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 & 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.  doi: 10.1002/nla.626.  Google Scholar

[2]

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

[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.  doi: 10.1007/s00607-010-0077-0.  Google Scholar

[4]

Z.-Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems,, Numer. Algor., 56 (2011), 297.  doi: 10.1007/s11075-010-9441-6.  Google Scholar

[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.  doi: 10.1093/imanum/drl017.  Google Scholar

[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.  doi: 10.1137/050623644.  Google Scholar

[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.  doi: 10.1137/S1064827503428114.  Google Scholar

[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.  doi: 10.1137/S0895479801395458.  Google Scholar

[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.  doi: 10.1002/nla.517.  Google Scholar

[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.  doi: 10.1016/j.laa.2007.02.018.  Google Scholar

[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.   Google Scholar

[12]

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

[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.  doi: 10.1023/B:BITN.0000014548.26616.65.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  doi: 10.1016/j.cam.2009.05.012.  Google Scholar

[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.  doi: 10.1016/j.camwa.2006.12.024.  Google Scholar

[19]

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

[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.  doi: 10.1137/0103003.  Google Scholar

[21]

Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2nd edition, (2003).  doi: 10.1137/1.9780898718003.  Google Scholar

[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.  doi: 10.1016/j.amc.2009.12.032.  Google Scholar

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.  doi: 10.1002/nla.626.  Google Scholar

[2]

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

[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.  doi: 10.1007/s00607-010-0077-0.  Google Scholar

[4]

Z.-Z. Bai, M. Benzi and F. Chen, On preconditioned MHSS iteration methods for complex symmetric linear systems,, Numer. Algor., 56 (2011), 297.  doi: 10.1007/s11075-010-9441-6.  Google Scholar

[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.  doi: 10.1093/imanum/drl017.  Google Scholar

[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.  doi: 10.1137/050623644.  Google Scholar

[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.  doi: 10.1137/S1064827503428114.  Google Scholar

[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.  doi: 10.1137/S0895479801395458.  Google Scholar

[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.  doi: 10.1002/nla.517.  Google Scholar

[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.  doi: 10.1016/j.laa.2007.02.018.  Google Scholar

[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.   Google Scholar

[12]

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

[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.  doi: 10.1023/B:BITN.0000014548.26616.65.  Google Scholar

[14]

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

[15]

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

[16]

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

[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.  doi: 10.1016/j.cam.2009.05.012.  Google Scholar

[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.  doi: 10.1016/j.camwa.2006.12.024.  Google Scholar

[19]

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

[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.  doi: 10.1137/0103003.  Google Scholar

[21]

Y. Saad, "Iterative Methods for Sparse Linear Systems,", 2nd edition, (2003).  doi: 10.1137/1.9780898718003.  Google Scholar

[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.  doi: 10.1016/j.amc.2009.12.032.  Google Scholar

[1]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[2]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[3]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[4]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[5]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[6]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054

[7]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[8]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[9]

Haoyu Li, Zhi-Qiang Wang. Multiple positive solutions for coupled Schrödinger equations with perturbations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020294

[10]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[11]

Yan'e Wang, Nana Tian, Hua Nie. Positive solution branches of two-species competition model in open advective environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021006

[12]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[13]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[14]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[15]

Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji, Byungjae Son. A diffusive weak Allee effect model with U-shaped emigration and matrix hostility. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020356

[16]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076

[17]

Yueh-Cheng Kuo, Huan-Chang Cheng, Jhih-You Syu, Shih-Feng Shieh. On the nearest stable $ 2\times 2 $ matrix, dedicated to Prof. Sze-Bi Hsu in appreciation of his inspiring ideas. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020358

 Impact Factor: 

Metrics

  • PDF downloads (111)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]