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A framework for symmetric band reduction

Authors:

Christian H. Bischof,

Bruno Lang, and

Xiaobai SunAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 26, Issue 4

Pages 581 - 601

https://doi.org/10.1145/365723.365735

Published: 01 December 2000 Publication History

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Abstract

We develop an algorithmic framework for reducing the bandwidth of symmetric matrices via orthogonal similarity transformations. This framework includes the reduction of full matrices to banded or tridiagonal form and the reduction of banded matrices to narrower banded or tridiagonal form, possibly in multiple steps. Our framework leads to algorithms that require fewer floating-point operations than do standard algorithms, if only the eigenvalues are required. In addition, it allows for space-time tradeoffs and enables or increases the use of blocked transformations.

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Zhang SShah ROotomo HYokota RWu PDehnavi MKulkarni MKrishnamoorthy S(2023)Fast Symmetric Eigenvalue Decomposition via WY Representation on Tensor CoreProceedings of the 28th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming10.1145/3572848.3577516(301-312)Online publication date: 25-Feb-2023
https://dl.acm.org/doi/10.1145/3572848.3577516
Manin VLang B(2023)Efficient parallel reduction of bandwidth for symmetric matricesParallel Computing10.1016/j.parco.2023.102998115:COnline publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1016/j.parco.2023.102998
Kobayashi MHirota YKudo SHoshi TYamamoto Y(2023)Automatic performance tuning using the ATMathCoreLib tool: Two experimental studies related to dense symmetric eigensolversConcurrency and Computation: Practice and Experience10.1002/cpe.784936:10Online publication date: 30-Jun-2023
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Index Terms

A framework for symmetric band reduction
1. Mathematics of computing
  1. Mathematical software
2. Software and its engineering
  1. Software notations and tools
    1. General programming languages
      1. Language types

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Reviews

Reviewer: Timothy R. Hopkins

The main paper proposes an algorithmic framework for reducing the bandwidth of symmetric matrices using orthogonal similarity transformations. The framework generalizes the ideas underlying the Householder tridiagonalization of full matrices, along with the Rutishauser's and, Murata and Horikoshi/Lang algorithms for banded matrices. The general multistep method proposed consists of the repeated application of a one-step band reduction algorithm which "peels off" a predefined number of subdiagonals. Various instances of this algorithm are discussed which both minimize the number of flops required given different storage constraints and improve data locality. The effect of computing eigenvectors is also considered. The improved execution speeds of these new algorithms over codes currently available in LAPACK is discussed. The companion algorithm paper describes the implementation of a software toolbox for reducing full symmetric matrices to banded form, banded matrices to narrower banded or tridiagonal form (with optional accumulation of orthogonal transformations) along with codes for converting matrix data from conventional or banded storage to more efficient packed storage schemes. The Fortran software, provided as a part of the Collected Algorithms, allows users to experiment with different reduction schemes with the aim of producing tailored code for particular hardware and applications.

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cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 26, Issue 4

Dec. 2000

155 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/365723

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 December 2000

Published in TOMS Volume 26, Issue 4

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Zhang SShah ROotomo HYokota RWu PDehnavi MKulkarni MKrishnamoorthy S(2023)Fast Symmetric Eigenvalue Decomposition via WY Representation on Tensor CoreProceedings of the 28th ACM SIGPLAN Annual Symposium on Principles and Practice of Parallel Programming10.1145/3572848.3577516(301-312)Online publication date: 25-Feb-2023
https://dl.acm.org/doi/10.1145/3572848.3577516
Manin VLang B(2023)Efficient parallel reduction of bandwidth for symmetric matricesParallel Computing10.1016/j.parco.2023.102998115:COnline publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1016/j.parco.2023.102998
Kobayashi MHirota YKudo SHoshi TYamamoto Y(2023)Automatic performance tuning using the ATMathCoreLib tool: Two experimental studies related to dense symmetric eigensolversConcurrency and Computation: Practice and Experience10.1002/cpe.784936:10Online publication date: 30-Jun-2023
https://doi.org/10.1002/cpe.7849
Li SLiao XLu YRoman JYue X(2022)A parallel structured banded DC algorithm for symmetric eigenvalue problemsCCF Transactions on High Performance Computing10.1007/s42514-022-00117-95:2(116-128)Online publication date: 11-Aug-2022
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Šušnjara AKressner D(2021)A fast spectral divide‐and‐conquer method for banded matricesNumerical Linear Algebra with Applications10.1002/nla.236528:4Online publication date: 8-Mar-2021
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Mohanty SSajith G(2019)An Input/Output Efficient Algorithm for Hessenberg ReductionInternational Journal of Foundations of Computer Science10.1142/S012905411950026630:08(1279-1300)Online publication date: 12-Dec-2019
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Abstract

References

Cited By

Index Terms

Recommendations

Algorithm 807: The SBR Toolbox—software for successive band reduction

Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs

Comparison of eigensolvers for symmetric band matrices

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