2012, 2(4): 811-821. doi: 10.3934/naco.2012.2.811

A generalization of the positive-definite and skew-Hermitian splitting iteration

1. 

School of Transportation, Nantong University, Nantong, 226019, China

2. 

School of Mathematical Sciences, Soochow University, Suzhou, 215006, China, China

Received  December 2011 Revised  September 2012 Published  November 2012

In this paper, a generalization of the positive-definite and skew-Hermitian splitting (GPSS) iteration is considered for solving non-Hermitian and positive definite systems of linear equations. Theoretical analysis shows that the GPSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrices. In some situations, this new scheme can outperform the Hermitian and skew-Hermitian splitting (HSS) method, the positive-definite and skew-Hermitian splitting (PSS) method, and the generalized HSS method (GHSS) and can be used as an efficient preconditioner. Numerical experiments using discretization of convection-diffusion-reaction equations demonstrate the effectiveness of the new method.
Citation: Yang Cao, Wei- Wei Tan, Mei-Qun Jiang. A generalization of the positive-definite and skew-Hermitian splitting iteration. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 811-821. doi: 10.3934/naco.2012.2.811
References:
[1]

Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle point problems, Numer. Linear Algebra Appl., 16 (2009), 447-479. doi: 10.1002/nla.626.

[2]

Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197. doi: 10.1007/s00607-010-0101-.

[3]

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.

[4]

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.

[5]

Z.-Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23. doi: 10.1093/imanum/drl017.

[6]

Z.-Z. Bai, G. H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), 583-603. doi: 10.1137/050623644.

[7]

Z.-Z. Bai, G. H. Golub, L.-Z. Lu and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 23 (2005), 844-863. doi: 10.1137/S1064827503428114.

[8]

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.

[9]

Z.-Z. Bai, G. H. Golub and M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), 319-335. doi: 10.1002/nla.517.

[10]

Z.-Z. Bai, G. H. Golub and M. K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), 413-440. doi: 10.1016/j.laa.2007.02.018.

[11]

Z.-Z. Bai, G. H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32.

[12]

M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), 360-374. doi: 10.1137/080723181.

[13]

M. Benzi, M. J. Gander and G. H Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43 (2003), 881-900. doi: 10.1023/B:BITN.0000014548.26616.65.

[14]

M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. doi: 10.1137/S0895479802417106.

[15]

M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61 (2011), 66-76. doi: 10.1016/j.apnum.2010.08.005.

[16]

L. C. Chan, M. K. Ng and N. K. Tsing, Spectral Analysis for HSS Preconditioners, Numer. Math. Theor. Meth. Appl., 1 (2008), 57-77.

[17]

M.-Q. Jiang and Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), 973-982. doi: 10.1016/j.cam.2009.05.012.

[18]

L. Li, T.-Z. Huang and X.-P. Liu, Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems, Comput. Math. Appl., 54 (2007), 147-159. doi: 10.1016/j.camwa.2006.12.024.

[19]

J.-Y. Pan, M. K. Ng and Z.-Z. Bai, New preconditioners for saddle point problems, Appl. Math. Comput., 172 (2006), 762-771. doi: 10.1016/j.amc.2004.11.016.

[20]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl.Math., 3 (1955), 28-41. doi: 10.1137/0103003.

[21]

Y. Saad, "Iterative Methods for Sparse Linear Systems," 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.

[22]

A.-L Yang, J. An and Y.-J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems, Appl. Math. Comput., 216 (2010), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

show all references

References:
[1]

Z.-Z. Bai, Optimal parameters in the HSS-like methods for saddle point problems, Numer. Linear Algebra Appl., 16 (2009), 447-479. doi: 10.1002/nla.626.

[2]

Z.-Z. Bai, On semi-convergence of Hermitian and skew-Hermitian splitting methods for singular linear systems, Computing, 89 (2010), 171-197. doi: 10.1007/s00607-010-0101-.

[3]

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.

[4]

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.

[5]

Z.-Z. Bai and G. H. Golub, Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems, IMA J. Numer. Anal., 27 (2007), 1-23. doi: 10.1093/imanum/drl017.

[6]

Z.-Z. Bai, G. H. Golub and C.-K. Li, Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices, SIAM J. Sci. Comput., 28 (2006), 583-603. doi: 10.1137/050623644.

[7]

Z.-Z. Bai, G. H. Golub, L.-Z. Lu and J.-F. Yin, Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. Sci. Comput., 23 (2005), 844-863. doi: 10.1137/S1064827503428114.

[8]

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.

[9]

Z.-Z. Bai, G. H. Golub and M. K. Ng, On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations, Numer. Linear Algebra Appl., 14 (2007), 319-335. doi: 10.1002/nla.517.

[10]

Z.-Z. Bai, G. H. Golub and M. K. Ng, On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Linear Algebra Appl., 428 (2008), 413-440. doi: 10.1016/j.laa.2007.02.018.

[11]

Z.-Z. Bai, G. H. Golub and J.-Y. Pan, Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems, Numer. Math., 98 (2004), 1-32.

[12]

M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl., 31 (2009), 360-374. doi: 10.1137/080723181.

[13]

M. Benzi, M. J. Gander and G. H Golub, Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems, BIT, 43 (2003), 881-900. doi: 10.1023/B:BITN.0000014548.26616.65.

[14]

M. Benzi and G. H. Golub, A preconditioner for generalized saddle point problems, SIAM J. Matrix Anal. Appl., 26 (2004), 20-41. doi: 10.1137/S0895479802417106.

[15]

M. Benzi and X.-P. Guo, A dimensional split preconditioner for Stokes and linearized Navier-Stokes equations, Appl. Numer. Math., 61 (2011), 66-76. doi: 10.1016/j.apnum.2010.08.005.

[16]

L. C. Chan, M. K. Ng and N. K. Tsing, Spectral Analysis for HSS Preconditioners, Numer. Math. Theor. Meth. Appl., 1 (2008), 57-77.

[17]

M.-Q. Jiang and Y. Cao, On local Hermitian and skew-Hermitian splitting iteration methods for generalized saddle point problems, J. Comput. Appl. Math., 231 (2009), 973-982. doi: 10.1016/j.cam.2009.05.012.

[18]

L. Li, T.-Z. Huang and X.-P. Liu, Asymmetric Hermitian and skew-Hermitian splitting methods for positive definite linear systems, Comput. Math. Appl., 54 (2007), 147-159. doi: 10.1016/j.camwa.2006.12.024.

[19]

J.-Y. Pan, M. K. Ng and Z.-Z. Bai, New preconditioners for saddle point problems, Appl. Math. Comput., 172 (2006), 762-771. doi: 10.1016/j.amc.2004.11.016.

[20]

D. W. Peaceman and H. H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations, J. Soc. Indust. Appl.Math., 3 (1955), 28-41. doi: 10.1137/0103003.

[21]

Y. Saad, "Iterative Methods for Sparse Linear Systems," 2nd edition, SIAM, Philadelphia, 2003. doi: 10.1137/1.9780898718003.

[22]

A.-L Yang, J. An and Y.-J. Wu, A generalized preconditioned HSS method for non-Hermitian positive definite linear systems, Appl. Math. Comput., 216 (2010), 1715-1722. doi: 10.1016/j.amc.2009.12.032.

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