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Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

Part of: Numerical linear algebra

Published online by Cambridge University Press:  31 January 2017

Jing Wang ,
Xue-Ping Guo  and
Hong-Xiu Zhong
Jing Wang*
Affiliation:
Shandong Computer Science Center, Jinan 250014, Shandong, P.R. China
Xue-Ping Guo*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200241, P.R. China
Hong-Xiu Zhong*
Affiliation:
School of Science, Jiangnan University, Wuxi 214122, P.R. China
*
*Corresponding author. Email addresses:[email protected] (J. Wang), [email protected] (X.-P. Guo), [email protected] (H.-X. Zhong)
*Corresponding author. Email addresses:[email protected] (J. Wang), [email protected] (X.-P. Guo), [email protected] (H.-X. Zhong)
*Corresponding author. Email addresses:[email protected] (J. Wang), [email protected] (X.-P. Guo), [email protected] (H.-X. Zhong)
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Abstract

Preconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β, the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method.

Keywords

Preconditioningcomplex systems of linear equationsPMHSS methodGPMHSS methodAGPMHSS method

MSC classification

Secondary: 65F10: Iterative methods for linear systems
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 1 , February 2017 , pp. 143 - 155
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Arridge, S. R., Optical Tomography in Medical Imaging, Inverse Problems, 15 (1999), pp. 41–93.CrossRefGoogle Scholar
[2] Bai, Z. Z., Benzi, M., and Chen, F., Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), pp. 93–111.Google Scholar
[3] Bai, Z. Z., Benzi, M., and Chen, F., On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), pp. 297–317.CrossRefGoogle Scholar
[4] Bai, Z. Z., Benzi, M., Chen, F., and Wang, Z. Q., Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with application to distributed control problems, IMA J. Numer. Anal., 33 (2013), pp. 343–369.CrossRefGoogle Scholar
[5] Bai, Z. Z. and Golub, G. H., Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), pp. 1–23.Google Scholar
[6] Bai, Z. Z., Golub, G. H., and Li, C. K., Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices, Math. Comput., 76 (2007), pp. 287–298.Google Scholar
[7] Bai, Z. Z., Golub, G. H., Lu, L. Z., and Yin, J. F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 26 (2005), pp. 844–863.Google Scholar
[8] Bai, Z. Z., Golub, G. H., and Ng, M. K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl., 24 (2003), pp. 603–626.Google Scholar
[9] Bai, Z. Z., Golub, G. H., and Ng, M. K., On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), pp. 319–335.Google Scholar
[10] Bai, Z. Z., Golub, G. H., and Pan, J. Y., Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), pp. 1–32.CrossRefGoogle Scholar
[11] Bai, Z. Z. and Guo, X. P., On Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math., 28 (2010), pp. 235–260.Google Scholar
[12] Bertaccini, D., Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electr. Trans. Numer. Anal., 18 (2004), pp. 49–64.Google Scholar
[13] Chen, M. H., Lin, R. F., and Wu, Q. B., Convergence analysis of the modified Newton-HSS method under the hölder continuous condition, J. Comput. Appl. Math., 264 (2014), pp. 115–130.Google Scholar
[14] Dehghan, M., Dehghani-Madiseh, M., and Hajarian, M., A generalized preconditioned MHSS method for a class of complex symmetric linear systems, Math. Model. Anal., 18 (2013), pp. 561–576.Google Scholar
[15] Feriani, A., Perotti, F., and Simoncini, V., Iterative system solvers for the frequency analysis of linear mechanical systems, Comput. Methods Appl. Mech. Eng., 190 (2000), pp. 1719–1739.Google Scholar
[16] Frommer, A., Lippert, T., Medeke, B., and Schilling, K. (EDS), Numerical Challenges in Lattice Quantum Chromodynamics, Springer, Berlin, 2000.Google Scholar
[17] Golub, G. H. and Van Loan, C. F., Matrix Computations, John Hopkins University Press, Baltimore, 2012.Google Scholar
[18] Guo, X. P. and Duff, I. S., Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl., 18 (2011), pp. 299–315.Google Scholar
[19] Kelly, C. T., Iterative Methods for Linear and Nonlinear Equations, SIAM, Philadelphia, 1995.Google Scholar
[20] Li, Y. and Guo, X. P., Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices, Numer. Algor., 10.1007/s11075-016-0196-6, 2016.Google Scholar
[21] Poirier, B., Efficient preconditioning scheme for block partitioned matrices with structured sparsity, Numer. Linear Algebra Appl., 7 (2000), pp. 715–726.Google Scholar
[22] Saad, Y., Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.Google Scholar
[23] Van, D. W. and Toyama, F. M., Accurate numerical solutions of the time-dependent Schrödinger equation, Phys. Rev. E, 75 (2007), pp. 036707.Google Scholar
[24] Wu, Q. B. and Chen, M. H., Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations, Numer. Algor., 64 (2013), pp. 659–683.Google Scholar
[25] Xu, W. W., A generalization of PMHSS iteration method for complex symmetric indefinite linear systems, Appl. Math. Comput., 219 (2013), pp. 10510–10517.Google Scholar
[26] Yang, A. L. and Wu, Y. J., Newton-MHSS methods for solving systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algebra, Control Optim., 2 (2012), pp. 839–853.Google Scholar
[27] Zhong, H. X., Chen, G. L., and Guo, X. P., On preconditioned modified Newton-MHSS method for systems of nonlinear equations with complex symmetric Jacobian matrices, Numer. Algor., 69 (2015), pp. 553–567.Google Scholar