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Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices

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Abstract

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the LDU factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.

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Notes

  1. Here the real case is cited for the sake of simplicity. An analogous analysis applies to the complex Hermitian case.

  2. Real matrices are used only for the sake of simplicity of the presentation.

  3. Here the matrix absolute value is defined element-wise.

  4. It is this mixing of large and small columns by orthogonal transformations oblivious of the difference in length that destroys the accuracy of the bidiagonalizaton. For illustrating examples see [44].

  5. We can also consider column-wise backward errors as in Sect. 4.2, but this involves the behavior of scaled condition numbers of the iterates.

  6. Alternatively, we may assume that X and Y are already well conditioned (thus properly scaled) and that the computed matrices satisfy \( \Vert \delta X\Vert _2 \le \epsilon _1 \Vert X\Vert _2,\quad \Vert \delta Y\Vert _2 \le \epsilon _2 \Vert Y\Vert _2,\quad |\delta D_{ii}| \le \epsilon _3 |D_{ii}|,\quad i=1,\ldots , p. \)

  7. The theory in [107] has been developed for Hermitian pencils \(H-\lambda M\) with positive definite M. Here we take \(M=I\) for the sake of simplicity.

  8. This term is used for typical errors occurring in the finite precision (computer) floating point arithmetic.

  9. Recall the example of the SVD of Vandermonde matrices in Sect. 4.6.3.

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Acknowledgements

The author wishes to thank Jesse Barlow, Jim Demmel, Froilán Martínez Dopico, Vjeran Hari, Plamen Koev, Juan Manuel Molera Molera, Eberhard Pietzsch, Ivan Slapničar, Ninoslav Truhar, Krešimir Veselić, for numerous exciting discussions on accurate matrix computations, and in particular to Julio Moro Carreño for encouragement to write this paper and for many useful comments that improved the presentation. Funding was provided by Hrvatska Zaklada za Znanost (IP-11-2013-9345, IP-2019-04-6268).

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    Zlatko Drmač

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Drmač, Z. Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices. SeMA 78, 53–92 (2021). https://doi.org/10.1007/s40324-020-00229-8

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