American Institute of Mathematical Sciences

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
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 [1] Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297 [2] David Maxwell. Kozlov-Maz'ya iteration as a form of Landweber iteration. Inverse Problems & Imaging, 2014, 8 (2) : 537-560. doi: 10.3934/ipi.2014.8.537 [3] 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 [4] 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 [5] 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 [6] Plamen Stefanov, Yang Yang. Multiwave tomography with reflectors: Landweber's iteration. Inverse Problems & Imaging, 2017, 11 (2) : 373-401. doi: 10.3934/ipi.2017018 [7] Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144 [8] Óscar Vega-Amaya, Joaquín López-Borbón. A perturbation approach to a class of discounted approximate value iteration algorithms with borel spaces. Journal of Dynamics & Games, 2016, 3 (3) : 261-278. doi: 10.3934/jdg.2016014 [9] Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165 [10] Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079 [11] Gemma Huguet, Rafael de la Llave, Yannick Sire. Fast iteration of cocycles over rotations and computation of hyperbolic bundles. Conference Publications, 2013, 2013 (special) : 323-333. doi: 10.3934/proc.2013.2013.323 [12] Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855 [13] Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683 [14] Scott Crass. New light on solving the sextic by iteration: An algorithm using reliable dynamics. Journal of Modern Dynamics, 2011, 5 (2) : 397-408. doi: 10.3934/jmd.2011.5.397 [15] Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control & Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033 [16] 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 [17] Scott Crass. Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168. Journal of Modern Dynamics, 2007, 1 (2) : 175-203. doi: 10.3934/jmd.2007.1.175 [18] Olivier Bokanowski, Maurizio Falcone, Roberto Ferretti, Lars Grüne, Dante Kalise, Hasnaa Zidani. Value iteration convergence of $\epsilon$-monotone schemes for stationary Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4041-4070. doi: 10.3934/dcds.2015.35.4041 [19] B. S. Lee, Arif Rafiq. Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 287-293. doi: 10.3934/naco.2014.4.287 [20] Byung-Soo Lee. Strong convergence theorems with three-step iteration in star-shaped metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 371-379. doi: 10.3934/naco.2011.1.371

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