Abstract
Several methods have been proposed to calculate a rigorous error bound of an approximate solution of a linear system by floating-point arithmetic. These methods are called ‘verification methods’. Applicable range of these methods are different. It depends mainly on the condition number and the dimension of the coefficient matrix whether such methods succeed to work or not. In general, however, the condition number is not known in advance. If the dimension or the condition number is large to some extent, then Oishi–Rump’s method, which is known as the fastest verification method for this purpose, may fail. There are more robust verification methods whose computational cost is larger than the Oishi–Rump’s one. It is not so efficient to apply such robust methods to well-conditioned problems. The aim of this paper is to choose a suitable verification method whose computational cost is minimum to succeed. First in this paper, four fast verification methods for linear systems are briefly reviewed. Next, a compromise method between Oishi–Rump’s and Ogita–Oishi’s one is developed. Then, an algorithm which automatically and efficiently chooses an appropriate verification method from five verification methods is proposed. The proposed algorithm does as much work as necessary to calculate error bounds of approximate solutions of linear systems. Finally, numerical results are presented.
Similar content being viewed by others
References
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM Publications, Philadelphia (2002)
Ogita, T., Oishi, S.: Fast verification for large-scale systems of linear equations. IPSJ Trans. 46(SIG10 TOM12), 10–18 (2005) (in Japanese)
Ogita, T., Oishi, S.: Fast verified solutions of linear systems. Jpn. J. Ind. Appl. Math. 26(2), 169–190 (2009)
Ogita, T., Oishi, S., Ushiro, Y.: Computation of sharp rigorous componentwise error bounds for the approximate solutions of systems of linear equations. Reliab. Comput. 9(3), 229–239 (2003)
Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. 26(6), 1955–1988 (2005)
Ogita, T., Rump, S.M., Oishi, S.: Verified Solution of Linear Systems Without Directed Rounding. Technical Report No. 2005-04, Advanced Research Institute for Science and Engineering, Waseda University (2005)
Oishi, S., Rump, S.M.: Fast verification of solutions of matrix equations. Numer. Math. 90(4), 755–773 (2002)
Oishi, S., Tanabe, K., Ogita, T., Rump, S.M.: Convergence of Rump’s method for inverting arbitrarily ill-conditioned matrices. J. Comput. Appl. Math. 205(1), 533–544 (2007)
Ozaki, K., Ogita, T., Oishi, S.: Adaptive verification method for dense linear systems. In: Proceedings of 2006 international symposium on nonlinear theory and its applications, pp. 323–326, Bologna, Italy (2006)
Rump, S.M.: Fast and parallel interval arithmetic. BIT 39(3), 534–554 (1999)
Rump, S.M.: Verification methods for dense and sparse systems of equations. In: Herzberger, J. (ed.) Topics in Validated Computations—Studies in Computational Mathematics, pp. 63–136. Elsevier, Amsterdam (1994)
Ozaki, K., Ogita, T., Miyajima, S., Oishi, S., Rump, S.M.: A method of obtaining verified solutions for linear systems suited for Java. J. Comput. Appl. Math. 199(2), 337–344 (2007)
Rump, S.M.: Inversion of extremely ill-conditioned matrices in floating-point. Jpn. J. Ind. Appl. Math. 26, 249–277 (2009)
LAPACK—Linear Algebra PACKage. http://www.netlib.org/lapack/
Alefeld, G., Herzberger, J.: Introduction to Interval Computations. Academic Press, New York (1983)
Rump, S.M.: Computer-assisted proofs and self-validating methods. In: Einarsson, B. (ed.) Handbook on Accuracy and Reliability in Scientific Computation, pp. 195–240. SIAM (2005)
MATLAB Programming version 7, the Mathworks
Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore and London (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ozaki, K., Ogita, T. & Oishi, S. An algorithm for automatically selecting a suitable verification method for linear systems. Numer Algor 56, 363–382 (2011). https://doi.org/10.1007/s11075-010-9389-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-010-9389-6