2024-02-26T15:07:32Z
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2022-12-27T04:15:05Z
136
NII Technical Report (NII-2004-006E)：Preconditioned GMRES Methods for Least Squares Problems
伊藤, 徳史
Ito, Tokushi
速水, 謙
Hayami, Ken
テクニカルレポート
Technical Report
For least squares problems of minimizing || b - A x ||_2 where A is a large sparse m x n (m >= n) matrix, the common method is to apply the conjugate gradient method to the normal equation A^T A x = A^T b. However, the condition number of A^T A is square of that of A, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning. In this paper, we propose two methods for applying the GMRES method to the least squares problem by using a n x m matrix B. We give the necessary and sufficient condition that B should satisfy in order that the proposed methods give a least squares solution. Then, for implementations for B, we propose an incomplete QR decomposition IMGS(l). Numerical experiments show that the simplest case l=0, which is equivalent to B= ( diag (A^T A) )^(-1) A^T, gives best results, and converges faster than previous methods for severely ill-conditioned problems.
departmental bulletin paper
国立情報学研究所
2004-05-07
application/pdf
NIIテクニカル・レポート
1
29
NII Technical Report
1346-5597
https://repository.nii.ac.jp/record/394/files/04-006E.pdf
eng