Abstract
Strassen's algorithm for fast matrix-matrix multiplication has been implemented for matrices of arbitrary shapes on the CRAY-2 and CRAY Y-MP supercomputers. Several techniques have been used to reduce the scratch space requirement for this algorithm while simultaneously preserving a high level of performance. When the resulting Strassen-based matrix multiply routine is combined with some routines from the new LAPACK library, LU decomposition can be performed with rates significantly higher than those achieved by conventional means. We succeeded in factoring a 2048 × 2048 matrix on the CRAY Y-MP at a rate equivalent to 325 MFLOPS.
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References
Bailey, D.H. 1988. Extra high speed matrix multiplication on the CRAY-2. SIAM J. Sci. Stat. Comp., 9, 3: 603–607.
Bischof, C., Demmel, J., Dongarra, J., DuCroz, J., Greenbaum, A., Hammarling, S., and Sorensen, D. 1988. LAPACK working note #5-Provisional contents. Tech. rept. ANL-88-38, Argonne Nat. Laboratory (Sept.).
Brent, R.P. 1970. Algorithms for matrix multiplication. Tech. rept. CS 157, Comp. Sci. Dept., Stanford Univ.
Coppersmith, D., and Winograd, S. 1987. Matrix multiplication via arithmetic progression. In Proc., 19th Annual ACM Symp. on the Theory of Computing, pp. 1–6.
Cray Research, Inc. 1989. UNICOS Math and Scientific Library Reference Manual. No. SR-2081, Version 5.0 (Mar.).
Dongarra, J.J., DuCroz, J., Duff, I., and Hammarling, S. 1988a. A set of Level 3 basic linear algebra subprograms. Tech. rept. MCS-P1-0888, MCSD, Argonne Nat. Laboratory (Aug.).
Dongarra, J.J., DuCroz, J., Duff, I., and Hammarling, S. 1988b. A set of Level 3 basic linear algebra subprograms: Model implementation and test programs. Tech. rept. MCS-P2-0888, MCSD, Argonne Nat. Laboratory (Aug.).
Gentleman, M.J. 1988. Private commun.
Higham, N.J. 1989. Exploiting fast matrix multiplication within the Level 3 BLAS. Tech. rept. TR 89–984, Dept. of Comp. Sci., Cornell Univ., Ithaca, N.Y. (Apr.). (To appear in ACM TOMS.)
Higham, N.J. 1990. Stability of a method for multiplying complex matrices with three real matrix multiplications. Numerical Analysis Rept. 181, Dept. of Math., Univ. of Manchester (Jan.).
Miller, W. 1975. Computational complexity and numerical stability. SIAM J. Computing, 4: 97–107.
Press, W.H., Flannery, B.P., Teukolsky, S.A., and Vetterling, T. 1986. Numerical Recipes. Cambridge Univ. Press, N.Y.
Strassen, V. 1969. Gaussian elimination is not optimal. Numer. Math., 13: 354–356.
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This work is supported through NASA Contract NAS 2-12961.
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Bailey, D.H., Lee, K. & Simon, H.D. Using Strassen's algorithm to accelerate the solution of linear systems. J Supercomput 4, 357–371 (1991). https://doi.org/10.1007/BF00129836
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DOI: https://doi.org/10.1007/BF00129836