A small note on the scaling of symmetric positive definite semiseparable matrices

Abstract

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix T satisfies property A, one can easily construct a diagonal matrix (D) over cap such that (D) over capT (D) over cap has the lowest condition number over all matrices DTD, for any choice of diagonal matrix D. Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. Article information• Vandebril, Raf; Golub, G.; Van Barel, Marc. A small note on the scaling of symmetric positive definite semiseparable matrices, Numerical Algorithms, volume 41, issue 3, pages 319-326, 2006.• The content of this article is identical to the content of the published paper, but without the final typesetting by the publisher.• Journal's homepage: http://link.springer.com/journal/11075 Abstract In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix T satisfies property A, one can easily construct a diagonal matrixD such thatDTD has the lowest condition number over all matrices DT D, for any choice of diagonal matrix D. Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices.Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.