**Numerical and Computational Challenges
in Science and Engineering Program**

*ARTICLE*

From Newsletter, December 2001

**Coxeter Lecture Series**

**Gene Golub, Department of Computer Science, Stanford University**

This fall's Coxeter Lecture Series was given by Gene Golub, Fletcher
Jones Professor of Computer Science at Stanford University.

Golub's major distinctions include the B. Bolzano Gold Medal for Merit
in the Field of Mathematical Sciences (1994), Member of the National
Academy of Sciences (1993) and of the National Academy of Engineering
(1990), SIAM President (1985-87), Director of the Scientific Computing
and Computational Mathematics Program at Stanford University (1988-98)
and Chairman of the Computer Science Department at Stanford University
(1981-84).

Prof. Golub's work in matrix computation devises and analyzes algorithms
for solving numerical problems arising in science and statistics. Specifically,
he develops algorithms for solving linear systems with special structure,
computes eigenvalues of sequences of matrices, and estimates functions
of matrices.

Golub's Coxeter Lectures focused on estimates for, and bounds on, the
quadratic form u' F(A) u / u' u, where A is an n x n symmetric, positive
definite matrix, u is a real vector and F is a differentiable function.
This problem arises in estimating errors of linear systems, computing
a parameter in a least squares problem with a quadratic constraint and
bounding elements of the inverse of a matrix.

He described a method based on the theory of moments and numerical quadrature
for estimating the quadratic form. A basic tool in this approach is
the Lanczos algorithm which can be used for computing the recursive
relationship for orthogonal polynomials.

He discussed several extensions of these ideas to problems in statistics
and to parameter estimation for solving ill-posed problems. The methods
developed are useful in conjunction with Monte Carlo computations.

In addition, methods for updating and downdating recurrences for orthogonal
polynomials using modified moments were discussed and some numerical
results showing the efficacy of these methods were presented.

For more details about the this fall's Coxeter Lectures,

see: www.fields.utoronto.ca./programs/scientific/01-02/numerical/coxeter/golub.html
.

Ken Jackson (Toronto)

Back to Thematic
Year Index