research-article

New Fast Algorithms for Matrix Operations

Author:

V. Ya. PanAuthors Info & Claims

SIAM Journal on Computing, Volume 9, Issue 2

Pages 321 - 342

https://doi.org/10.1137/0209027

Published: 01 May 1980 Publication History

Abstract

A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear noncommutative algorithms for matrix multiplication. The result is an asymptotic improvement of Strassen’s famous algorithms for matrix operations.

References

[1]

Alfred V. Aho, John E. Hopcroft, Jeffrey D. Ullman, The design and analysis of computer algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975x+470

[2]

Allan Borodin, Ian Munro, The computational complexity of algebraic and numeric problems, American Elsevier Publishing Co., Inc., New York-London-Amsterdam, 1975x+174

[3]

Roger W. Brockett, David Dobkin, On the optimal evaluation of a set of bilinear forms, Fifth Annual ACM Symposium on Theory of Computing (Austin, Tex., 1973), Assoc. Comput. Mach., New York, 1973, 88–95

[4]

R. W. Brockett, D. Dobkin, On the number of multiplications required for matrix multiplication, SIAM J. Comput., 5 (1976), 624–628

Digital Library

[5]

Roger W. Brockett, David Dobkin, On the optimal evaluation of a set of bilinear forms, Linear Algebra and Appl., 19 (1978), 207–235

[6]

C. M. Fiduccia, Fast matrix multiplication, Proc. Third Ann. ACM Symp. on Theory of Computing, 1971, 45–49

[7]

Charles M. Fiduccia, R. E. Miller, J. W. Thatcher, On obtaining upper bounds on the complexity of matrix multiplicationComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 31–40, 187–212

[8]

C. M. Fiduccia, Ph.D. Thesis, On algebraic complexity of matrix multiplication, Brown University, Providence, RI, 1973

[9]

N. Gastinel, Sur le calcul des produits de matrices, Numer. Math., 17 (1971), 222–229

Digital Library

[10]

H. F. deGroote, On varieties of optimal algorithms for the computation of bilinear mappings II. Optimal algorithms for 2×2 matrix multiplication, Tech. Rep., Mathematisches Institute, Universität Tübingen, 1978

[11]

J. E. Hopcroft, L. R. Kerr, Some techniques for proving certain simple programs optimal, Proc. of the 1969 Tenth Ann. Symp. on Switching and Automata Theory, 1969, 36–45

[12]

J. E. Hopcroft, L. R. Kerr, On minimizing the number of multiplications necessary for matrix multiplication, SIAM J. Appl. Math., 20 (1971), 30–36

Digital Library

[13]

J. E. Hopcroft, J. Musinski, Duality applied to the complexity of matrix multiplication and other bilinear forms, SIAM J. Comput., 2 (1973), 159–173

Digital Library

[14]

Z. M. Kedem, D. G. Kirkpatrick, Addition requirements for rational functions, SIAM J. Comput., 6 (1977), 188–199

Digital Library

[15]

D. G. Kirkpatrick, On the additions necessary to compute certain functions, Proc. 4th Ann. ACM Symp. on Theory of Computing, 1972, 94–101

[16]

Julian D. Laderman, A noncommutative algorithm for multiplying $3\times 3$ matrices using $23$ muliplications, Bull. Amer. Math. Soc., 82 (1976), 126–128

[17]

V. Ya Pan, On some methods of computing polynomial values, Problemy Kibernet., (1962), 21–30, Transl.Problems of Cybernetics, edited by A. A. Lyapunov., U.S.S.R., (1962), 7, pp. 20–30, U.S. Department of Commerce

[18]

V. Ya Pan, Ph.D. Thesis, Methods for computing polynomials, Department of Mechanics and Mathematics, Moscow State University, 1964, (In Russian.)

[19]

V. Ya Pan, On schemes for the computation of products and inverses of matrices, Russian Math. Surveys, 27 (1972), 249–250

[20]

V. Ya. Pan, Strassen's algorithm is not optimal. Trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations19th Annual Symposium on Foundations of Computer Science (Ann Arbor, Mich., 1978), IEEE, Long Beach, Calif., 1978, 166–176

[21]

R. L. Probert, On the composition of matrix multiplication algorithms, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), Utilitas Math., Winnipeg, Man., 1977, 357–366. Congress. Numer., XVIII

[22]

G. Schachtel, A noncommutative algorithm for multiplying $5\times 5$ matrices using $103$ multiplications, Inform. Process. Lett., 7 (1978), 180–182

[23]

Volker Strassen, Gaussian elimination is not optimal, Numer. Math., 13 (1969), 354–356

Digital Library

[24]

Volker Strassen, R. E. Miller, J. W. Thatcher, Evaluation of rational functionsComplexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), Plenum, New York, 1972, 1–10, 187–212

[25]

O. Sykora, A fast non-commutative algorithm for matrix multiplicationMathematical foundations of computer science (Proc. Sixth Sympos., Tatranská Lomnica, 1977), Springer, Berlin, 1977, 504–512. Lecture Notes in Comput. Sci., Vol. 53

[26]

Volker Strassen, Vermeidung von Divisionen, J. Reine Angew. Math., 264 (1973), 184–202

[27]

S. Winograd, A new algorithm for inner product, IEEE Trans. Computers, C-17 (1968), 693–694

[28]

Shmuel Winograd, On the number of multiplications necessary to compute certain functions, Comm. Pure Appl. Math., 23 (1970), 165–179

[29]

S. Winograd, On multiplication of $2\times 2$ matrices, Linear Algebra and Appl., 4 (1971), 381–388

[30]

S. Winograd, to appear

Cited By

Kauers MMoosbauer J(2023)Flip Graphs for Matrix MultiplicationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597120(381-388)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597120
Hadas TSchwartz O(2023)Towards Practical Fast Matrix Multiplication based on Trilinear AggregationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597099(289-297)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597099
Moran YSchwartz OAgrawal KShun J(2023)Multiplying 2 × 2 Sub-Blocks Using 4 MultiplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591083(379-390)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591083
Show More Cited By

Index Terms

New Fast Algorithms for Matrix Operations

Index terms have been assigned to the content through auto-classification.

Recommendations

Fast sparse matrix multiplication

Let A and B two n×n matrices over a ring R (e.g., the reals or the integers) each containing at most m nonzero elements. We present a new algorithm that multiplies A and B using O(m^0.7n^1.2+n^2+o(1)) algebraic operations (i.e., multiplications, additions ...
Fast Polar Decomposition of an Arbitrary Matrix

The polar decomposition of an $m \times n$ matrix A of full rank, where $m \geqq n$, can be computed using a quadratically convergent algorithm of Higham [SIAM J. Sci. Statist. Comput., 7(1986), pp. 1160–1174]. The algorithm is based on a Newton ...
Fast commutative matrix algorithms
Abstract
We show that the product of an n × 3 matrix and a 3 × 3 matrix over a commutative ring can be computed using 6 n + 3 multiplications. For two 3 × 3 matrices this gives us an algorithm using 21 multiplications. This is an improvement ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 9, Issue 2

May 1980

221 pages

ISSN:0097-5397

DOI:10.1137/smjcat.1980.9.issue-2

Issue’s Table of Contents

Copyright © 1980 Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 May 1980

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 12 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Kauers MMoosbauer J(2023)Flip Graphs for Matrix MultiplicationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597120(381-388)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597120
Hadas TSchwartz O(2023)Towards Practical Fast Matrix Multiplication based on Trilinear AggregationProceedings of the 2023 International Symposium on Symbolic and Algebraic Computation10.1145/3597066.3597099(289-297)Online publication date: 24-Jul-2023
https://dl.acm.org/doi/10.1145/3597066.3597099
Moran YSchwartz OAgrawal KShun J(2023)Multiplying 2 × 2 Sub-Blocks Using 4 MultiplicationsProceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures10.1145/3558481.3591083(379-390)Online publication date: 17-Jun-2023
https://dl.acm.org/doi/10.1145/3558481.3591083
Williams VWilliams R(2018)Subcubic Equivalences Between Path, Matrix, and Triangle ProblemsJournal of the ACM10.1145/318689365:5(1-38)Online publication date: 29-Aug-2018
https://dl.acm.org/doi/10.1145/3186893
Schönhage A(1981)Partial and Total Matrix MultiplicationSIAM Journal on Computing10.1137/021003210:3(434-455)Online publication date: 1-Aug-1981
https://dl.acm.org/doi/10.1137/0210032

View Options

View options

Get Access

Login options

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

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents