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 nonÂorthogonal rankÂone modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and nonÂorthogonal rankÂone 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.
|Translated title of the contribution||Simultaneous tridiagonalisation of two symmetric matrices|
|Pages (from-to)||1643 - 1660|
|Number of pages||18|
|Journal||International Journal for Numerical Methods in Engineering|
|Publication status||Published - Jul 2003|