Applications of Mathematics, Vol. 66, No. 2, pp. 269-285, 2021

Verified numerical computations for large-scale linear systems

Katsuhisa Ozaki, Takeshi Terao, Takeshi Ogita, Takahiro Katagiri

Received November 25, 2019.   Published online January 15, 2021.

Abstract:  This paper concerns accuracy-guaranteed numerical computations for linear systems. Due to the rapid progress of supercomputers, the treatable problem size is getting larger. The larger the problem size, the more rounding errors in floating-point arithmetic can accumulate in general, and the more inaccurate numerical solutions are obtained. Therefore, it is important to verify the accuracy of numerical solutions. Verified numerical computations are used to produce error bounds on numerical solutions. We report the implementation of a verification method for large-scale linear systems and some numerical results using the RIKEN K computer and the Fujitsu PRIMEHPC FX100, which show the high performance of the verified numerical computations.
Keywords:  verified numerical computation; floating-point arithmetic; high-performance computing; large-scale linear system
Classification MSC:  65G20, 65G50, 65Y05
DOI:  10.21136/AM.2021.0318-19

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Affiliations:   Katsuhisa Ozaki (corresponding author), Takeshi Terao, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama-shi, Saitama 337-8570, Japan, e-mail:,; Takeshi Ogita, Tokyo Woman's Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan, e-mail:; Takahiro Katagiri, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8601, Japan, e-mail:

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