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NII Technical Report (NII-2001-002E):On the Behaviour of the Conjugate Residual Method for Singular Systems
https://doi.org/10.20736/0000000354
https://doi.org/10.20736/0000000354ac21fc4a-91a2-458c-9fbf-5b1aa3e08e2e
名前 / ファイル | ライセンス | アクション |
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NII Technical Report (NII-2001-002E):On the Behaviour of the Conjugate Residual Method for Singular Systems (163.3 kB)
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Item type | レポート / Report(1) | |||||||||
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公開日 | 2001-07-04 | |||||||||
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言語 | en | |||||||||
タイトル | NII Technical Report (NII-2001-002E):On the Behaviour of the Conjugate Residual Method for Singular Systems | |||||||||
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言語 | eng | |||||||||
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言語 | ja | |||||||||
主題Scheme | Other | |||||||||
主題 | テクニカルレポート | |||||||||
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言語 | en | |||||||||
主題Scheme | Other | |||||||||
主題 | Technical Report | |||||||||
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資源 | http://purl.org/coar/resource_type/c_6501 | |||||||||
タイプ | departmental bulletin paper | |||||||||
ID登録 | ||||||||||
ID登録 | 10.20736/0000000354 | |||||||||
ID登録タイプ | JaLC | |||||||||
著者 |
速水, 謙
× 速水, 謙
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抄録 | ||||||||||
内容記述タイプ | Abstract | |||||||||
内容記述 | 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). $ | |||||||||
言語 | en | |||||||||
書誌情報 |
ja : NIIテクニカル・レポート en : NII Technical Report 発行日 2001-07-04 |
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言語 | ja | |||||||||
出版者 | 国立情報学研究所 | |||||||||
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収録物識別子タイプ | ISSN | |||||||||
収録物識別子 | 1346-5597 |