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On deflated CGW methods for solving nonsymmetric positive definite linear systems.

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Title: On deflated CGW methods for solving nonsymmetric positive definite linear systems.
Authors: Du, Kui1 (AUTHOR) kuidu@xmu.edu.cn, Fan, Jia-Jun2 (AUTHOR) jiajunfan@stu.xmu.edu.cn, Wang, Fang2 (AUTHOR) fangwang@stu.xmu.edu.cn
Source: Calcolo. Jun2025, Vol. 62 Issue 2, p1-21. 21p.
Subjects: Linear systems, Lanczos method, Iterative methods (Mathematics), Empirical research, Matrices (Mathematics), Linear algebra, Eigenvalue equations
Abstract: We propose a deflated CGW method (D-CGW) to approximate the solution of nonsymmetric positive definite linear systems. D-CGW utilizes some eigenspaces, and can be useful in cases when the slow convergence is due to a small number of eigenvalues with large magnitude in the generalized eigenvalue problem S x = λ H x , where S and H represent the skew-symmetric and symmetric parts of the coefficient matrix of nonsymmetric positive definite linear systems, respectively. Additionally, we introduce a skew-symmetric Lanczos method with deflated restarting, referred to as S 2 Lan-DR, which simultaneously solves nonsymmetric positive definite linear systems and computes eigenspaces. The computed eigenspaces can be used in D-CGW to solve successive linear systems. Numerical experiments are given to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]
Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="AR" term="%22Du%2C+Kui%22">Du, Kui</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> kuidu@xmu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Fan%2C+Jia-Jun%22">Fan, Jia-Jun</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> jiajunfan@stu.xmu.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Wang%2C+Fang%22">Wang, Fang</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> fangwang@stu.xmu.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Calcolo%22">Calcolo</searchLink>. Jun2025, Vol. 62 Issue 2, p1-21. 21p.
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  Data: <searchLink fieldCode="DE" term="%22Linear+systems%22">Linear systems</searchLink><br /><searchLink fieldCode="DE" term="%22Lanczos+method%22">Lanczos method</searchLink><br /><searchLink fieldCode="DE" term="%22Iterative+methods+%28Mathematics%29%22">Iterative methods (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Empirical+research%22">Empirical research</searchLink><br /><searchLink fieldCode="DE" term="%22Matrices+%28Mathematics%29%22">Matrices (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+algebra%22">Linear algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Eigenvalue+equations%22">Eigenvalue equations</searchLink>
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  Data: We propose a deflated CGW method (D-CGW) to approximate the solution of nonsymmetric positive definite linear systems. D-CGW utilizes some eigenspaces, and can be useful in cases when the slow convergence is due to a small number of eigenvalues with large magnitude in the generalized eigenvalue problem S x = λ H x , where S and H represent the skew-symmetric and symmetric parts of the coefficient matrix of nonsymmetric positive definite linear systems, respectively. Additionally, we introduce a skew-symmetric Lanczos method with deflated restarting, referred to as S 2 Lan-DR, which simultaneously solves nonsymmetric positive definite linear systems and computes eigenspaces. The computed eigenspaces can be used in D-CGW to solve successive linear systems. Numerical experiments are given to illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Calcolo is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – SubjectFull: Lanczos method
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      – SubjectFull: Iterative methods (Mathematics)
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      – SubjectFull: Eigenvalue equations
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              M: 06
              Text: Jun2025
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