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 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.

scholarly journals On the numerical solutions for nonlinear Volterra-Fredholm integral equations

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Parviz Darania ◽  
Saeed Pishbin
Keyword(s):  
Nonlinear System ◽  
Integral Equations ◽  
Numerical Experiments ◽  
Numerical Solutions ◽  
Coefficient Matrix ◽  
Collocation Methods ◽  
Fredholm Integral Equations ◽  
Stability Estimates ◽  
Lower Triangular ◽  
Theoretical Results

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

scholarly journals H-Matrices in Fuzzy Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Linear System ◽  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Fuzzy Linear System ◽  
Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an H-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.

scholarly journals NUMERICAL SOLUTION OF HIGHER INDEX NONLINEAR INTEGRAL ALGEBRAIC EQUATIONS OF HESSENBERG TYPE USING DISCONTINUOUS COLLOCATION METHODS

2014 ◽  
Vol 19 (1) ◽  
pp. 99-117 ◽  
Author(s):  
Babak Shiri
Keyword(s):  
Numerical Solution ◽  
Convergence Analysis ◽  
Large Class ◽  
Numerical Experiments ◽  
Collocation Methods ◽  
Algebraic Equations ◽  
Linear And Nonlinear ◽  
Theoretical Results

In this paper, we deal with a system of linear and nonlinear integral algebraic equations (IAEs) of Hessenberg type. Convergence analysis of the discontinuous collocation methods is investigated for the large class of IAEs based on the new definitions. Finally, some numerical experiments are provided to support the theoretical results.

scholarly journals Preconditioning Techniques Based on the Birkhoff–von Neumann Decomposition

2017 ◽  
Vol 17 (2) ◽  
pp. 201-215 ◽  
Author(s):  
Michele Benzi ◽  
Bora Uçar
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Sparse Matrices ◽  
Stochastic Matrices ◽  
Preconditioning Techniques ◽  
Von Neumann ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices ◽  
Theoretical Results

AbstractWe introduce a class of preconditioners for general sparse matrices based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices. These preconditioners are aimed primarily at solving challenging linear systems with highly unstructured and indefinite coefficient matrices. We present some theoretical results and numerical experiments on linear systems from a variety of applications.

Convergence of the Preconditioned AOR Method for an H-Matrix

2013 ◽  
Vol 756-759 ◽  
pp. 2615-2619
Author(s):  
Jie Jing Liu
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Numerical Experiments ◽  
Convergence Rates ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Aor Method ◽  
Upper Triangular Matrix ◽  
Theoretical Results

Linear system with H-matrix often appears in a wide variety of areas and is studied by many numerical researchers. In order to improve the convergence rates of iterative method solving the linear system whose coefficient matrix is an H-matrix. In this paper, a preconditioned AOR iterative method with a multi-parameters preconditioner with a general upper triangular matrix is proposed. In addition, the convergence of the coressponding iterative method are established. Lastly, we provide numerical experiments to illustrate the theoretical results.

scholarly journals Limiting Distributions for the Minimum-Maximum Models

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 719
Author(s):  
Ling Peng ◽  
Lei Gao
Keyword(s):  
Convergence Analysis ◽  
Numerical Experiments ◽  
Random Sequence ◽  
Extreme Value ◽  
Distribution Functions ◽  
Extreme Value Distributions ◽  
Limiting Distributions ◽  
Theoretical Results ◽  
Independent And Identically Distributed

We consider the extreme value problem of the minimum-maximum models for the independent and identically distributed random sequence and stationary random sequence, respectively. By invoking some probability formulas and Taylor’s expansions of the distribution functions, the limiting distributions for these two kinds of sequences are obtained. Moreover, convergence analysis is carried out for those extreme value distributions. Several numerical experiments are conducted to validate our theoretical results.

scholarly journals On applying weighted seed techniques to GMRES algorithm for solving multiple linear systems

2018 ◽  
Vol 36 (3) ◽  
pp. 155-172
Author(s):  
Lakhdar Elbouyahyaoui ◽  
Mohammed Heyouni
Keyword(s):  
Linear Systems ◽  
Sparse Matrix ◽  
Coefficient Matrix ◽  
University Of Florida ◽  
Block Arnoldi ◽  
The Matrix ◽  
Multiple Linear Systems ◽  
The University ◽  
Theoretical Results ◽  
Methods Numerical

In the present paper, we are concerned by weighted Arnoldi like methods for solving large and sparse linear systems that have different right-hand sides but have the same coefficient matrix. We first give detailed descriptions of the weighted Gram-Schmidt process and of a Ruhe variant of the weighted block Arnoldi algorithm. We also establish some theoretical results that links the iterates of the weighted block Arnoldi process to those of the non weighted one. Then, to accelerate the convergence of the classical restarted block and seed GMRES methods, we introduce the weighted restarted block and seed GMRES methods. Numerical experiments that are done with different matrices coming from the Matrix Market repository or from the university of Florida sparse matrix collection are reported at the end of this work in order to compare the performance and show the effectiveness of the proposed methods.

scholarly journals More on Generalizations and Modifications of Iterative Methods for Solving Large Sparse Indefinite Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Jen-Yuan Chen ◽  
David R. Kincaid ◽  
Yu-Chien Li
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Linear Equations ◽  
Indefinite Systems ◽  
Systems Of Linear Equations ◽  
Indefinite Linear Systems

Continuing from the works of Li et al. (2014), Li (2007), and Kincaid et al. (2000), we present more generalizations and modifications of iterative methods for solving large sparse symmetric and nonsymmetric indefinite systems of linear equations. We discuss a variety of iterative methods such as GMRES, MGMRES, MINRES, LQ-MINRES, QR MINRES, MMINRES, MGRES, and others.

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