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Three optimization models for multisplitting preconditioner. (English) Zbl 1273.65074

Summary: We use matrix multisplitting with weighting parameters as the preconditioner of \(A\). The optimal weighting parameters are determined by the approaching theory, and the scale of approaching is defined by \(F\)-norm, 2-norm, and \(\infty\)-norm, respectively. Base on these three minimize models, three algorithms are presented and the convergence theories are established. Finally, numerical examples show that the preconditioner with the optimal weighting parameters, which are obtained from minimize \(F\)-norm and 2-norm models, can improve the condition number of \(A\) effectively. Besides, general weighting parameters are more effective than non-negative weighting parameters.

MSC:

65K05 Numerical mathematical programming methods
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[1] DOI: 10.1007/BF01934996 · Zbl 0566.65017 · doi:10.1007/BF01934996
[2] DOI: 10.1017/CBO9780511624100 · doi:10.1017/CBO9780511624100
[3] DOI: 10.1006/jcph.2002.7176 · Zbl 1015.65018 · doi:10.1006/jcph.2002.7176
[4] DOI: 10.1137/S1064827599356900 · Zbl 0985.65035 · doi:10.1137/S1064827599356900
[5] DOI: 10.1137/S1064827594271421 · Zbl 0856.65019 · doi:10.1137/S1064827594271421
[6] DOI: 10.1137/S1064827595294691 · Zbl 0930.65027 · doi:10.1137/S1064827595294691
[7] DOI: 10.1016/S0168-9274(98)00118-4 · Zbl 0949.65043 · doi:10.1016/S0168-9274(98)00118-4
[8] DOI: 10.1137/S1064827598339372 · Zbl 0959.65047 · doi:10.1137/S1064827598339372
[9] Demyanov V. F., Introduction to Minimax (1974)
[10] Huang T.-Z., Special Matrix Analysis and Application (2007)
[11] Li D.-H., Algorithm and Theorem for Numerical Optimization (2010)
[12] Li Q.-Y., Numerical Analysis (2008)
[13] DOI: 10.1016/j.camwa.2008.10.096 · Zbl 1186.65039 · doi:10.1016/j.camwa.2008.10.096
[14] Yuan Y.-X., Optimization Theorem and Method (1997)
[15] Zhou J., Matrix Analysis and Application (2008)
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