scholarly journals A relaxed block splitting preconditioner for complex symmetric indefinite linear systems

2018 ◽  
Vol 16 (1) ◽  
pp. 561-573
Author(s):  
Yunying Huang ◽  
Guoliang Chen
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Minimal Polynomial ◽  
Coefficient Matrix ◽  
Subspace Iteration ◽  
Preconditioned Matrix ◽  
Complex Symmetric ◽  
Indefinite Linear Systems

AbstractIn this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the original block two-by-two coefficient matrix. We study the spectral properties and the eigenvector distributions of the corresponding preconditioned matrix. In addition, the degree of the minimal polynomial of the preconditioned matrix is also derived. Finally, some numerical experiments are presented to illustrate the effectiveness of the relaxed splitting preconditioner.

On Preconditioned MHSS Real-Valued Iteration Methods for a Class of Complex Symmetric Indefinite Linear Systems

2016 ◽  
Vol 6 (2) ◽  
pp. 192-210 ◽  
Author(s):  
Zhi-Ru Ren ◽  
Yang Cao ◽  
Li-Li Zhang
Keyword(s):  
Linear Systems ◽  
Iteration Method ◽  
Spectral Properties ◽  
Convergence Theory ◽  
Numerical Examples ◽  
Preconditioned Matrix ◽  
Iteration Methods ◽  
Complex Symmetric ◽  
Indefinite Linear Systems

AbstractA generalized preconditioned modified Hermitian and skew-Hermitian splitting (GPMHSS) real-valued iteration method is proposed for a class of complex symmetric indefinite linear systems. Convergence theory is established and the spectral properties of an associated preconditioned matrix are analyzed. We also give several variants of the GPMHSS preconditioner and consider the spectral properties of the preconditioned matrices. Numerical examples illustrate the effectiveness of our proposed method.

A Convergence Analysis of the MINRES Method for Some Hermitian Indefinite Systems

2017 ◽  
Vol 7 (4) ◽  
pp. 827-836
Author(s):  
Ze-Jia Xie ◽  
Xiao-Qing Jin ◽  
Zhi Zhao
Keyword(s):  
Convergence Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
Indefinite Systems ◽  
Indefinite Linear Systems ◽  
Theoretical Results

AbstractSome convergence bounds of the minimal residual (MINRES) method are studied when the method is applied for solving Hermitian indefinite linear systems. The matrices of these linear systems are supposed to have some properties so that their spectra are all clustered around ±1. New convergence bounds depending on the spectrum of the coefficient matrix are presented. Some numerical experiments are shown to demonstrate our theoretical results.

scholarly journals A Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1302
Author(s):  
Hong-Xiu Zhong ◽  
Xian-Ming Gu ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Coefficient Matrix ◽  
Complex Symmetric Linear Systems ◽  
Right Hand ◽  
Orthogonality Conditions ◽  
Free Block ◽  
Complex Symmetric ◽  
The Right ◽  
Conjugate Residual

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

scholarly journals Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Cui-Xia Li ◽  
Yan-Jun Liang ◽  
Shi-Liang Wu
Keyword(s):  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Krylov Subspace Methods ◽  
Point Of View ◽  
Subspace Methods ◽  
Practical Point ◽  
Complex Symmetric Linear Systems ◽  
Complex Symmetric

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

On Euler preconditioned SHSS iterative method for a class of complex symmetric linear systems

2019 ◽  
Vol 53 (5) ◽  
pp. 1607-1627 ◽  
Author(s):  
Cheng-Liang Li ◽  
Chang-Feng Ma
Keyword(s):  
Iterative Method ◽  
Linear Systems ◽  
Krylov Subspace ◽  
Broad Class ◽  
Single Step ◽  
Convergence Properties ◽  
Complex Symmetric Linear Systems ◽  
Preconditioned Matrix ◽  
Complex Symmetric ◽  
Preconditioned Krylov Subspace Method

In this paper, we propose an Euler preconditioned single-step HSS (EP-SHSS) iterative method for solving a broad class of complex symmetric linear systems. The proposed method can be applied not only to the non-singular complex symmetric linear systems but also to the singular ones. The convergence (semi-convergence) properties of the proposed method are carefully discussed under suitable restrictions. Furthermore, we consider the acceleration of the EP-SHSS method by preconditioned Krylov subspace method and discuss the spectral properties of the corresponding preconditioned matrix. Numerical experiments verify the effectiveness of the EP-SHSS method either as a solver or as a preconditioner for solving both non-singular and singular complex symmetric linear systems.

scholarly journals Fast collocation method for a two-dimensional variable-coefficient linear nonlocal diffusion model

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xuhao Zhang ◽  
Aijie Cheng
Keyword(s):  
Collocation Method ◽  
Diffusion Model ◽  
Variable Coefficient ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Coefficient Matrix ◽  
Nonlocal Diffusion ◽  
Two Dimensional ◽  
Subspace Iteration ◽  
Dimensional Variable

Abstract In this paper, a fast collocation method is developed for a two-dimensional variable-coefficient linear nonlocal diffusion model. By carefully dealing with the variable coefficient in the integral operator and then analyzing the structure of the coefficient matrix, we can reduce the computational operations in each Krylov subspace iteration from $O(N^{2})$ O ( N 2 ) to $O(N\log N)$ O ( N log N ) and the memory requirement for the coefficient matrix from $O(N^{2})$ O ( N 2 ) to $O(N)$ O ( N ) . Numerical experiments are carried out to show the utility of the fast collocation method.

scholarly journals A Polynomial Preconditioned Global CMRH Method for Linear Systems with Multiple Right-Hand Sides

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ke Zhang ◽  
Chuanqing Gu
Keyword(s):  
Theoretical Result ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Krylov Subspace ◽  
Polynomial Degree ◽  
Right Hand

The restarted global CMRH method (Gl-CMRH(m)) (Heyouni, 2001) is an attractive method for linear systems with multiple right-hand sides. However, Gl-CMRH(m) may converge slowly or even stagnate due to a limited Krylov subspace. To ameliorate this drawback, a polynomial preconditioned variant of Gl-CMRH(m) is presented. We give a theoretical result for the square case that assures that the number of restarts can be reduced with increasing values of the polynomial degree. Numerical experiments from real applications are used to validate the effectiveness of the proposed method.

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