Convergence analysis on iterative methods for semidefinite systems

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

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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].

Original languageEnglish (US)
Pages (from-to)797-815
Number of pages19
JournalJournal of Computational Mathematics
Volume26
Issue number6
StatePublished - Nov 2008

All Science Journal Classification (ASJC) codes

  • Computational Mathematics

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