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 Preconditioning Complex Symmetric Linear Systems

2015 ◽  
Vol 2015 ◽  
pp. 1-20 ◽  
Author(s):  
Enrico Bertolazzi ◽  
Marco Frego
Keyword(s):  
Linear Systems ◽  
Computational Cost ◽  
Coefficient Matrix ◽  
Iterative Solvers ◽  
Complex Symmetric Linear Systems ◽  
Symmetric Complex ◽  
Complex Linear ◽  
Complex Symmetric ◽  
Incomplete Cholesky Decomposition ◽  
Conjugate Residual

A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.

An Extension of the COCR Method to Solving Shifted Linear Systems with Complex Symmetric Matrices

2011 ◽  
Vol 1 (2) ◽  
pp. 97-107 ◽  
Author(s):  
Tomohiro Sogabe ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Large Scale ◽  
Electronic Structure Calculations ◽  
Symmetric Matrices ◽  
Complex Symmetric Linear Systems ◽  
Shifted Linear Systems ◽  
Complex Symmetric ◽  
Complex Symmetric Matrices ◽  
Structure Calculations ◽  
Conjugate Residual

AbstractThe Conjugate Orthogonal Conjugate Residual (COCR) method [T. Sogabe and S.-L. Zhang, JCAM, 199 (2007), pp. 297-303.] has recently been proposed for solving complex symmetric linear systems. In the present paper, we develop a variant of the COCR method that allows the efficient solution of complex symmetric shifted linear systems. Some numerical examples arising from large-scale electronic structure calculations are presented to illustrate the performance of the variant.

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.

A COMINRES-QLP Method for Solving Complex Symmetric Linear Systems

2020 ◽  
Vol 140 (12) ◽  
pp. 832-841
Author(s):  
Lijun Liu ◽  
Kazuaki Sekiya ◽  
Masao Ogino ◽  
Koki Masui
Keyword(s):  
Linear Systems ◽  
Complex Symmetric Linear Systems ◽  
Complex Symmetric

A method for solving linear programming with interval-valued trapezoidal fuzzy variables

2018 ◽  
Vol 52 (3) ◽  
pp. 955-979 ◽  
Author(s):  
Ali Ebrahimnejad
Keyword(s):  
Linear Programming ◽  
Fuzzy Numbers ◽  
Auxiliary Problem ◽  
Coefficient Matrix ◽  
Uncertain Parameters ◽  
Fuzzy Variables ◽  
Right Hand ◽  
Fuzzy Cost ◽  
The Right ◽  
Interval Valued

An efficient method to handle the uncertain parameters of a linear programming (LP) problem is to express the uncertain parameters by fuzzy numbers which are more realistic, and create a conceptual and theoretical framework for dealing with imprecision and vagueness. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand side, and/or the elements of the coefficient matrix. The aim of this article is to introduce a formulation of FLP problems involving interval-valued trapezoidal fuzzy numbers for the decision variables and the right-hand-side of the constraints. We propose a new method for solving this kind of FLP problems based on comparison of interval-valued fuzzy numbers by the help of signed distance ranking. To do this, we first define an auxiliary problem, having only interval-valued trapezoidal fuzzy cost coefficients, and then study the relationships between these problems leading to a solution for the primary problem. It is demonstrated that study of LP problems with interval-valued trapezoidal fuzzy variables gives rise to the same expected results as those obtained for LP with trapezoidal fuzzy variables.

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