Super-fast validated solution of linear systems

Authors:
Siegfried M. Rump;Takeshi Ogita
Affiliations:
Institute for Realible Computing, Hamburg University of Technology, Hamburg, Germany and Graduate School of Science and Engineering, Waseda University, Tokyo, Japan;Graduate School of Science and Engineering, Waseda University, Tokyo, Japan
Venue:
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Year:
2007

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Abstract

Validated solution of a problem means to compute error bounds for a solution in finite precision. This includes the proof of existence of a solution. The computed error bounds are to be correct including all possible effects of rounding errors. The fastest known validation algorithm for the solution of a system of linear equations requires twice the computing time of a standard (purely) numerical algorithm. In this paper we present a super-fast validation algorithm for linear systems with symmetric positive definite matrix. This means that the entire computing time for the validation algorithm including computation of an approximated solution is the same as for a standard numerical algorithm. Numerical results are presented.