A Variant of the ORTHOMIN(2) Method for Singular Linear Systems |
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Authors: | Kuniyoshi Abe Shao-Liang Zhang Taketomo Mitsui Cheng-Hai Jin |
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Institution: | (1) Faculty of Economics and Information, Gifu Shotoku University, Gifu, 500-8288, Japan;(2) Graduate School of Engineering, University of Tokyo, Tokyo, 113-8656, Japan;(3) Graduate School of Information Science, Nagoya University, Nagoya, 464-8601, Japan;(4) Faculty of Engineering, University of Tokushima, Tokushima, 770-8506, Japan |
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Abstract: | For singular linear systems A
x=b, ORTHOMIN(2) is known theoretically to attain the minimum residual min
x R
n b–A
x 2 under a certain condition. However, in the actual computation with finite precision arithmetic, the residual is often observed to be reduced further than the theoretically expected level. Therefore, we propose a variant of ORTHOMIN(2), which is mathematically equivalent to the original ORTHOMIN(2) method, but uses recurrence formulas that are different from those of ORTHOMIN(2); they contain alternative expressions for the auxiliary vector and the recurrence coefficients. Although our implementation has the same computational costs as ORTHOMIN(2), numerical experiments on singular systems show that our implementation is more accurate and less affected by rounding errors than ORTHOMIN(2). |
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Keywords: | Krylov subspace method Orthomin(2) method singular systems two-dimensional minimization minimum residual norm |
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