# American Institute of Mathematical Sciences

• Previous Article
Numerical solutions of Volterra integro-differential equations using General Linear Method
• NACO Home
• This Issue
• Next Article
Numerical solution with analysis of HIV/AIDS dynamics model with effect of fusion and cure rate
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, doi: 10.3934/naco.2019033
##### References:

show all references

##### References:
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
 [1] Erchuan Zhang, Lyle Noakes. Riemannian cubics and elastica in the manifold $\operatorname{SPD}(n)$ of all $n\times n$ symmetric positive-definite matrices. Journal of Geometric Mechanics, 2019, 11 (2) : 277-299. doi: 10.3934/jgm.2019015 [2] Yi Xu, Jinjie Liu, Liqun Qi. A new class of positive semi-definite tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2018186 [3] Ai-Li Yang, Yu-Jiang Wu. Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 839-853. doi: 10.3934/naco.2012.2.839 [4] Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811 [5] Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141 [6] Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047 [7] El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449 [8] Yuhong Dai, Nobuo Yamashita. Convergence analysis of sparse quasi-Newton updates with positive definite matrix completion for two-dimensional functions. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 61-69. doi: 10.3934/naco.2011.1.61 [9] Sihem Guerarra. Positive and negative definite submatrices in an Hermitian least rank solution of the matrix equation AXA*=B. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 15-22. doi: 10.3934/naco.2019002 [10] Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061 [11] Giuseppe Geymonat, Françoise Krasucki. Hodge decomposition for symmetric matrix fields and the elasticity complex in Lipschitz domains. Communications on Pure & Applied Analysis, 2009, 8 (1) : 295-309. doi: 10.3934/cpaa.2009.8.295 [12] Haibin Chen, Liqun Qi. Positive definiteness and semi-definiteness of even order symmetric Cauchy tensors. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1263-1274. doi: 10.3934/jimo.2015.11.1263 [13] Antonio Ambrosetti, Massimiliano Berti. Homoclinics and complex dynamics in slowly oscillating systems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 393-403. doi: 10.3934/dcds.1998.4.393 [14] Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91 [15] Alexandra Skripchenko. Symmetric interval identification systems of order three. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 643-656. doi: 10.3934/dcds.2012.32.643 [16] Núria Fagella, Àngel Jorba, Marc Jorba-Cuscó, Joan Carles Tatjer. Classification of linear skew-products of the complex plane and an affine route to fractalization. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3767-3787. doi: 10.3934/dcds.2019153 [17] Raffaele D’Ambrosio, Giuseppe De Martino, Beatrice Paternoster. A symmetric nearly preserving general linear method for Hamiltonian problems. Conference Publications, 2015, 2015 (special) : 330-339. doi: 10.3934/proc.2015.0330 [18] Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883 [19] Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141 [20] Ali K. Unver, Christian Ringhofer, M. Emir Koksal. Parameter extraction of complex production systems via a kinetic approach. Kinetic & Related Models, 2016, 9 (2) : 407-427. doi: 10.3934/krm.2016.9.407

Impact Factor:

## Tools

Article outline

Figures and Tables