Computing the eigenvalues of symmetric tridiagonal matrices via a Cayley transformation

Jared L. Aurentz, Thomas Mach, Raf Vandebril, and David S. Watkins

Abstract

In this paper we present a new algorithm for solving the symmetric tridiagonal eigenvalue problem that works by first using a Cayley transformation to convert the symmetric matrix into a unitary one and then uses Gragg's implicitly shifted unitary QR algorithm to solve the resulting unitary eigenvalue problem. We prove that under reasonable assumptions on the symmetric matrix this algorithm is backward stable. It is comparable to other algorithms in terms of accuracy. Although it is not the fastest algorithm, it is not conspicuously slow either. It is approximately as fast as the symmetric tridiagonal QR algorithm.

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Key words

eigenvalue, unitary QR, symmetric matrix, core transformations, rotations

AMS subject classifications

65F15, 65H17, 15A18, 15B10

Links to the cited ETNA articles

 [3] Vol. 44 (2015), pp. 327-341 Jared L. Aurentz, Thomas Mach, Raf Vandebril, and David S. Watkins: Fast and stable unitary QR algorithm

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