Simultaneous Tridiagonalisation of Two Symmetric Matrices

SD Garvey (Nottingham University), F Tisseur (Manchester University), MI Friswell (University of Wales Swansea), JET Penny (Aston University) & U Prells (University of Wales Swansea)

International Journal for Numerical Methods in Engineering, Vol. 57, No. 12, July 2003, pp. 1643-1660

Abstract

We show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the nonsingularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non­orthogonal rank­one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non­orthogonal rank­one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n^3) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments.

Paper Availability

This material has been published in the International Journal for Numerical Methods in Engineering, Vol. 57, No. 12, July 2003, pp. 1643-1660. Unfortunately the copyright agreement with Wiley InterScience does not allow for the PDF file of the paper to be available on this website. However the paper is available from the Wiley website - see the link below.


Link to paper using doi:10.1002/nme.733

International Journal for Numerical Methods in Engineering on Wiley InterScience