scholarly journals Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Yu-Chien Li ◽  
Jen-Yuan Chen ◽  
David R. Kincaid
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Methods ◽  
Arnoldi Process ◽  
Indefinite Linear Systems

We discuss a variety of iterative methods that are based on the Arnoldi process for solving large sparse symmetric indefinite linear systems. We describe the SYMMLQ and SYMMQR methods, as well as generalizations and modifications of them. Then, we cover the Lanczos/MSYMMLQ and Lanczos/MSYMMQR methods, which arise from a double linear system. We present pseudocodes for these algorithms.

scholarly journals New iterative methods for dense linear systems

2021 ◽  
Vol 293 ◽  
pp. 02013
Author(s):  
Jinmei Wang ◽  
Lizi Yin ◽  
Ke Wang
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Methods ◽  
Parallel Implementation ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Linear System Of Equations ◽  
Dense Linear System ◽  
Linear Systems Of Equations

Solving dense linear systems of equations is quite time consuming and requires an efficient parallel implementation on powerful supercomputers. Du, Zheng and Wang presented some new iterative methods for linear systems [Journal of Applied Analysis and Computation, 2011, 1(3): 351-360]. This paper shows that their methods are suitable for solving dense linear system of equations, compared with the classical Jacobi and Gauss-Seidel iterative methods.

scholarly journals Block Preconditioned SSOR Methods for -Matrices Linear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhao-Nian Pu ◽  
Xue-Zhong Wang
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Methods ◽  
Spectral Radius ◽  
Numerical Examples ◽  
Comparison Results ◽  
Block Preconditioner

We present a block preconditioner and consider block preconditioned SSOR iterative methods for solving linear system . When is an -matrix, the convergence and some comparison results of the spectral radius for our methods are given. Numerical examples are also given to illustrate that our methods are valid.

Convergence Analysis of Preconditioned SSOR Iterative Method for Linear Systems

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.

scholarly journals Analysis of Some Nonstationary Iterative Methods Using the Pascal and Vandermonde Linear Systems

2021 ◽  
pp. 1-3
Author(s):  
Azizu S ◽  
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Methods ◽  
Programming Language ◽  
Numerical Experiments ◽  
Polynomial Function ◽  
Paper Analysis ◽  
Vandermonde Matrix ◽  
Cubic Polynomial ◽  
Matlab Programming

In this paper, analysis of some nonstationary iterative methods using the Vandermonde and Pascal linear system is reported. The nonstationary iterative methods selected were GMRES and QMR to assess their performance on the identified linear systems. The paper focused on the convergence relative residual and number of iteration for each type of chosen linear system. The Vandermonde matrix is mostly applied to interpolation of both quadratic and cubic polynomial function. The resulting polynomial has the form: p(x) = an xn + an-1xn-1 +...+ a1 x + a0 . From the numerical experiments conducted using the matlab programming language, the GMRES is recommended when solving the identified linear systems

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.

scholarly journals Multipreconditioned GMRES for simulating stochastic automata networks

2018 ◽  
Vol 16 (1) ◽  
pp. 986-998
Author(s):  
Chun Wen ◽  
Ting-Zhu Huang ◽  
Xian-Ming Gu ◽  
Zhao-Li Shen ◽  
Hong-Fan Zhang ◽  
...  
Keyword(s):  
Steady State ◽  
Probability Distribution ◽  
Linear Systems ◽  
Iterative Methods ◽  
Communication Systems ◽  
Queueing Systems ◽  
Steady State Probability ◽  
Stochastic Automata ◽  
Automata Networks ◽  
Generator Matrices

AbstractStochastic Automata Networks (SANs) have a large amount of applications in modelling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seidel are inefficient due to the huge size of the generator matrices. In this paper, the multipreconditioned GMRES (MPGMRES) is considered by using two or more preconditioners simultaneously. Meanwhile, a selective version of the MPGMRES is presented to overcome the rapid increase of the storage requirements and make it practical. Numerical results on two models of SANs are reported to illustrate the effectiveness of these proposed methods.

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