2023-03-31T13:47:07Z
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2022-12-05T06:50:15Z
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NII Technical Report (NII-2001-002E)：On the Behaviour of the Conjugate Residual Method for Singular Systems
速水, 謙
Hayami, Ken
テクニカルレポート
Technical Report
Consider applying the Conjugate Residual (CR) method, which is a Krylov subspace type iterative solver, to systems of linear equations $ A {\bf x} = {\bf b} $ or least squares problems $ {\displaystyle \min_{ {\bf x} \in \rn} \| {\bf b} - A {\bf x} \|_2 } $, where $ A $ is singular and nonsymmetric. We will show that when $R(A)^\perp=\ker A$, the CR method can be decomposed into the $ R(A) $ and $ \ker A $ components, and the necessary and sufficient condition for the CR method to converge to the least squares solution without breaking down for arbitrary $ {\bf b} $ and initial approximate solution $ {\bf x}_0, $ is that the symmetric part $ M(A) $ of $ A $ is semi-definite and $ \rank \, M(A) = \rank A $. Furthermore, when ${\bf x}_0 \in R(A), $ the approximate solution converges to the pseudo inverse solution. Next, we will also derive the necessary and sufficient condition for the CR method to converge to the least squares solution without breaking down for arbitrary initial approximate solutions, for the case when $ R(A) \oplus \ker A = \rn $ and $ {\bf b} \in R(A). $
departmental bulletin paper
国立情報学研究所
2001-07-04
application/pdf
NIIテクニカル・レポート
NII Technical Report
1346-5597
https://repository.nii.ac.jp/record/354/files/01-002E.pdf
eng