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

Tahereh Salimi Siahkolaei; Davod Khojasteh Salkuyeh

doi:10.3934/naco.2019033

Article Contents

2019, Volume 9, Issue 4: 483-492. Doi: 10.3934/naco.2019033

This issue Previous Article

$ \theta $

scheme with two dimensional wavelet-like incremental unknowns for a class of porous medium diffusion-type equations Next Article A hybrid parametrization approach for a class of nonlinear optimal control problems

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

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

^*Corresponding author
^*Corresponding author

Received: August 2018

Revised: April 2019

Published: November 2019

Abstract / Introduction Full Text(HTML) Figure(0) / Table(4) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 65F10, 65F50, 65F08.

Citation:

FullText(HTML)

Table 1. 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

| Show Table

DownLoad: CSV

Table 2. 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

| Show Table

DownLoad: CSV

Table 3. 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

| Show Table

DownLoad: CSV

Table 4. 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

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

[1]	O. Axelsson and A. Kucherov, Real valued iterative methods for solving complex symmetric linear systems, Numer. Linear Algebra Appl., 7 (2000), 197-218. doi: 10.1002/1099-1506(200005)7:4<197::AID-NLA194>3.0.CO;2-S.
[2]	Z.-Z. Bai, M. Benzi and F. Chen, Modified HSS iteration methods for a class of complex symmetric linear systems, Computing, 87 (2010), 93-111. doi: 10.1007/s00607-010-0077-0.
[3]	Z.-Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems, Numer. Algor., 56 (2011), 297-317. doi: 10.1007/s11075-010-9441-6.
[4]	Z.-Z. Bai, Rotated block triangular preconditioning based on PMHSS, Sci. China Math., 56 (2013), 2523-2538. doi: 10.1007/s11425-013-4695-9.
[5]	Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM. J. Matrix Anal. Appl., 24 (2003), 603-626. doi: 10.1137/S0895479801395458.
[6]	Z.-Z. Bai, M. Benzi, F. Chen and Z.-Q. Wang, Preconditioned MHSS iteration methods for a class of block two-by-two linear systems with applications to distributed control problems, IMA J. Numer. Anal., 33 (2013), 343-369. doi: 10.1093/imanum/drs001.
[7]	Z.-Z. Bai, B. N. Parlett and Z.-Q. Wang, On generalized successive overrelaxation methods for augmented linear systems, Numer. Math., 102 (2005), 1-38. doi: 10.1007/s00211-005-0643-0.
[8]	M. Benzi and D. Bertaccini, Block preconditioning of real-valued iterative algorithms for complex linear systems, IMA J. Numer. Anal., 28 (2008), 598-618. doi: 10.1093/imanum/drm039.
[9]	M. Benzi and G. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. doi: 10.1137/S0895479802417106.
[10]	M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numerica, 14 (2005), 1-137. doi: 10.1017/S0962492904000212.
[11]	A. Bunse-Gerstner and R. Stover, On a conjugate gradient-type method for solving complex symmetric linear systems, Linear Algebra Appl., 287 (1999), 105-123. doi: 10.1016/S0024-3795(98)10091-5.
[12]	M. Dehghan, M. M. Dehghani and M. Hajarian, A generalized preconditioned MHSS method for a class of complex symmetric linear systems, J. Math. Model. Anal., 18 (2013), 561-576. doi: 10.3846/13926292.2013.839964.
[13]	V. Edalatpour, D. Hezari and D. K. Salkuyeh, Two efficient inexact algorithms for a class of large sparse complex linear systems, Mediterr. J. Math., 13 (2016), 2301-2318. doi: 10.1007/s00009-015-0621-4.
[14]	G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
[15]	D. Hezari, D. K. Salkuyeh and V. Edalatpour, A new iterative method for solving a class of complex symmetric system of linear equathions, Numer. Algor., 73 (2016), 927-955. doi: 10.1007/s11075-016-0123-x.
[16]	D. Hezari, V. Edalatpour and D. K. Salkuyeh, Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations, Numer. Linear Algebera Appl., 22 (2015), 761-776. doi: 10.1002/nla.1987.
[17]	Z.-Z. Liang and G.-F. Zhang, On SSOR iteration method for a class of block two-by-wo linear systems, Numer. Algor., 71 (2016), 655-671. doi: 10.1007/s11075-015-0015-5.
[18]	D. K. Salkuyeh, D. Hezari and V. Edalatpour, Generalized successive overrelaxation iterative method for a class of complex symmetric linear system of equations, Int. J. Comput. Math., 92 (2015), 802-815. doi: 10.1080/00207160.2014.912753.
[19]	D. K. Salkuyeh and T. S. Siahkolaei, Two-parameter TSCSP method for solving complex symmetric system of linear equations, Calcolo, 55 (2018), 8. doi: 10.1007/s10092-018-0252-9.
[20]	G.-F. Zhang and Z. Zheng, A parameterized splitting iteration methods for complex symmetric linear systems, Jpn. J. Indust. Appl. Math., 31 (2014), 265-278. doi: 10.1007/s13160-014-0140-x.