Matrices and Moments: perturbation for least squares

Gene H Golub

Given a matrix A, (mxn) a vector b, and an approximate solution vector, we are interested in determining 
approximate error bounds induced by the approximate solution. We are able to obtain bounds for the perturbation 
using the Theory of Momnents. For  an nxn symmetric, positive definite matrix A and a real vector u, we study a 
method  to estimate and bound the quadratic form u' F(A)u/ u'u where F is a differentiable function. This 
problem arises in many applications in least squares theory eg computing a parameter in a least squares problem 
with a quadratic constraint, regularization and estimating backward perturbations of linear least squares 
problems. 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. We will present some numerical results showing the efficacy of our methods and will 
discuss various extensions of the method.

(Joint work with Zheng Su)