July 24, 2024

Numerical and Computational Challenges in Science and Engineering Program

Coxeter Lecture Series
October 29, 30 and November 1, 2001

Gene Golub, Department of Computer Science, Stanford University

Matrices, moments and quadrature

Audio of the Talks -- Lecture 1-- Lecture 2 -- Lecture 3
Photos of the event
Article about the event

Given an nxn symmetric, positive definite matrix A and a real vector u, it is desirable to estimate and bound the quadratic form
u' F(A) u/ u'u
where 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.

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

There are many extensions of these ideas to problems in statistics and to parameter estimation for solving ill-posed problems. The method which we develop will be useful in conjunction with Monte Carlo computations.

In addition. we are able to develop methods for updating and downdating recurrences for orthogonal polynomials using modified moments. We will present some numerical results showing the efficacy of our methods.