December 14, 2018

Numerical and Computational Challenges in Science and Engineering Program

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: .

Ken Jackson (Toronto)

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