Bisection eigenvalue method for Hermitian matrices with quasiseparable representation and a related inverse problem

Abstract

We study the bisection method for Hermitian matrices with quasiseparable representations.We extend here our results (published in ETNA, 44, 342–366 (2015)) for quasiseparable matrices of order one to an essentially wider class of matrices with quasiseparable representation of any order. To perform numerical tests with our algorithms we need a set of matrices with prescribed spectrum from which to build their quasiseparable generators, without building the whole matrix. We develop a method to solve this inverse problem. Our algorithm for quasiseparable of Hermitian matrices of any order is used to compute singular values of a matrix A₀, in the general case not a Hermitian one, with given quasiseparable representation via definition, i.e., as the eigenvalues of the Hermitian matrix A = A^*₀A₀. We also show that after the computation of an eigenvalue one can compute the corresponding eigenvector easily. The performance of the developed algorithms is illustrated by a series of numerical tests.

Dedicated to Rien Kaashoek on the occasion of his 80^th birthday