# 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.

## Geek Guides

Practical (and free!) books for the most technical people on the planet.

- NEW: Get in the Fast Lane with NVMe
- NEW: Take Control of Growing Redis NoSQL Server Clusters
- Linux in the Time of Malware
- Apache Web Servers and SSL Encryption
- Build a Private Cloud for Less Than $10,000!

**Plus many more.**

## Trending Topics

## Upcoming Webinar

### Getting Started with DevOps - Including New Data on IT Performance from Puppet Labs 2015 State of DevOps Report

August 27, 2015

12:00 PM CDT

DevOps represents a profound change from the way most IT departments have traditionally worked: from siloed teams and high-anxiety releases to everyone collaborating on uneventful and more frequent releases of higher-quality code. It doesn't matter how large or small an organization is, or even whether it's historically slow moving or risk averse — there are ways to adopt DevOps sanely, and get measurable results in just weeks.

Free to *Linux Journal* readers.

Three More Lessons | Aug 04, 2015 |

August 2015 Issue of Linux Journal: Programming | Aug 03, 2015 |

August 2015 Video Preview | Aug 03, 2015 |

Django Models and Migrations | Jul 30, 2015 |

Secure Server Deployments in Hostile Territory, Part II | Jul 29, 2015 |

Hacking a Safe with Bash | Jul 28, 2015 |

- Three More Lessons
- Django Models and Migrations
- August 2015 Issue of Linux Journal: Programming
- Hacking a Safe with Bash
- Secure Server Deployments in Hostile Territory, Part II
- The Controversy Behind Canonical's Intellectual Property Policy
- Shashlik - a Tasty New Android Simulator
- Huge Package Overhaul for Debian and Ubuntu
- General Relativity in Python
- Embed Linux in Monitoring and Control Systems

## 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.