### CISC 822 : Algebraic Algorithms

Working list of lectures (continuously under construction)

Catalog Description:
Algorithms for exact symbolic computation with integers of arbitrary size, polynomials, matrices, fields of quotients, and modular domains. Key algorithmic problmes: GCD, factorization, and solution of equations. Key techniques: Chinese remaindering, Hensel lifting, and transforms (FFT).

Current Texts:

The Art of Computer Programming Volume 2, Seminumeric Algorithms (3rd Edition),
D. E. Knuth, Addison Wesley, 1998

Saclib Users' Guide George Collins, et.al. and papers from the literature.

Goals:

Systems of polynomial equations and inequalities arise in all scientific and engineering disciplines. In computer science itself, for example, the areas of theorem proving and robot motion planning particularly involve modelling of problems as quantified polynomial systems. Algorithms for the solution of many kinds of problems involving such systems will be the core subject of this course.

Among the tools used are algorithms for manipulating mathematical objects such as arbitrary length integers, univariate and multivariate polynomials, rational functions, and matrices. We will discuss correctness issues, analyze algorithm costs, and consider implentation issues.

Implementations of these algorithms are the core components of computer algebra systems such as Axiom, the 3 M's (Macsyma, Maple, Mathematica), Reduce, and Saclib. With these systems computer science has changed the nature of computational science. Computer algebra systems have revolutionized the working environments of scientists and engineers everywhere.

Contents:
First semester:

• (3 weeks) Representation and algorithms for arithmetic of large integers (multiplication and division algorithms including Karatsuba, FFT based, greatest common divisor).
• (2 weeks) Representation and algorithms for polynomial arithmetic (multiplication and division (quotient/remainder) algorithms).
• (4 weeks) Polynomial greatest common divisors, resultants, discriminants - includes modular methods, Chinese remainder, Hensel lifting).
• (4 weeks) Polynomial factorization.
• (1 week) Groebner bases.

Second Semester:
• (3 weeks) Real root isolation and refinement, interval arithmetic.
• (2 weeks) Computation with algebraic numbers.
• (5 weeks) Cylindrical algebraic decomposition algorithm and quantifier elimination.
• (4 weeks) Matrix topics (fraction free elimination, Krylov space methods, matrix normal forms - Smith, Hermite, Frobenius, Jordan).

Required Background: Consent of instructor.

Restrictions: Offered in alternate years.

Helpful Background: Courses in linear or abstract algebra (eg. Math 650), or in analysis of algorithms (eg. CISC 621), or in computer algebra (eg. CISC 623).

Recent Instructors: Collins and Saunders[Fall 98], Saunders [Fall 95 and Fall 93 as CISC 821, Spring 90 as CISC 829]
Lakshman [Spring 91 as CISC 829]