Symmetric-diagonal reductions as preprocessing for symmetric positive definite generalized eigenvalue solvers

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Ahmadnasab Morad

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Journal Paper

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Journal: Journal of Mathematical Modeling Year:2023 | Volume:11 | Issue:2 Page(s): 301-322

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Title

Symmetric-diagonal reductions as preprocessing for symmetric positive definite generalized eigenvalue solvers

Author(s)

Ahmadnasab Morad | Issue Writer Certificate

Keywords

Symmetric definite generalized eigenvalue problem‎

‎Cholesky-QR method‎

Schur-QR method

QZ method

‎rounding error analysis

Abstract

We discuss some potential advantages of the orthogonal symmetric-diagonal reduction in two main versions of the Schur-QR method for symmetric positive definite generalized eigenvalue problems. We also advise and use the appropriate reductions as preprocessing on the solvers, mainly the Cholesky-QR method, of the considered problems. We discuss numerical stability of the methods via providing upper bound for backward error of the computed eigenpairs and via investigating two kinds of scaled residual errors. We also propose and apply two kinds of symmetrizing which improve the stability and the performance of the methods. Numerical experiments show that the implemented versions of the Schur-QR method and the preprocessed versions of the Cholesky-QR method are usually more stable than the Cholesky-QR method.

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