A survey of preconditioned iterative methods for linear systems of algebraic equations

Many iterative linear solver packages focus on real-valued systems and do not deal well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc and Aztec tend to focus on the real-valued systems, while complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERF’s that can be used to solve complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigenvalues in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the previously mentioned approaches, and show that the interleaved form achieves a better outcome than either separate ERF. [This article is the second part of a sequence of reports. See the December 2002 issue for Part 1—Editor.]

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Sparse Preconditioned Iterative Methods for Dense Linear Systems

SIAM Journal on Scientific Computing ◽

10.1137/0915073 ◽

1994 ◽

Vol 15 (5) ◽

pp. 1190-1200 ◽

Cited By ~ 10

Author(s):

Yi Yan

Keyword(s):

Linear Systems ◽

Iterative Methods ◽

Preconditioned Iterative ◽

Preconditioned Iterative Methods

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Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations

Communications in Computational Physics ◽

10.4208/cicp.120314.230115a ◽

2015 ◽

Vol 18 (2) ◽

pp. 469-488 ◽

Cited By ~ 28

Author(s):

Xiao-Qing Jin ◽

Fu-Rong Lin ◽

Zhi Zhao

Keyword(s):

Linear Systems ◽

Iterative Methods ◽

Dimensional Space ◽

Fractional Diffusion ◽

Second Order ◽

Diffusion Equations ◽

Two Dimensional ◽

Fractional Diffusion Equations ◽

Preconditioned Iterative ◽

Preconditioned Iterative Methods

AbstractIn this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

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Convergence Analysis of Preconditioned SSOR Iterative Method for Linear Systems

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.644-650.1988 ◽

2014 ◽

Vol 644-650 ◽

pp. 1988-1991

Author(s):

Ting Zhou

Keyword(s):

Linear System ◽

Iterative Method ◽

Convergence Analysis ◽

Linear Systems ◽

Iterative Methods ◽

Convergence Result ◽

Numerical Example ◽

Preconditioned Iterative ◽

Preconditioned Iterative Methods

For solving the linear system, different preconditioned iterative methods have been proposed by many authors. In this paper, we present preconditioned SSOR iterative method for solving the linear systems. Meanwhile, we apply the preconditioner to H-matrix and obtain the convergence result. Finally, a numerical example is also given to illustrate our results.

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A modified AOR-type iterative method for L-matrix linear systems

The ANZIAM Journal ◽

10.1017/s1446181100012840 ◽

2007 ◽

Vol 49 (2) ◽

pp. 281-292 ◽

Cited By ~ 2

Author(s):

Shiliang Wu ◽

Tingzhu Huang

Keyword(s):

Iterative Method ◽

Linear Systems ◽

Iterative Methods ◽

Convergence Rates ◽

Comparison Theorems ◽

Iterative Schemes ◽

Seidel Method ◽

Comparison Results ◽

Preconditioned Iterative ◽

Preconditioned Iterative Methods

AbstractBoth Evans et al. and Li et al. have presented preconditioned methods for linear systems to improve the convergence rates of AOR-type iterative schemes. In this paper, we present a new preconditioner. Some comparison theorems on preconditioned iterative methods for solving L-matrix linear systems are presented. Comparison results and a numerical example show that convergence of the preconditioned Gauss-Seidel method is faster than that of the preconditioned AOR iterative method.

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