An orthogonal similarity reduction of a matrix into semiseparable form

An algorithm to reduce a symmetric matrix to a similar semiseparable one of semiseparability rank 1, using orthogonal similarity transformations, is proposed in this paper. It is shown that, while running to completion, the proposed algorithm gives information on the spectrum of the similar initial matrix. In fact, the proposed algorithm shares the same properties of the Lanczos method and the Householder reduction to tridiagonal form. Furthermore, at each iteration, the proposed algorithm performs a step of the QR method without shift to a principal submatrix to retrieve the semiseparable structure. The latter step can be considered a kind of subspace-like iteration method, where the size of the subspace increases by one dimension at each step of the algorithm. Hence, when during the execution of the algorithm the Ritz values approximate the dominant eigenvalues closely enough, diagonal blocks will appear in the semiseparable part where the norm of the corresponding subdiagonal blocks goes to zero in the subsequent iteration steps, depending on the corresponding gap between the eigenvalues. A numerical experiment is included, illustrating the properties of the new algorithm.status: publishe

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.nrpages: 28status: publishe