Iterative Refinement of Schur decompositions

The Schur decomposition of a square matrix \(A\) is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of \(A...

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Bibliographic Details
Published inarXiv.org
Main Authors Bujanović, Zvonimir, Kressner, Daniel, Schröder, Christian
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 21.03.2022
Subjects
Algorithms
Convergence
Decomposition
Eigenvalues
Iterative methods
Matrix
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Summary:The Schur decomposition of a square matrix \(A\) is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (more) accurate Schur decomposition of \(A\) from a given approximate Schur decomposition. This task arises, for example, in the context of parameter-dependent eigenvalue problems and mixed precision computations. We have developed a Newton-like algorithm that requires the solution of a triangular matrix equation and an approximate orthogonalization step in every iteration. We prove local quadratic convergence for matrices with mutually distinct eigenvalues and observe fast convergence in practice. In a mixed low-high precision environment, our algorithm essentially reduces to only four high-precision matrix-matrix multiplications per iteration. When refining double to quadruple precision, it often needs only 3-4 iterations, which reduces the time of computing a quadruple precision Schur decomposition by up to a factor of 10-20.
ISSN:2331-8422