Parallel algorithm for solving some spectral problems of linear algebra

Malyshev, Alexander

doi:10.1016/0024-3795(93)90477-6

Cited by 65 publications

(49 citation statements)

References 18 publications

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“…In contrast, using the matrix sign function to split the spectrum along an arbitrary line or circle will generally require a matrix inversion. This also eliminates the need for matrix exponentiation in Malyshev's algorithm [42], which was used to split along lines. We note that if the chosen circle is centered on the real axis, or if the chosen line is vertical, then all arithmetic will be real if A and B are real.…”

Section: Other Kinds Of Regionsmentioning

confidence: 99%

“…3), and an inverse-free method based on original algorithms of Bulgakov, Godunov and Malyshev [30,16,40,41,42], which is the main topic of this paper. Both kinds of algorithms are easy to parallelize because they require only large matrix operations which have been successfully parallelized on most existing machines: matrix-matrix multiplication, QR decomposition and (for the sign function) matrix inversion.…”

Section: Introductionmentioning

confidence: 99%

“…This paper is primarily concerned with an algorithm based on the pioneering work of Godunov, Bulgakov and Malyshev [30,16,40], in particular on the work of Malyshev [41,42]. We have made the following improvements on their work:…”

Section: Introductionmentioning

confidence: 99%

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An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Bai

Demmel

1997

Numerische Mathematik

116

164

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Summary.We discuss an inverse-free, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB . This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm. Mathematics Subject Classification (1991): 65F15

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Section: Other Kinds Of Regionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Bai

Demmel

1997

Numerische Mathematik

116

164

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“…Malyshev presents a scheme based on the matrix disc function [18] for the spectral division of matrix pairs. "Malyshev's iteration", in Figure 2, is first applied to the matrix pair.…”

Section: Introductionmentioning

confidence: 99%

Spectral division methods for block generalized Schur decompositions

Sun

Quintana–Ort́ı

2004

Math. Comp.

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Abstract. We provide a different perspective of the spectral division methods for block generalized Schur decompositions of matrix pairs. The new approach exposes more algebraic structures of the successive matrix pairs in the spectral division iterations and reveals some potential computational difficulties. We present modified algorithms to reduce the arithmetic cost by nearly 50%, remove inconsistency in spectral subspace extraction from different sides (left and right), and improve the accuracy of subspaces. In application problems that only require a single-sided deflating subspace, our algorithms can be used to obtain a posteriori estimates on the backward accuracy of the computed subspaces with little extra cost.

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“…However, it is dubious whether such an approach offers any advantage over methods that tackle CARE directly. For example, numerical experiments in [10] reveal that Malyshev's inverse free iteration [34] applied to the symplectic pencil requires significantly more execution time than the sign function iteration.…”

Section: Other Methodsmentioning

confidence: 99%

A parallel Schur method for solving continuous-time algebraic Riccati equations

Granat

Kågström

Kreßner

2008

2008 IEEE International Conference on Computer-Aided Control Systems

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Abstract-Numerical algorithms for solving the continuoustime algebraic Riccati matrix equation on a distributed-memory parallel computer are considered. In particular, it is shown that the Schur method, based on computing the stable invariant subspace of a Hamiltonian matrix, can be parallelized in an efficient and scalable way. Our implementation employs the state-of-the-art library ScaLAPACK as well as recently developed parallel methods for reordering the eigenvalues in a real Schur form. Some experimental results are presented, confirming the scalability of our implementation and comparing it with an existing implementation of the matrix sign iteration from the PLiCOC library.

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Parallel algorithm for solving some spectral problems of linear algebra

Cited by 65 publications

References 18 publications

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Spectral division methods for block generalized Schur decompositions

A parallel Schur method for solving continuous-time algebraic Riccati equations

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