Proceedings of the 2013 International Symposium on Nonlinear Theory and its Applications

2013

Session Number:C3L-A

Session:

Number:438

Verified Solutions of Sparse Linear Systems with Special Matrices

Takeshi Ogita,

pp.438-438

Publication Date:

Online ISSN:2188-5079

Summary:

To solve linear systems is ubiquitous since it is one of the basic and significant tasks in scientific computing. When solving a linear system by the use of floating-point arithmetic, rounding errors are included in the computed solution. In order to verify the quality of the computed solution, there are so-called verified numerical computations. In this talk we discuss several methods of calculating error bounds of computed solutions of large sparse linear systems whose coefficient matrices have special structures such as M-matrix, H-matrix, positive definite and so forth. Numerical results are also presented.

References:

[1] A. Minamihata, K. Sekine, T. Ogita, S. Oishi: Fast verified solutions of sparse linear systems with H-matrices, submitted for publication.

[2] A. Neumaier: Grand challenges and scientific standards in interval analysis, Reliable Computing, 8 (2002), 313-320.

[3] A. Neumaier: Interval Methods for Systems of Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1990.

[4] T. Ogita, S. Oishi, Y. Ushiro: Fast verification of solutions for sparse monotone matrix equations, Computing, Suppl. 15 (2001), 175-187.

[5] 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, 63-136, 1994.

[6] S. M. Rump: Verification of positive definiteness, BIT Numerical Mathematics, 46 (2006), 433-452.

[7] S. M. Rump, T. Ogita: Super-fast validated solution of linear systems, J. Comp. Appl. Math., 199:2 (2007), 199-206.