Developing algorithms and software for the parallel solution of the symmetric eigenvalue problem

Auckenthaler, T.; Bungartz, Hans‐Joachim; Huckle, Thomas; Krämer, Lukas; Lang, Bruno; Willems, Paul R.

doi:10.1016/j.jocs.2011.05.002

Cited by 18 publications

(15 citation statements)

References 13 publications

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“…This fact has been used for the first parallelization approach, where the k eigenvectors are distributed uniformly across the p processes. For the second approach [45] the k eigenvectors are seen as a matrix of size n Â k, which is distributed in a 2D blocked manner across a 2D processor grid with p r rows and p c columns. Each process transforms its local part of the matrix.…”

Section: Reduction From Banded To Tridiagonal Formmentioning

confidence: 99%

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

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a b s t r a c tThe computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.

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Section: Reduction From Banded To Tridiagonal Formmentioning

confidence: 99%

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

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“…Auckenthaler et al [Auckenthaler et al 2011a;Auckenthaler et al 2011b;Auckenthaler 2012] have implemented a two-step distributed-memory tridiagonalization algorithm as part of a solver for the generalized symmetric eigenproblem. Their band-to-tridiagonal step uses an improved version of Lang's algorithm [Lang 1993], which performs one sweep.…”

Section: Related Workmentioning

confidence: 99%

“…For brevity, we will not present this algorithm and its variants, but instead refer the reader to the detailed complexity analysis (and performance modeling) in [Auckenthaler 2012] (summarized in the papers [Auckenthaler et al 2011a] and [Auckenthaler et al 2011b]). We present their complexity results in asymptotic notation; the hidden constant factors vary…”

Section: Eigenvalues Onlymentioning

confidence: 99%

Avoiding Communication in Successive Band Reduction

Ballard

Demmel

Knight

2015

ACM Trans. Parallel Comput.

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The running time of an algorithm depends on both arithmetic and communication (i.e., data movement) costs, and the relative costs of communication are growing over time. In this work, we present sequential and parallel algorithms for tridiagonalizing a symmetric band matrix that asymptotically reduce communication compared to previous approaches.The tridiagonalization of a symmetric band matrix is a key kernel in solving the symmetric eigenvalue problem for both full and band matrices. In order to preserve sparsity, tridiagonalization routines use annihilate-and-chase procedures that previously have suffered from poor data locality. We improve data locality by reorganizing the computation and obtain asymptotic improvements. We consider the cases of computing eigenvalues only and of computing eigenvalues and all eigenvectors.

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“…Как следствие, время решения экстремальных за-дач (вычислительная стоимость) также существенно возросло. Одним из путей преодоления возникших затруднений является разработка параллельных методов решения как прямых, так и оптимизационных задач, и их эффективная реализация в среде параллельных вычисле-ний [34,35]. Предложен новый параллельный гибридный алгоритм M-PCASFC, который объединяет стохастический алгоритм M-PCA, используемый при сканировании области поиска, и детерминирован-ный метод кривой, заполняющей пространство, при локальном поис-ке [29].…”

Section: P1unclassified

Methodology of Extremal Problems Solving for Material and Hydromechanical Systems

Сулимов¹,

Шкапов²

2012

EJSI

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Developing algorithms and software for the parallel solution of the symmetric eigenvalue problem

Cited by 18 publications

References 13 publications

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Avoiding Communication in Successive Band Reduction

Methodology of Extremal Problems Solving for Material and Hydromechanical Systems

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