Abstract
This paper is concerned with the problem of verifying the accuracy of an approximate solution of a linear system. A fast method of calculating both lower and upper error bounds of the approximate solution is proposed. By the proposed method, it is possible to obtain the error bounds which are as tight as needed. As a result, it can be verified that the obtained error bounds are of high quality. Numerical results are presented elucidating properties and efficiencies of the proposed verification method.
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References
ANSI/ IEEE, IEEE Standard for Binary Floating Point Arithmetic, Std 754-1985 edition, IEEE, New York, 1985.
G.H. Golub, C.F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore and London, 1996.
N.J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM Publications, Philadelphia, PA, 2002.
S. Miyajima, T. Ogita, S. Oishi, A method of generating linear systems with an arbitrarily ill-conditioned matrix and an arbitrary solution, Proceedings of 2005 International Symposium on Nonlinear Theory and its Applications, Bruges (2005), 741–744.
T. Ogita, S. Oishi, Y. Ushiro, Fast verification of solutions for sparse monotone matrix equations, Topics in Numerical Analysis: With Special Emphasis on Nonlinear Problems (Computing, Supplement 15, G. Alefeld and X. Chen eds.), Springer Wien, New York, Austria (2001), 175–187.
T. Ogita, S. Oishi, Y. Ushiro, Computation of sharp rigorous componentwise error bounds for the approximate solutions of systems of linear equations, Reliable Computing, 9(3) (2003), 229–239.
T. Ogita, S.M. Rump, S. Oishi, Accurate Sum and Dot Product, SIAMJ. Sci. Comput. 26(6) (2005), 1955–1988.
S. Oishi, S.M. Rump, Fast verification of solutions of matrix equations, Numer.Math. 90(4) (2002), 755–773.
S.M. Rump, T. Ogita, S. Oishi Accurate floating-point summation, Part I and II, SIAM J. Sci. Comput., to appear.
S.M. Rump, Fast and parallel interval arithmetic, BIT 39(3) (1999), 534–554.
S.M. Rump, INTLAB — INTerval LABoratory, Developments in Reliable Computing (T. Csendes ed.), Kluwer Academic Publishers, Dordrecht, 1999, 77–104. http://www.ti3.tu-harburg.de/rump/intlab/
S.M. Rump, Kleine Fehlerschranken bei Matrixproblemen, Universität Karlsruhe, PhD thesis, 1980.
S.M. Rump, Verification methods for dense and sparse systems of equations, Topics in Validated Computations — Studies in Computational Mathematics (J. Herzberger ed.), Elsevier, Amsterdam, 1994, 63–136.
T. Yamamoto, Error bounds for approximate solutions of systems of equations, Japan J. Appl. Math. 1(1) (1984), 157–171.
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© 2008 Birkhäuser Verlag Basel/Switzerland
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Ogita, T., Oishi, S. (2008). Tight Enclosures of Solutions of Linear Systems. In: Bandle, C., Losonczi, L., Gilányi, A., Páles, Z., Plum, M. (eds) Inequalities and Applications. International Series of Numerical Mathematics, vol 157. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8773-0_16
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DOI: https://doi.org/10.1007/978-3-7643-8773-0_16
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