Computer Algebra
Computer Algebra
See also the
course
description at the mastermath site.
Start: | 6 Februari 2007 |
| Location: | Amsterdam, VU: WG S205 | |
| Day and Time: | Tuesday, 14:00 - 17:45 |
The course material will consist of the pdf files for the slides,
handouts (copies of Chapters of books) and exercises sets that will
be handed out and posted here as well.
In the twelfth (and thirteenth) lecture we discussed some ideas behind
the factorization of integer polynomials (Berlekamp-Hensel) and the relation
with Newton iteration for the location of real zeroes of a polynomial,
as well as Newton poytopes.
The eleventh
lecture (May 1, 2007) will be concerned with Gröbner bases of
noncommutative polynomials; handout available.
We will also discuss the more
general Knuth-Bendix methods;
recommmended reading: A.J.J. Dick, An introduction to Knuth-Bendix completion.
Exercises
The
tenth
lecture (April 24, 2007) is concerned with group theoretic algorithms.
Recommended reading: Chapter 8 of the TAPAS book.
Exercises.
In the ninth lecture we covered the basic theory for (commutative)
Gröbner bases.
The
eighth
lecture was concerned with some applications of LLL, including
those mentioned in the
original
LLL paper.
Also, a few exercises.
In the
seventh
lecture the LLL algorithm was treated, along the lines of sections
of Henri Cohen's book on Computational Algebraic Number Theory.
The
fifth
and sixth lectures dealt with the Gaussian elimination algorithm,
in particular PRS-like variants to compute fraction free determinants
(Bareiss), and Hermite and Smith normal forms.
We followed essentially Chapter 10 of Yap, which was handed out.
The
fourth
lecture dealt with the Chinese Remainder Algorithm, and with
Polynomial Remainder Sequences. I handed out part of a Chapter on this
topic by Geddes, Czapor, Labahn. Also handed out were some elementary
exercises to get going with Magma.
The
third and fourth sets of exercises were also handed out.
In the
first part of the third lecture I discussed Euclidean domains and the Euclidean
algorithm; the second part was mainly about the integer case, in the guise
of continued fractions.
I handed out (in week 4) a copy of my notes on continued fractions (in Dutch).
For the hands-on sessions I handed out the Appendix by Geoff Bailey on the
Magma language (from Bosma/Cannon: Discovering Mathematics with Magma)
The
second
lecture was concerned with the Fast Fourier transform. Some sections
of Modern Computer Algebra were handed out.
There are three exercises; two on FFT and one on Huffman coding. Here is the
missing page
from Johnsonbaugh's description of Huffman coding.
The
first
lecture concerned a general introduction, as well as some examples.
I handed out copies of Chapter 0 of Yap.
Here is the first exercise set.
Last update: 11 april 2007