Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs

Abstract

We consider the reduction of a symmetric indefinite matrix pair (A,B), with B nonsingular, to tridiagonal-diagonal form by congruence transformations. This is an important reduction in solving polynomial eigenvalue problems with symmetric coefficient matrices and in frequency response computations. The pair is first reduced to symmetric-diagonal form. We describe three methods for reducing the symmetric-diagonal pair to tridiagonal-diagonal form. Two of them employ more stable versions of Brebner and Grad's pseudosymmetric Givens and pseudosymmetric Householder reductions, while the third is new and based on a combination of Householder reflectors and hyperbolic rotations. We prove an optimality condition for the transformations used in the third reduction. We present numerical experiments that compare the different approaches and show improvements over Brebner and Grad's reductions.