2023-03-25T05:26:38Z
https://repository.nii.ac.jp/oai
oai:repository.nii.ac.jp:02000369
2023-01-11T05:56:38Z
136
NII Technical Report (NII-2020-001E)：A Stabilized GMRES Method for Solving Underdetermined Least Squares Problems
LIAO, Zeyu
速水, 謙
Hayami, Ken
保國, 惠一
Morikuni, Keiichi
YIN, Jun-Feng
テクニカルレポート
Technical Report
Consider using the right-preconditioned generalized minimal residual (AB-GMRES) method, which is an efficient method for solving underdetermined least squares problems. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and illconditioned problems, the iterates of the AB-GMRES method may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. Thus, the process becomes numerically stable and the system becomes consistent, rendering better convergence and a more accurate solution. Numerical experiments show that the proposed method is robust and efficient for solving inconsistent and ill-conditioned underdetermined least squares problems. The method can be considered as a way of making the GMRES stable for highly ill-conditioned inconsistent problems.
departmental bulletin paper
国立情報学研究所
2020-03-30
application/pdf
NIIテクニカル・レポート
1
20
NII Technical Report
1346-5597
https://repository.nii.ac.jp/record/2000369/files/20-001E.pdf
eng