The Lanczos-Ritz values appearing in an orthogonal similarity reduction of a matrix into semiseparable form

It is well known how any symmetric matrix can be reduced by an orthogonal similarity transformation into tridiagonal form. Once the tridiagonal matrix has been computed, several algorithms can be used to compute either the whole spectrum or part of it. In this paper, we propose an algorithm to reduce any symmetric matrix into a similar semiseparable one of semiseparability rank 1, by orthogonal similarity transformations. It turns out that partial execution of this algorithm computes a semiseparable matrix whose eigenvalues are the Ritz-values obtained by the Lanczos’ process applied to the original matrix. Moreover, it is shown that at the same time a type of nested subspace iteration is performed. These properties allow to design different algorithms to compute the whole or part of the spectrum. Numerical experiments illustrate the properties of the new algorithm.