Assignment 1 (.pdf)
Assignment 2 (.pdf)
Assignment 3 (.pdf)
Assignment 4 (.pdf)
Assignment 5 (.pdf)
Topics in Computer Algebra, Summer 2013
We will meet in K9509 on Tuesdays and Thursdays from 9:30-11:30pm.
To register for the course, please contact Diane Pogue in Mathematics
- The Fast Fourier Transform.
Fast multiplication and fast division.
The Schoenhage-Strassen multiplication algorithm.
The Fast Euclidean algorithm.
- Polynomial Interpolation.
Brown's GCD algorithm using dense interpolation.
Zippel's sparse interpolation algorithm.
Ben-Or and Tiwari sparse interpolation.
- Symbolic Linear Algebra.
The Bareiss fraction-free algorithm for solving Ax=b over an integral domain.
Division free algorithms: minor expansion and the Berkowitz algorithms.
Rational number reconstruction and solving Ax=b over Q using p-adic lifting.
- Algebraic Number Fields.
Minimal polynomials, norms, cyclotomic fields.
The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x].
Solving Ax=b over Q(alpha) using a modular method.
- Polynomial Data Structures.
Polynomial multiplication and division using geobuckets.
Polynomial multiplication and division using heaps.
Generic data structures and generic coding.
We will use Maple for programming and calculations. The following Maple worksheet has some notes for programming in Maple: MapleNotes.mws
Five assigments worth 15% each plus a course project worth 25%.
Algorithms for Computer Algebra by Geddes, Czapor and Labahn
Modern Computer Algebra by von zur Gathen and Gerhard
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors, W.S. Brown, 1971.
On the Design and Implementation of Brown’s Algorithm over the Integers and Number Fields, Monagan and Wittkopf, 2000.
A Deterministic Algorithm for Interpolating Sparse Multivariate Polynomials by M. Ben-Or and P. Tiwari.
P-adic Reconstruction of Rational Numbers by Guy, Davenport and Wang.
Maximal Quotient Rational Reconstruction by Monagan.
Algebraic Factoring and Rational Function Integration by B. Trager.
Sparse Polynomial Arithmetic by Stephen Johnson.
Analysis of Algorithms, A Case Study: Determinants of Matrices with Polynomial Entries by Gentleman and Johnson
Polynomial division using dynamic arrays, heaps and packed exponent vectors by Monagan and Pearce.
Probabilistic algorithms for interpolating polynomials. by R. Zippel, 1979.
Interpolating polynomials from their values. by R. Zippel, 1990.