On parallelizable eigensolvers

Auslander, Louis; Tsao, Anna

doi:10.1016/0196-8858(92)90011-k

Cited by 30 publications

(20 citation statements)

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“…Lin and Zmijewski [38] develop a more numerically stable algorithm that employs orthogonal bases for the projectors and they implement it on a parallel computer. Similar ideas were developed independently by Auslander and Tsao [3] and, for generalized eigenvalue problems, by Bulgakov and Godunov [15], [23] and Malyshev [39], [40], [41] (see also [24]). …”

Section: Introductionmentioning

confidence: 75%

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Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

Nakatsukasa¹,

Higham²

2013

SIAM J. Sci. Comput.

135

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Abstract. Spectral divide and conquer algorithms solve the eigenvalue problem for all the eigenvalues and eigenvectors by recursively computing an invariant subspace for a subset of the spectrum and using it to decouple the problem into two smaller subproblems. A number of such algorithms have been developed over the last 40 years, often motivated by parallel computing and, most recently, with the aim of achieving minimal communication costs. However, none of the existing algorithms has been proved to be backward stable, and they all have a significantly higher arithmetic cost than the standard algorithms currently used. We present new spectral divide and conquer algorithms for the symmetric eigenvalue problem and the singular value decomposition that are backward stable, achieve lower bounds on communication costs recently derived by Ballard, Demmel, Holtz, and Schwartz, and have operation counts within a small constant factor of those for the standard algorithms. The new algorithms are built on the polar decomposition and exploit the recently developed QR-based dynamically weighted Halley algorithm of Nakatsukasa, Bai, and Gygi, which computes the polar decomposition using a cubically convergent iteration based on the building blocks of QR factorization and matrix multiplication. [24]).An important step in the practical development of spectral divide and conquer algorithms was the toolbox of Bai and Demmel [5], [6], which provides the building blocks for constructing algorithms via the Newton iteration for the matrix sign function. A parallel implementation based on these ideas is described in [8].

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Section: Introductionmentioning

confidence: 75%

“…2 Compute the orthogonal polar factor U p of A − σI by the QDWH algorithm. 3 Use subspace iteration to compute an orthogonal…”

Section: Invariant Subspaces Via Polar Decomposition Let a ∈ Rmentioning

confidence: 99%

Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

Nakatsukasa¹,

Higham²

2013

SIAM J. Sci. Comput.

135

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“…The PRISM project, with which this work is associated, is also producing algorithms for the symmetric case; see [6,12] for more details.…”

Section: Introductionmentioning

confidence: 99%

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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“…All these methods suffer from the use of fine-grain parallelism, instability, slow or misconvergence in the presence of clustered eigenvalues of the original problem or some constructed subproblems [16]. The other algorithms most closely related to the approach used here may be found in [2,6,24], where symmetric matrices or, more generally, matrices with real spectra are treated.…”

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confidence: 99%

The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers

Bai¹,

Demmel²,

Dongarra³

et al. 1997

SIAM J. Sci. Comput.

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Abstract. The implementation and performance of a class of divide-and-conquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divide-and-conquer (SDC) algorithm with Newton iteration. Although the algorithm requires several times as many floating point operations as the best serial QR algorithm, it can be simply constructed from a small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some ill-conditioned cases, the algorithm may lose numerical stability, but this can easily be detected and compensated for.The algorithm reached 31% efficiency with respect to the underlying PUMMA matrix multiplication and 82% efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41% efficiency with respect to the matrix multiplication in CMSSL on a 32 node Thinking Machines CM-5 with vector units. Our performance model predicts the performance reasonably accurately.To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.

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On parallelizable eigensolvers

Cited by 30 publications

References 0 publications

Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

Stable and Efficient Spectral Divide and Conquer Algorithms for the Symmetric Eigenvalue Decomposition and the SVD

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

The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers

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