any and all open source work is now dead.

## Thankyou for a well written a

Thankyou for a well written article. TeXmacs acts as an excellent interface to mupad. I assume that the TeXmacs screen display generated by TeX. The graphics is generated by javaview. The combination of TeXmacs and javaview greatly enhance the mupad experience.