1997
DOI: 10.1007/s002110050264
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An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems

Abstract: 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. Simil… Show more

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Cited by 116 publications
(168 citation statements)
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“…For this reason, the Reference [32] recommends not to use the Jordan decomposition whenever it is possible and use instead the more reliable Schur form, which we do in the current paper. We note that this view is also shared by [27] in which the standard serial algorithm for the SDC problem is proposed to be the ordered Schur decomposition due to its well-established numerical stability. The ordered Schur form implementations are available in various platforms in LAPACK [34], OCTAVE [35], MATLAB 7.0 [36], and as a public add-on to MATLAB [31].…”
Section: Preliminaries and Notationmentioning
confidence: 80%
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“…For this reason, the Reference [32] recommends not to use the Jordan decomposition whenever it is possible and use instead the more reliable Schur form, which we do in the current paper. We note that this view is also shared by [27] in which the standard serial algorithm for the SDC problem is proposed to be the ordered Schur decomposition due to its well-established numerical stability. The ordered Schur form implementations are available in various platforms in LAPACK [34], OCTAVE [35], MATLAB 7.0 [36], and as a public add-on to MATLAB [31].…”
Section: Preliminaries and Notationmentioning
confidence: 80%
“…where the eigenvalues of A DD are exactly the same as the eigenvalues of A in D. This problem is called the ordinary SDC problem [28]. A real square matrix A of size n can be transformed via an orthogonal transformation U into the so-called real Schur form by writing U T AU = R where R is quasi-upper triangular, which means that the matrix R has either 1-by-1 or 2-by-2 diagonal blocks on the diagonal corresponding to the real and complex eigenvalues, respectively, of the matrix A [29].…”
Section: Preliminaries and Notationmentioning
confidence: 99%
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“…To show how to solve eigenvalue problems quickly and stably, we use an algorithm from [5], modified slightly to use only the randomized rank revealing decomposition from the last section. As described in Section 6.1, it can compute either an invariant subspace of a matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB, using only QRR, RURV and matrix multiplication.…”
Section: Eigenvalue Problemsmentioning
confidence: 99%