article

Free access

Matrix Inversion Using Parallel Processing

Author:

Marshall C. PeaseAuthors Info & Claims

Journal of the ACM (JACM), Volume 14, Issue 4

Pages 757 - 764

https://doi.org/10.1145/321420.321434

Published: 01 October 1967 Publication History

PDF eReader

Abstract

Two general methods of matrix inversion, Gauss's algorithm and the method of bordering, are analyzed from the viewpoint of their adaptability for parallel computation. The analysis is not based on any specific type of parallel processor; its purpose is rather to see if parallel capabilities could be used effectively in matrix inversion.

It is shown that both methods are indeed able to make effective use of parallel capability. With reasonable assumptions on the parallelism that is available, the speeds of the two methods are roughly comparable. The two methods, however, make use of different kinds of parallelism.

To implement Gauss's algorithm we would like to have (a) parallel transfer capability for n numbers, if the matrix is n X n, (b) the capability for parallel multiplication of the accessed numbers by a common multiplier, and (c) parallel additive read-in capability. For the method of bordering, we need, primarily, the capability of forming the Euclidean inner product of two n-dimensional real vectors. The latter seems somewhat harder to implement, but, because it is an operation that is fundamental to linear algebra in general, it is one that might be made available for other purposes. If so, then the method of bordering becomes of interest.

References

[1]

PEASE, M. C, Meshed of Matrix Algebra Academic Press, New York, 1965. (This is a good source for geerM matrix theory.)

Google Scholar

[2]

GANTHAMACHER, F. R., The Theory of Matrices (2 vols.). Chelsea, New York, 1960. (This is a good source for genera{ mrix theory,)

Google Scholar

[3]

CHANE, B. A, uo GITHENS, J. A. Balk processing in distributed logic memory. IEEE Tras EC46, 2 (April 1965), 186-I96.

Google Scholar

[4]

FADDERVA, V. N, Computational Methods of Linear Algebra. Dover, New York, 1959.

Google Scholar

[5]

WIOKINSON, J, H. The Algebraic Eigenvaluc Problem. Clarendon Press, Oxford, 1965. (This gives a detailed discussioN of the stability problem in general and of the effect of various pwtiag procedures.)

Crossref

Google Scholar

Cited By

View all

Bertilsson EIngemarsson CGustafsson O(2021)Low-Latency Parallel Hermitian Positive-Definite Matrix Inversion for Massive MIMO2021 IEEE Workshop on Signal Processing Systems (SiPS)10.1109/SiPS52927.2021.00013(23-28)Online publication date: Oct-2021
https://doi.org/10.1109/SiPS52927.2021.00013
Heller D(2020)A Survey of Parallel Algorithms in Numerical Linear AlgebraSIAM Review10.1137/102009620:4(740-777)Online publication date: 30-Jun-2020
https://dl.acm.org/doi/10.1137/1020096
Yuan LLiu CTang CHuang SChattopadhyay AAscheid GXing Z(2017)A Flexible Divide-and-Conquer MPSoC Architecture for MIMO Interference CancellationIEEE Transactions on Very Large Scale Integration (VLSI) Systems10.1109/TVLSI.2017.272860925:10(2789-2802)Online publication date: 25-Sep-2017
https://dl.acm.org/doi/10.1109/TVLSI.2017.2728609
Show More Cited By

Index Terms

Matrix Inversion Using Parallel Processing
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Theory of computation
  1. Models of computation
    1. Concurrency
      1. Parallel computing models

Recommendations

A Note On Parallel Matrix Inversion

We present one-sweep parallel algorithms for the inversion of general and symmetric positive definite matrices. The algorithms feature simple programming and performance optimization while maintaining the same arithmetic cost and numerical properties of ...
Scalable matrix inversion using MapReduce
HPDC '14: Proceedings of the 23rd international symposium on High-performance parallel and distributed computing

Matrix operations are a fundamental building block of many computational tasks in fields as diverse as scientific computing, machine learning, and data mining. Matrix inversion is an important matrix operation, but it is difficult to implement in today'...
Generalized matrix inversion is not harder than matrix multiplication

Starting from the Strassen method for rapid matrix multiplication and inversion as well as from the recursive Cholesky factorization algorithm, we introduced a completely block recursive algorithm for generalized Cholesky factorization of a given ...

Comments

Information & Contributors

Information

Published In

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 October 1967

Published in JACM Volume 14, Issue 4

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

31
Total Citations
View Citations
2,875
Total Downloads

Downloads (Last 12 months)322
Downloads (Last 6 weeks)58

Reflects downloads up to 06 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Bertilsson EIngemarsson CGustafsson O(2021)Low-Latency Parallel Hermitian Positive-Definite Matrix Inversion for Massive MIMO2021 IEEE Workshop on Signal Processing Systems (SiPS)10.1109/SiPS52927.2021.00013(23-28)Online publication date: Oct-2021
https://doi.org/10.1109/SiPS52927.2021.00013
Heller D(2020)A Survey of Parallel Algorithms in Numerical Linear AlgebraSIAM Review10.1137/102009620:4(740-777)Online publication date: 30-Jun-2020
https://dl.acm.org/doi/10.1137/1020096
Yuan LLiu CTang CHuang SChattopadhyay AAscheid GXing Z(2017)A Flexible Divide-and-Conquer MPSoC Architecture for MIMO Interference CancellationIEEE Transactions on Very Large Scale Integration (VLSI) Systems10.1109/TVLSI.2017.272860925:10(2789-2802)Online publication date: 25-Sep-2017
https://dl.acm.org/doi/10.1109/TVLSI.2017.2728609
Kim JCandan KSapino M(2016)Locality-sensitive and Re-use Promoting Personalized PageRank computationsKnowledge and Information Systems10.1007/s10115-015-0843-647:2(261-299)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1007/s10115-015-0843-6
Shi LZhao YTang J(2012)Batch Mode Active Learning for Networked DataACM Transactions on Intelligent Systems and Technology10.1145/2089094.20891093:2(1-25)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1145/2089094.2089109
D'Amato ARuiz LSilva AFoleiss JCosta JHubner R(2012)The UNB-RISC16 Object File FormatProceedings of the 2012 31st International Conference of the Chilean Computer Science Society10.1109/SCCC.2012.28(181-189)Online publication date: 12-Nov-2012
https://dl.acm.org/doi/10.1109/SCCC.2012.28
Wright DCobbold R(2010)Two-dimensional phononic crystals with time-varying properties: a multiple scattering analysisSmart Materials and Structures10.1088/0964-1726/19/4/04500619:4(045006)Online publication date: 25-Feb-2010
https://doi.org/10.1088/0964-1726/19/4/045006
ADELI HVISHNUBHOTLA P(2008)Parallel ProcessingComputer-Aided Civil and Infrastructure Engineering10.1111/j.1467-8667.1987.tb00150.x2:3(257-269)Online publication date: 6-Nov-2008
https://doi.org/10.1111/j.1467-8667.1987.tb00150.x
Cha JGupta S(2008)Data Partitioning and Placement Schemes for Matrix Multiplications on a PIM ArchitectureProceedings of the 2008 International Symposium on Parallel and Distributed Computing10.1109/ISPDC.2008.7(309-316)Online publication date: 1-Jul-2008
https://dl.acm.org/doi/10.1109/ISPDC.2008.7
Lau KKumar MVenkatesh R(1996)Parallel matrix inversion techniquesProceedings of 1996 IEEE Second International Conference on Algorithms and Architectures for Parallel Processing, ICA/sup 3/PP '9610.1109/ICAPP.1996.562917(515-521)Online publication date: 1996
https://doi.org/10.1109/ICAPP.1996.562917
Show More Cited By

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Abstract

References

Cited By

Index Terms

Recommendations

A Note On Parallel Matrix Inversion

Scalable matrix inversion using MapReduce

Generalized matrix inversion is not harder than matrix multiplication

Comments

Information

Published In

Publisher

Publication History

Permissions

Check for updates

Qualifiers

Contributors

Other Metrics

Bibliometrics

Article Metrics

Other Metrics

Citations

Cited By

View options

PDF

eReader

Get Access

Login options

Full Access

Figures

Other

Share

Share this Publication link

Share on social media

Affiliations