Foundations of Computer Algebra:
Introduction

Marc Moreno Maza
University of Western Ontario
AM 475 / AM 563 and CS424 / CS556 - Winter 2007

Date: January 7, 2008

Computer Algebra

Objectives
A breif history
Challenges and Achievements

A First Session with AXIOM

Conclusions

A First Session with MAPLE
A Second Session with AXIOM

Operations on rational numbers
Operations on real numbers
Operations on matrices
Operations on univariate polynomials
Algebraic systems
Conclusions

A review of complexity notions

Orders of magnitude
The complexity of divide-and-conquer algorithms

Implementing Computer Algebra: basic ideas

Which mathematical types do we need to implement?
How to encode the elements of these types?

Integers.
Rational numbers.
Modular Integers.
Real algebraic numbers.
Univariate polynomials.
Multivariate polynomials.

What are our requirements for the language?

Bounded genericity with dependent types
Inheritance
Post-facto extensions
Efficiency

Asymptotically fast algorithms

The Karatsuba trick

Experimentation

Acknowledgments
Bibliography
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Next: Computer Algebra

Marc Moreno Maza
2008-01-07

Foundations of Computer Algebra: Introduction

Foundations of Computer Algebra:
Introduction