Title | ||
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Error-free transformations of matrix multiplication by using fast routines of matrix multiplication and its applications |
Abstract | ||
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This paper is concerned with accurate matrix multiplication in floating-point arithmetic. Recently, an accurate summation algorithm was developed by Rump et al. (SIAM J Sci Comput 31(1):189---224, 2008). The key technique of their method is a fast error-free splitting of floating-point numbers. Using this technique, we first develop an error-free transformation of a product of two floating-point matrices into a sum of floating-point matrices. Next, we partially apply this error-free transformation and develop an algorithm which aims to output an accurate approximation of the matrix product. In addition, an a priori error estimate is given. It is a characteristic of the proposed method that in terms of computation as well as in terms of memory consumption, the dominant part of our algorithm is constituted by ordinary floating-point matrix multiplications. The routine for matrix multiplication is highly optimized using BLAS, so that our algorithms show a good computational performance. Although our algorithms require a significant amount of working memory, they are significantly faster than `gemmx' in XBLAS when all sizes of matrices are large enough to realize nearly peak performance of `gemm'. Numerical examples illustrate the efficiency of the proposed method. |
Year | DOI | Venue |
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2012 | 10.1007/s11075-011-9478-1 | Numerical Algorithms |
Keywords | Field | DocType |
Matrix multiplication,Accurate computations,Floating-point arithmetic,Error-free transformation | Matrix analysis,Multiplication algorithm,Matrix (mathematics),Matrix chain multiplication,Arithmetic,Algorithm,Freivalds' algorithm,Matrix multiplication,Block matrix,Matrix splitting,Mathematics | Journal |
Volume | Issue | ISSN |
59 | 1 | 1017-1398 |
Citations | PageRank | References |
3 | 0.47 | 10 |
Authors | ||
4 |
Name | Order | Citations | PageRank |
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Katsuhisa Ozaki | 1 | 13 | 4.70 |
Takeshi Ogita | 2 | 231 | 23.39 |
Shin'ichi Oishi | 3 | 280 | 37.14 |
Siegfried M. Rump | 4 | 774 | 102.83 |