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Avoiding Communication in Successive Band Reduction

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Nicholas KnightAuthors Info & Claims

ACM Transactions on Parallel Computing (TOPC), Volume 1, Issue 2

Article No.: 11, Pages 1 - 37

https://doi.org/10.1145/2686877

Published: 18 February 2015 Publication History

Abstract

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 distributed-memory parallel algorithms for tridiagonalizing full symmetric and symmetric band matrices 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 structure, tridiagonalization routines use annihilate-and-chase procedures that previously have suffered from poor data locality and high parallel latency cost. We improve both by reorganizing the computation and obtain asymptotic improvements. We also propose new algorithms for reducing a full symmetric matrix to band form in a communication-efficient manner. In this article, we consider the cases of computing eigenvalues only and of computing eigenvalues and all eigenvectors.

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Index Terms

Avoiding Communication in Successive Band Reduction
1. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices

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Published In

cover image ACM Transactions on Parallel Computing

ACM Transactions on Parallel Computing Volume 1, Issue 2

Special Issue on PPOPP 2012

January 2015

224 pages

ISSN:2329-4949

EISSN:2329-4957

DOI:10.1145/2737841

Editor:
Phillip B. Gibbons
Intel Labs, Pittsburgh, USA

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Copyright © 2015 ACM.

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Publication History

Published: 18 February 2015

Accepted: 01 July 2014

Revised: 01 July 2014

Received: 01 April 2013

Published in TOPC Volume 1, Issue 2

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https://doi.org/10.1109/PDP62718.2024.00037
Manin VLang B(2023)Efficient parallel reduction of bandwidth for symmetric matricesParallel Computing10.1016/j.parco.2023.102998115:COnline publication date: 1-Feb-2023
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Mele VLaccetti G(2023)Algorithm and Software Overhead: A Theoretical Approach to Performance PortabilityParallel Processing and Applied Mathematics10.1007/978-3-031-30445-3_8(89-100)Online publication date: 27-Apr-2023
https://doi.org/10.1007/978-3-031-30445-3_8
Poulson J(2020)High-performance sampling of generic determinantal point processesPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences10.1098/rsta.2019.0059378:2166(20190059)Online publication date: 20-Jan-2020
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Dongarra JGates MHaidar AKurzak JLuszczek PTomov SYamazaki I(2018)The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme ScaleSIAM Review10.1137/17M111773260:4(808-865)Online publication date: 8-Nov-2018
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Solomonik EBallard GDemmel JHoefler TScheideler CHajiaghayi M(2017)A Communication-Avoiding Parallel Algorithm for the Symmetric Eigenvalue ProblemProceedings of the 29th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3087556.3087561(111-121)Online publication date: 24-Jul-2017
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Ishigami HHasegawa HKimura KNakamura Y(2017)A Parallel Bisection and Inverse Iteration Solver for a Subset of Eigenpairs of Symmetric Band MatricesEigenvalue Problems: Algorithms, Software and Applications in Petascale Computing10.1007/978-3-319-62426-6_3(31-50)Online publication date: 28-Sep-2017
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