高精度内積計算アルゴリズムを用いた連立一次方程式の精度保証付き数値計算法(<論文特集>ロバスト計算と精度保証)

概要

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IEEE standard 754 is widely used as a standard of floating-point arithmetic. Most of CPUs in today's computers support IEEE standard 754. Using double precision arithmetic following IEEE standard 754, the authors have proposed fast methods of verifying the accuracy of a numerical solution of a linear system. In this paper, an accurate and fast verification method for a linear system is developed using residual iteration method. The residual iteration requires the availability of high precision computation. Up to now, extended precision, i.e. multiple precision and quadruple precision are used for the accurate computation of the residual. However, such higher precision arithmetic systems are not necessarily available on all computers. Therefore, the residual iteration using such systems does not have the portability. In this paper, an accurate, fast and portable method for a linear system using the fact that an algorithm of accurate dot product can portably be implemented and applied to the residual iteration. Finally, numerical results are presented showing the effectiveness of the proposed verification method.
2006-09-19