How Can We Speed Up Matrix Multiplication?

作者: Victor Pan

DOI: 10.1137/1026076

关键词: Arithmetic functionDiscrete mathematicsLinear algebraMathematicsMultiplication algorithmAlgebraAlgebraic numberDiagonal matrixMatrix multiplicationMatrix chain multiplicationComputational problemTheoretical computer scienceApplied mathematicsComputational mathematics

摘要: Due to the new algebraic methods of algorithm design, recently it became possible perform multiplication and inversion $N \times N$ matrices using $O(N^{2.496} )$ rather than $O(N^3 arithmetical operations. Consequently, algorithms for several other computational problems linear algebra combinatorics have been accelerated. The major ideas techniques that led progress are surveyed.

参考文章(16)
Vermeidung von Divisionen. Crelle's Journal. ,vol. 1973, pp. 184- 202 ,(1973) , 10.1515/CRLL.1973.264.184
Shmuel Winograd, Arithmetic Complexity of Computations Society for Industrial and Applied Mathematics. ,(1980) , 10.1137/1.9781611970364
Allan Borodin, Ian Munro, THE COMPUTATIONAL COMPLEXITY OF ALGEBRAIC AND NUMERIC PROBLEMS. American Elsevier. ,(1975)
Aho AV, JE Hopcroft, JD Ullman, The Design and Analysis of Computer Algorithms ,(1974)
D. Coppersmith, S. Winograd, On the Asymptotic Complexity of Matrix Multiplication SIAM Journal on Computing. ,vol. 11, pp. 472- 492 ,(1982) , 10.1137/0211038
Julian D. Laderman, A noncommutative algorithm for multiplying $3 \times 3$ matrices using 23 multiplications Bulletin of the American Mathematical Society. ,vol. 82, pp. 126- 129 ,(1976) , 10.1090/S0002-9904-1976-13988-2
A. Schönhage, Partial and Total Matrix Multiplication SIAM Journal on Computing. ,vol. 10, pp. 434- 455 ,(1981) , 10.1137/0210032
Francesco Romani, Some Properties of Disjoint Sums of Tensors Related to Matrix Multiplication SIAM Journal on Computing. ,vol. 11, pp. 263- 267 ,(1982) , 10.1137/0211020
W.L. Miranker, V.Ya. Pan, Methods of aggregation Linear Algebra and its Applications. ,vol. 29, pp. 231- 257 ,(1980) , 10.1016/0024-3795(80)90245-1