A multiple shift QR-step for structured rank matrices☆

Dedicated to Bill Gragg on the occasion of his 70th birthday.
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Abstract

Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, QR-algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth.

Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored.

A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations.

This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift QR-step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices.

Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach.

Keywords

Multishift
QR-algorithm
Structured rank matrices
Implicit QR-algorithms

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The research of the first two authors, was partially supported by the Research Council K.U.Leuven, project OT/ 05/40 (Large rank structured matrix computations), CoE EF/ 05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research–Flanders (Belgium), G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé-Hermite approximation), G.0423.05 (RAM: Rational modelling: optimal conditioning and stable algorithms), and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization). The first author has a grant as “Postdoctoraal Onderzoeker” from the Fund for Scientific Research–Flanders (Belgium). The work of the third author was partially supported by MIUR, grant number 2004015437. The scientific responsibility rests with the authors.

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