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
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.
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