×

zbMATH — the first resource for mathematics

Improvement of error-free splitting for accurate matrix multiplication. (English) Zbl 1320.65070
Summary: Recently, new algorithms for accurate matrix multiplication have been developed by the authors. A characteristic of the algorithms is a high dependency on level-3 BLAS routines, which are highly optimized for several architectures. An error-free splitting for floating-point matrices is a key technique in the algorithms. In this paper, an improvement of the error-free splitting is focused on. It is shown by numerical examples that the accuracy of computed results of matrix products can be improved by the modified error-free splitting, compared to that by the previous algorithms.
MSC:
65F99 Numerical linear algebra
65Y99 Computer aspects of numerical algorithms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ANSI/IEEE 754-1985: IEEE standard for binary floating-point arithmetic. New York, 1985.
[2] ANSI/IEEE 754-2008: IEEE standard for floating-point arithmetic. New York, 2008.
[3] The GNU Multiple Precision Arithmetic Library: http://gmplib.org/.
[4] The MPFR Library: http://www.mpfr.org/.
[5] exflib—extend precision floating-point arithmetic library: http://www-an.acs.i.kyoto-u.ac.jp/ fujiwara/exflib/exflib-index.html.
[6] The MPACK; Multiple precision arithmetic BLAS (MBLAS) and LAPACK (MLAPACK) http://mplapack.sourceforge.net/.
[7] Li, X.; Demmel, J.; Bailey, D.; Henry, G.; Hida, Y.; Iskandar, J.; Kahan, W.; Kang, S.; Kapur, A.; Martin, M.; Thompson, B.; Tung, T.; Yoo, D., Design, implementation and testing of extended and mixed precision BLAS, ACM Trans. Math. Software, 28, 2, 152-205, (2002) · Zbl 1070.65523
[8] Bailey, D., A Fortran-90 based multiprecision system, ACM Trans. Math. Software, 21, 4, 379-387, (1995) · Zbl 0883.68017
[9] Ogita, T.; Rump, S. M.; Oishi, S., Accurate sum and dot product, SIAM J. Sci. Comput., 26, 1955-1988, (2005) · Zbl 1084.65041
[10] Rump, S. M., Ultimately fast accurate summation, SIAM J. Sci. Comput., 31, 5, 3466-3502, (2009) · Zbl 1202.65033
[11] Rump, S. M.; Ogita, T.; Oishi, S., Accurate floating-point summation part i: faithful rounding, SIAM J. Sci. Comput., 31, 1, 189-224, (2008) · Zbl 1185.65082
[12] Demmel, J.; Hida, Y., Accurate and efficient floating point summation, SIAM J. Sci. Comput., 25, 4, 1214-1248, (2003) · Zbl 1061.65039
[13] Rump, S. M.; Ogita, T.; Oishi, S., Fast high precision summation, Nonlinear Theory and Its Applications (NOLTA), IEICE, 1, 1, 2-24, (2010)
[14] Ozaki, K.; Ogita, T.; Oishi, S., Tight and efficient enclosure of matrix multiplication by using optimized BLAS, Numer. Linear Algebra Appl., 18, 2, 237-248, (2011) · Zbl 1249.65098
[15] Ozaki, K.; Ogita, T.; Oishi, S.; Rump, S. M., Error-free transformation of matrix multiplication by using fast routines of matrix multiplication and its applications, Numer. Algorithms, 59, 1, 95-118, (2012) · Zbl 1244.65062
[16] OpenBLAS: http://xianyi.github.com/OpenBLAS/.
[17] Goto, K.; Geijn, R. V.D., High-performance implementation of the level-3 BLAS, ACM Trans. Math. Software, 35, 1, (2008), Article No. 4
[18] Golub, G. H.; Van Loan, C. F., Matrix computations, (2012), Johns Hopkins
[19] Rump, S. M., Error estimation of floating-point summation and dot product, BIT, 52, 1, 201-220, (2012) · Zbl 1243.65047
[20] MATLAB Programming version 7, the mathworks.
[21] Rump, S. M., INTLAB—interval laboratory, (Csendes, Tibor, Developments in Reliable Computing, (1999), Kluwer Academic Publishers Dordrecht), 77-104 · Zbl 0949.65046
[22] Ogita, T.; Oishi, S., Fast inclusion of interval matrix multiplication, Reliab. Comput., 11, 3, 191-205, (2005) · Zbl 1072.65063
[23] CRlibm: Correctly Rounded mathematical library: http://lipforge.ens-lyon.fr/www/crlibm/.
[24] Neumaier, A., Interval methods for systems of equations, (2008), Cambridge University Press
[25] Frommer, A.; Hashemi, B., Verified error bounds for solutions of Sylvester matrix equations, Linear Algebra Appl., 436, 2, 405-420, (2012) · Zbl 1236.65045
[26] Miyajima, S., Fast enclosure for solutions of Sylvester equations, Linear Algebra Appl., 439, 4, 856-878, (2013) · Zbl 1281.65069
[27] Jeannerod, C.-P.; Rump, S. M., Improved error bounds for inner products in floating-point arithmetic, SIAM J. Matrix Anal. Appl., 34, 2, 338-344, (2013) · Zbl 1279.65052
[28] W. Kahan, J.D. Darcy, How Java’s Floating-Point Hurts Everyone Everywhere, http://www.cs.berkeley.edu/ wkahan/.
[29] Dekker, T. J., A floating-point technique for extending the available precision, Numer. Math., 18, 224-242, (1971) · Zbl 0226.65034
[30] Ogita, T.; Rump, S. M.; Oishi, S., Verified solutions of linear systems without directed rounding, technical report 2005-04, (2005), Advanced Research Institute for Science and Engineering, Waseda University
[31] Morikura, Y.; Ozaki, K.; Oishi, S., Verification methods for linear systems using ufp estimation with rounding-to-nearest, Nonlinear Theory and its Applications, IEICE, 4, 1, 12-22, (2013)
[32] Rump, S. M.; Zimmermann, P.; Boldo, S.; Melquiond, G., Computing predecessor and successor in rounding to nearest, BIT, 49, 2, 419-431, (2009) · Zbl 1196.65089
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.