The QR-algorithm for arrowhead matrices

Nicola Mastronardi1,2, Marc Van Barel1, Ellen Van Camp1, Raf Vandebril1

Keywords: Arrowhead matrices, QR-algorithm, diagonal-plus-semi- separable matrices

Abstract

One of the most fundamental problems in numerical linear algebra is the computation of the eigenspectrum of matrices. Of special interest are the symmetric eigenproblems. The classical algorithm for computing the whole spectral decomposition first reduces the symmetric matrix into a symmetric tridiagonal matrix by means of orthogonal similarity transformations and secondly, the QR-algorithm is applied to this tridiagonal matrix. Another frequently used technique to find the eigendecomposition of this tridiagonal matrix is a divide and conquer strategy. Using this strategy, quite often the eigendecomposition of arrowhead matrices is needed. An arrowhead matrix consists of a diagonal matrix with one extra nonzero row and column.

The most common way of computing the eigenvalues of an arrowhead matrix is solving a secular equation. In this talk, we will propose an al- ternative way for computing the eigendecomposition of arrowhead matrices, more precisely we will apply the QR-algorithm to them. When exploiting the special structure of arrowhead matrices, which actually is the structure of diagonal-plus-semiseparable matrices because an arrowhead matrix be- longs to the latter class of matrices, the QR-algorithm will be speeded up. A comparison between this QR-algorithm and solving the secular equation will be presented.

1 Department of Computer Science, Katholieke Universiteit Leuven, Celestij- nenlaan 200A, 3001 Heverlee, Belgium, email: [marc.vanbarel, ellen.vancamp, raf. vandebril, nicola.mastronardi]@cs.kuleuven.ac.be 2 Istituto per le Applicazioni del Calcolo. ”M. Picone”, sez. Bari Consiglio Nazionale delle Ricerche, via G. Amendola, 122/I 70126 Bari, Italy

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