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December  2019, 9(4): 483-492. doi: 10.3934/naco.2019033

## A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

 Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

*Corresponding author

Received  August 2018 Revised  April 2019 Published  May 2019

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

Citation: Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033
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The optimal parameters for MHSS, GSOR, SSOR, ASSOR and PSSOR
 $m$ Method $16$ $32$ $64$ $128$ $256$ $512$ Example 1 PMHSS $\alpha_{opt}$ 1.09 1.36 1.35 1.05 1.05 1.05 GSOR $\alpha_{opt}$ 0.551 0.495 0.457 0.432 0.418 0.412 SSOR $\omega_{opt}$ 0.33 0.29 0.26 0.24 0.24 0.23 ASSOR $\omega_{opt}$ 0.80 0.77 0.75 0.74 0.72 0.72 PSSOR $\alpha_{opt}$ 0.47 0.48 0.54 0.54 0.55 0.55 $\omega_{opt}$ 0.83 0.83 0.82 0.82 0.82 0.82 Example 2 PMHSS $\alpha _{opt}$ 1.43 1.53 1.38 1.26 1.24 1.24 GSOR $\alpha_{opt}$ 0.189 0.190 0.190 0.190 0.190 0.190 SSOR $\omega_{opt}$ 0.09 0.09 0.10 0.10 0.10 0.10 ASSOR $\omega_{opt}$ 0.64 0.64 0.64 0.64 0.64 0.64 PSSOR $\alpha_{opt}$ 0.08 0.09 0.09 0.09 0.09 0.09 $\omega_{opt}$ 0.89 0.89 0.89 0.89 0.89 0.89 Example 3 PMHSS $\alpha_{opt}$ 0.61 0.42 0.57 0.78 0.73 0.73 GSOR $\alpha_{opt}$ 0.908 0.776 0.566 0.354 0.199 0.105 SSOR $\omega_{opt}$ 0.69 0.52 0.34 0.19 0.10 0.05 ASSOR $\omega_{opt}$ 0.62 0.62 0.62 0.61 0.61 0.61 PSSOR $\alpha_{opt}$ 1.93 1.50 1.31 1.02 0.90 0.90 $\omega_{opt}$ 0.82 0.74 0.68 0.62 0.61 0.61
 $m$ Method $16$ $32$ $64$ $128$ $256$ $512$ Example 1 PMHSS $\alpha_{opt}$ 1.09 1.36 1.35 1.05 1.05 1.05 GSOR $\alpha_{opt}$ 0.551 0.495 0.457 0.432 0.418 0.412 SSOR $\omega_{opt}$ 0.33 0.29 0.26 0.24 0.24 0.23 ASSOR $\omega_{opt}$ 0.80 0.77 0.75 0.74 0.72 0.72 PSSOR $\alpha_{opt}$ 0.47 0.48 0.54 0.54 0.55 0.55 $\omega_{opt}$ 0.83 0.83 0.82 0.82 0.82 0.82 Example 2 PMHSS $\alpha _{opt}$ 1.43 1.53 1.38 1.26 1.24 1.24 GSOR $\alpha_{opt}$ 0.189 0.190 0.190 0.190 0.190 0.190 SSOR $\omega_{opt}$ 0.09 0.09 0.10 0.10 0.10 0.10 ASSOR $\omega_{opt}$ 0.64 0.64 0.64 0.64 0.64 0.64 PSSOR $\alpha_{opt}$ 0.08 0.09 0.09 0.09 0.09 0.09 $\omega_{opt}$ 0.89 0.89 0.89 0.89 0.89 0.89 Example 3 PMHSS $\alpha_{opt}$ 0.61 0.42 0.57 0.78 0.73 0.73 GSOR $\alpha_{opt}$ 0.908 0.776 0.566 0.354 0.199 0.105 SSOR $\omega_{opt}$ 0.69 0.52 0.34 0.19 0.10 0.05 ASSOR $\omega_{opt}$ 0.62 0.62 0.62 0.61 0.61 0.61 PSSOR $\alpha_{opt}$ 1.93 1.50 1.31 1.02 0.90 0.90 $\omega_{opt}$ 0.82 0.74 0.68 0.62 0.61 0.61
Numerical results for Example 1
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 21 21 21 21 21 20 CPU 0.02 0.03 0.08 0.36 1.94 1.48 GSOR IT 20 22 24 26 27 27 CPU 0.02 0.02 0.06 0.39 2.05 11.27 SSOR IT 19 21 23 26 26 27 CPU 0.02 0.03 0.09 0.55 3.02 16.89 ASSOR IT 5 5 6 6 6 6 CPU 0.01 0.02 0.09 0.16 0.91 4.82 PSSOR IT 4 4 4 4 4 4 CPU 0.01 0.02 0.03 0.13 0.63 3.31
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 21 21 21 21 21 20 CPU 0.02 0.03 0.08 0.36 1.94 1.48 GSOR IT 20 22 24 26 27 27 CPU 0.02 0.02 0.06 0.39 2.05 11.27 SSOR IT 19 21 23 26 26 27 CPU 0.02 0.03 0.09 0.55 3.02 16.89 ASSOR IT 5 5 6 6 6 6 CPU 0.01 0.02 0.09 0.16 0.91 4.82 PSSOR IT 4 4 4 4 4 4 CPU 0.01 0.02 0.03 0.13 0.63 3.31
Numerical results for Example 2
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 34 37 38 38 38 38 CPU 0.02 0.04 0.09 0.60 3.21 26.73 GSOR IT 80 76 72 69 68 68 CPU 0.03 0.04 0.16 1.04 4.85 27.01 SSOR IT 74 74 66 66 66 66 CPU 0.03 0.06 0.19 1.33 7.61 41.39 ASSOR IT 7 7 7 7 7 7 CPU 0.01 0.01 0.03 0.18 0.96 5.09 PSSOR IT 3 3 3 3 3 3 CPU 0.02 0.02 0.03 0.11 0.51 2.64
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 34 37 38 38 38 38 CPU 0.02 0.04 0.09 0.60 3.21 26.73 GSOR IT 80 76 72 69 68 68 CPU 0.03 0.04 0.16 1.04 4.85 27.01 SSOR IT 74 74 66 66 66 66 CPU 0.03 0.06 0.19 1.33 7.61 41.39 ASSOR IT 7 7 7 7 7 7 CPU 0.01 0.01 0.03 0.18 0.96 5.09 PSSOR IT 3 3 3 3 3 3 CPU 0.02 0.02 0.03 0.11 0.51 2.64
Numerical results for Example 3
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 30 30 30 30 30 32 CPU 0.02 0.04 0.16 1.09 6.33 34.32 GSOR IT 7 11 20 44 71 131 CPU 0.02 0.02 0.08 0.97 8.19 90.83 SSOR IT 6 10 17 33 66 135 CPU 0.02 0.02 0.10 1.12 11.86 140.55 ASSOR IT 8 8 8 8 8 8 CPU 0.01 0.01 0.05 0.30 1.62 9.34 PSSOR IT 4 5 6 7 7 7 CPU 0.02 0.02 0.05 0.27 1.48 8.46
 Method $m=16$ $m=32$ $m=64$ $m=128$ $m=256$ $m=512$ PMHSS IT 30 30 30 30 30 32 CPU 0.02 0.04 0.16 1.09 6.33 34.32 GSOR IT 7 11 20 44 71 131 CPU 0.02 0.02 0.08 0.97 8.19 90.83 SSOR IT 6 10 17 33 66 135 CPU 0.02 0.02 0.10 1.12 11.86 140.55 ASSOR IT 8 8 8 8 8 8 CPU 0.01 0.01 0.05 0.30 1.62 9.34 PSSOR IT 4 5 6 7 7 7 CPU 0.02 0.02 0.05 0.27 1.48 8.46
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