Colloquium Mathematics, Professor Michiel Hochstenbach

Join us for coffee and tea at 15.30 p.m.

Date:                           Friday, April 26^th 2013
Speaker:                     Prof. M. Hochstenbach, TU Eindhoven
Room:                         5161.0105 (Bernoulliborg)
Time:                          16.00

Title: Probabilistic upper bounds for the matrix two-norm

Abstract:

Lanczos bidiagonalization is a popular tool for the approximation of the two-norm of a large sparse matrix. This method provides a guaranteed lower bound. The initial vector is often chosen randomly. However, if an unlucky choice is made, the true value of ||A|| can be arbitrarily larger.

In this talk we first give an overview of the method, and then show how we can derive probabilistic upper bounds, using adaptive polynomials that are implicitly generated in the Krylov process. These bounds are correct with a user-requested probability, for instance, 99% or 99.9%. The techniques are very fast (e.g., 0.2 sec for a matrix of dimension 25000) and yield quite tight bounds.

Similar methods can also be used for probabilistic upper/lower bounds for the largest/smallest eigenvalue of a symmetric matrix.