Volume 26, Issue 6
Convergence Analysis on Iterative Methods for Semidefinite Systems

Jinbiao Wu, Young-Ju Lee, Jinchao Xu & Ludmil Zikatanov

DOI:

J. Comp. Math., 26 (2008), pp. 797-815

Published online: 2008-12

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  • Abstract

The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sufficient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regraded as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8,9].

  • Keywords

Semidefinite systems Subspace correction methods Iterative methods Energy norm convergence

  • AMS Subject Headings

65F10 65N22 65N55

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COPYRIGHT: © Global Science Press

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@Article{JCM-26-797, author = {}, title = {Convergence Analysis on Iterative Methods for Semidefinite Systems}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {6}, pages = {797--815}, abstract = { The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sufficient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regraded as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8,9].}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8660.html} }
TY - JOUR T1 - Convergence Analysis on Iterative Methods for Semidefinite Systems JO - Journal of Computational Mathematics VL - 6 SP - 797 EP - 815 PY - 2008 DA - 2008/12 SN - 26 DO - http://dor.org/ UR - https://global-sci.org/intro/jcm/8660.html KW - Semidefinite systems KW - Subspace correction methods KW - Iterative methods KW - Energy norm convergence AB - The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sufficient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regraded as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8,9].
Jinbiao Wu, Young-Ju Lee, Jinchao Xu & Ludmil Zikatanov. (1970). Convergence Analysis on Iterative Methods for Semidefinite Systems. Journal of Computational Mathematics. 26 (6). 797-815. doi:
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