An introduction to Axiom (4): Lists and matrices

Following on from http://amca01.wordpress.com/2008/05/26/an-introduction-to-axiom-3/,
here’s the fourth installment – lists, matrices and linear algebra.
Lists are a standard data structure in most CAS’s, and Axiom is no exception. A list is delineated with square brackets, and can either be defined by listing it in full, or by iterating a function over a range of integers.
Given a [...]

An introduction to Axiom (3): Calculus

This is the third in a sequence of posts about the open source CAS Axiom. The previous post can be found at http://amca01.wordpress.com/2008/05/25/an-introduction-to-axiom-2/.
In this post I shall be looking at Axiom’s calculus abilities, starting with every beginning student’s love, limits:

Note here the use of %plusInfinity which is positive infinity. Axiom also has %minusInfinity. [...]

An introduction to Axiom (2): Algebra

This is the second in a sequence of posts about the open source CAS Axiom. The first can be found at http://amca01.wordpress.com/2008/05/25/an-introduction-to-axiom-1/
Axiom variables are created using the “colon equals” method of many other computer languages, and user defined functions with a double equals:

Note that for the last example, the expression involving radicals can be [...]

An introduction to Axiom (1): The basics

This is the first of what I intend to be a series of posts, exploring the open source computer algebra system Axiom. Maxima has already been extremely well described on math-blog. I hope to do something similar for Axiom, but over several posts.
Obtaining and using Axiom
First: go to http://www.axiom-developer.org/ for a description and [...]

Calculating pi

Here’s a lovely iteration which converges quadratically to . Start with:

.
Then:
for
for
for
This remarkable iteration comes from Borwein and Borwein’s “Pi and the AGM“; this is the first formula for of the many in the book. It is based on the arithmetic-geometric mean, which is defined as follows:
Given [...]

Numerical computation of square roots

If you calculate by applying the Newton iteration

to the equation , you obtain
.
The trouble with this particular iteration is that even though it’s quadratically convergent, each step requires a division by the current estimate , which is a slow operation.
There’s a neat trick to overcome this, which I either found as a [...]

A really beautiful animation

Every now and again I come across a mathematical image so powerful that it quite stops me in my tracks. So it was with this magnificent animation:

which shows a “tesseract” (a four dimensional hypercube), being rotated. I wrote to the creator of this astonishing image, Jason Hise, asking how he created it. [...]

“Down with Determinants”

It’s always pleasant to find a pedagogical decision being supported by a higher authority. In my case, the decision was to eliminate determinants from an elementary linear algebra course, and the higher authority is Sheldon Axler from Michigan State University, whose magnificent article “Down with Determinants” (American Mathematical Monthly, 102, 1995, pp139-154); online at [...]