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Title of Journal: Numer Algor

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Abbravation: Numerical Algorithms

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Springer US

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DOI

10.1002/pssc.200461455

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1572-9265

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A small note on the scaling of symmetric positive

Authors: Raf Vandebril Gene Golub Marc Van Barel
Publish Date: 2006/02/09
Volume: 41, Issue: 3, Pages: 319-326
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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 hatD such that hatDThatD 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 formThe research was partially supported by the Research Council KU Leuven project OT/00/16 SLAP Structured Linear Algebra Package by the Fund for Scientific Research–Flanders Belgium projects G007801 SMA Structured Matrices and their Applications G017602 ANCILA Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra G018402 CORFU Constructive study of Orthogonal Functions and G04550 RHPH Riemann–Hilbert problems random matrices and Padé–Hermite approximation and by the Belgian Programme on Interuniversity Poles of Attraction initiated by the Belgian State Prime Ministers Office for Science Technology and Culture project IUAP V22 Dynamical Systems and Control Computation Identification Modelling The scientific responsibility rests with the authors The second author participates in the SCCM program Gates 2B Stanford University CA USA and is also partially supported by the NSF The first author visited the second one with a grant by the Fund for Scientific Research–Flanders Belgium

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