Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers

Demmel, James; Marques, Osni; Parlett, Beresford Ν.; Vömel, Christof

doi:10.1137/070688778

Cited by 71 publications

(69 citation statements)

References 24 publications

(23 reference statements)

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“…In the ''all eigenvectors'' case, D&C typically is much faster than QR/QL, at the cost of 2n 2 additional workspace. The same is true for D&C vs. B&I, as is well documented in the literature [12,58]. In practice D&C is therefore often used to compute all eigenvectors, even if only a subset of them is needed, and we restrict ourselves to this case here for space reasons.…”

Section: Partial Eigensystems Of Symmetric Tridiagonal Matricesmentioning

confidence: 89%

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: Partial Eigensystems Of Symmetric Tridiagonal Matricesmentioning

confidence: 89%

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

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“…LAPACK is a high performance linear algebra computing library written by Fortran. LAPACK is widely used in eigen problems and it is one of the underlying algorithms Library in MATLAB and Spark (Demmel et al, 2009). The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-2/W7, 2017 ISPRS Geospatial Week 2017, 18-22 September 2017 In the experiment, we compare the runtime of Lanczos and LAPACK on the same datasets and calculate the speedup using Lanczos (k = 10).…”

Section: Runtime and Speedupmentioning

confidence: 99%

Parallel Spatiotemporal Spectral Clustering With Massive Trajectory Data

Qin

Chen

et al. 2017

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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ABSTRACT:Massive trajectory data contains wealth useful information and knowledge. Spectral clustering, which has been shown to be effective in finding clusters, becomes an important clustering approaches in the trajectory data mining. However, the traditional spectral clustering lacks the temporal expansion on the algorithm and limited in its applicability to large-scale problems due to its high computational complexity. This paper presents a parallel spatiotemporal spectral clustering based on multiple acceleration solutions to make the algorithm more effective and efficient, the performance is proved due to the experiment carried out on the massive taxi trajectory dataset in Wuhan city, China.

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“…We refer to this process as the band reduction phase. We assume the eigendecomposition of the tridiagonal matrix T is computed via an efficient algorithm such as Bisection/Inverse Iteration, MRRR, Divide-and-Conquer, or QR Iteration (see, e.g., [Demmel et al 2008]), and we ignore the computation and communication costs of this phase. desired, then a back-transformation phase is needed to reconstruct the eigenvectors of A from the eigenvectors of T .…”

Section: Eigendecomposition Of Band Matricesmentioning

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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Performance and Accuracy of LAPACK's Symmetric Tridiagonal Eigensolvers

Cited by 71 publications

References 24 publications

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Parallel Spatiotemporal Spectral Clustering With Massive Trajectory Data

Avoiding Communication in Successive Band Reduction

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