Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal ¹

Raf Vandebril², Ellen Van Camp³, Marc Van Barel⁴, Nicola Mastronardi⁵

Abstract:

In this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal-plus-semiseparable matrix, where we can freely choose the diagonal. Very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones. This reduction is strongly connected to the reduction to tridiagonal form. The class of semiseparable matrices can be considered as a subclass of the diagonal-plus-semiseparable matrices. Therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form.

A numerical experiment is performed comparing thereby the accuracy of this reduction algorithm with respect to the accuracy of the traditional reduction to tridiagonal form, and the reduction to semiseparable form. The experiment indicates that all three reduction algorithms are equally accurate. Moreover it is shown in the experiments that asymptotically all the three approaches have the same complexity, i.e. that they have the same factor preceding the $n^3$ term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior.
Keywords: Orthogonal similarity transformation, semiseparable plus diagonal matrix, symmetric matrix