Simultaneous tridiagonalization of two symmetric matrices

Abstract

SUMMARYWe 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 non-orthogonal rank-one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non-orthogonal rank-one 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.

Co-Ordinate Transformations for Second Order Systems. Part Ii: Elementary Structure-Preserving Transformations

“…Such transformations will be a generalization of transformations which have been reported by the present authors [6] for the simultaneous tridiagonalization of two symmetric matrices.…”

“…Such a transformation is expected to follow a close parallel to a process already developed for simultaneously tridiagonalizing the two matrices of an undamped system [6]. Tridiagonal system form may be especially useful for dynamic substructuring applications.…”

Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs

“…However, when J = I, the choice of k is crucial for the stability of the reduction. Indeed, using a result of Ostrowski [15, p. 224] one can show that inherent relative errors in a symmetric matrix A can be magnified by as much as κ(Q) 2 in passing to Q T AQ for any nonsingular Q [8]. Clearly, κ(G) = 1 for Givens rotations, but for hyperbolic rotations [4] …”

“…A robust tridiagonal-diagonal reduction is therefore of prime importance before one can consider using any of the methods cited above. We note that Garvey et al [8] have considered a less compact form that allows the second matrix to be in tridiagonal form. One feature of their approach is that no assumption is made on the nonsingularity of the two matrices.…”

“…One feature of their approach is that no assumption is made on the nonsingularity of the two matrices. The simultaneous tridiagonalization is convenient if one needs to solve linear systems of the form (A − ωB)x = b for many values of ω, as is required in frequency response computations [8], but it is less attractive than the tridiagonal-diagonal form for eigenvalue computations.…”