ScaLAPACK's MRRR algorithm

Vömel, Christof

doi:10.1145/1644001.1644002

Cited by 26 publications

(22 citation statements)

References 48 publications

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“…LAPACK 3.2.2 [31] and release 3.2 of the PLAPACK library [53] also provide the new MRRR algorithm [54,12], which will be included in a future ScaLAPACK release as well [55].…”

Section: Partial Eigensystems Of Symmetric Tridiagonal Matricesmentioning

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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“…LAPACK 3.2.2 [31] and release 3.2 of the PLAPACK library [53] also provide the new MRRR algorithm [54,12], which will be included in a future ScaLAPACK release as well [55].…”

Section: Partial Eigensystems Of Symmetric Tridiagonal Matricesmentioning

confidence: 99%

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

et al. 2011

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“…The estimate of parallelism assumes that clusters are processed sequentially, while in reality the bulk of the work (the refinement of the eigenvalues and the final computation of eigenpairs) can be parallelized. Nonetheless, matrices with high clustering still pose difficulties to MRRR as they introduce load-balancing issues and communication, which considerably reduce the parallel scalability [38,37,30]. Therefore, even if we did not have the desire to guarantee improved accuracy of the method, we could use the mixed precision approach to significantly enhance parallelism.…”

Section: Adjusting the Algorithmmentioning

confidence: 99%

“…12 Criterion I is used in LAPACK [12] and in results of mr3smp in [32], which usually uses II. Criterion II is used in ScaLAPACK [38] and Elemental [33]. In massively parallel computing environments, criteria III and IV can (and should) be additionally complemented with the splitting based on absolute gaps; see also [34].…”

Section: Adjusting the Algorithmmentioning

confidence: 99%

Improved Accuracy and Parallelism for MRRR-Based Eigensolvers---A Mixed Precision Approach

Petschow¹,

Quintana–Ort́ı²,

Bientinesi³

2014

SIAM J. Sci. Comput.

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The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to real tridiagonal form. For its solution, the algorithm of multiple relatively robust representations (MRRR) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only as accurate as or more accurate than the best available methods, but also-under most circumstances-faster and more scalable than the competition.

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“…We cannot give an exhaustive overview but want to point to some publications concerning relative perturbation theory [26,27,28,30], about issues of relevance for a robust implementation [10,11,22], and addressing problematic aspects in the original version and how they can be overcome [2,34].…”

Section: A743mentioning

confidence: 99%

A Framework for the $MR^3$ Algorithm: Theory and Implementation

Willems¹,

Lang²

2013

SIAM J. Sci. Comput.

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This paper provides a streamlined and modular presentation of the MR 3 algorithm for computing selected eigenpairs of symmetric tridiagonal matrices, thus disentangling the principles driving MR 3 and the (recursive) "core" algorithm from the specific (e.g., twisted) decompositions used to represent the matrices at different recursion depths and from the (dqds) transformations converting between them. Our approach allows a modular full proof for the correctness of the MR 3 algorithm. This proof is based on five requirements concerning the interplay between the core algorithm and its subcomponents. These requirements can also guide in implementing the algorithm, because they expose quantities that can and should be monitored at runtime. Our new implementation XMR, which is based on the above analysis, is described and compared to xSTEMR from Lapack 3.2.2. Numerical experiments comparing the robustness and performance of both implementations are given.1. Introduction. The established strategy for solving a dense real symmetric or complex Hermitian eigenproblem numerically is to reduce the matrix to real tridiagonal form by an orthogonal similarity transformation [1,16] and then solve the tridiagonal problem. Before there was MR 3 , usually one of the following three standard algorithms was employed for the second part.QR iteration has been the workhorse since the 1960's [14,15]. The current implementation xSTEQR in Lapack [1] is rock solid. At the time of this writing, the best method for computing all eigenvalues is the dqds algorithm [13,31], which can be considered to be a specially tailored QR iteration. But for computing eigenvectors, QR exhibits a true O(n 3 ) complexity, making it very slow in practice. In fact, for the dense symmetric problem, the solution of the tridiagonal problem using QR can outweigh the reduction phase by far [9]. On the other hand, QR is the only method that allows accumulating the eigenvectors of the original dense problem in place, thus saving a final matrix-matrix multiplication. Together with recently proposed high-performance realizations of sequences of rotations [21,33], this can make QR (almost) competitive with the approaches discussed below if most or all eigenvectors are required.Divide and conquer (DC) has been known quite a long time [3], but it took nearly 15 years until a stable implementation was found [18]. Complexity is O(n 3 ) in theory, but in practice DC behaves more like O(n 2.5 ) on average [4, 6], due to effective use of BLAS3-operations and a large amount of work that can be saved via deflation.

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ScaLAPACK's MRRR algorithm

Cited by 26 publications

References 48 publications

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Improved Accuracy and Parallelism for MRRR-Based Eigensolvers---A Mixed Precision Approach

A Framework for the $MR^3$ Algorithm: Theory and Implementation

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