On the solution of a class of fuzzy system of linear equations

In this paper, we consider the system of linear equations $Ax=b$, where $A\in \Bbb{R}^{n \times n}$ is a crisp H-matrix and $b$ is a fuzzy $n$-vector. We then investigate the existence and uniqueness of a fuzzy solution to this system. The results can also be used for the class of M-matrices and strictly diagonally dominant matrices. Finally, some numerical examples are given to illustrate the presented theoretical results.Comment: 7 page

Two-step scale-splitting method for solving complex symmetric system of linear equations

Based on the Scale-Splitting (SCSP) iteration method presented by Hezari et al. in (A new iterative method for solving a class of complex symmetric system linear of equations, Numerical Algorithms 73 (2016) 927-955), we present a new two-step iteration method, called TSCSP, for solving the complex symmetric system of linear equations $(W+iT)x=b$, where $W$ and $T$ are symmetric positive definite and symmetric positive semidefinite matrices, respectively. It is shown that if the matrices $W$ and $T$ are symmetric positive definite, then the method is unconditionally convergent. The optimal value of the parameter, which minimizes the spectral radius of the iteration matrix is also computed. Numerical {comparisons} of the TSCSP iteration method with the SCSP, the MHSS, the PMHSS and the GSOR methods are given to illustrate the effectiveness of the method.Comment: 13 pages. Current status: Unsubmitted. arXiv admin note: text overlap with arXiv:1403.5902, arXiv:1611.0370

A Generalization of the 2D-DSPM for Solving Linear System of Equations

In [7], a new iterative method for solving linear system of equations was presented which can be considered as a modification of the Gauss-Seidel method. Then in [4] a different approach, say 2D-DSPM, and more effective one was introduced. In this paper, we improve this method and give a generalization of it. Convergence properties of this kind of generalization are also discussed. We finally give some numerical experiments to show the efficiency of the method and compare with 2D-DSPM.Comment: 11 pages, submitte

A modification of the generalized shift-splitting method for singular saddle point problems

A modification of the generalized shift-splitting (GSS) method is presented for solving singular saddle point problems. In this kind of modification, the diagonal shift matrix is replaced by a block diagonal matrix which is symmetric positive definite. Semi-convergence of the proposed method is investigated. The induced preconditioner is applied to the saddle point problem and the preconditioned system is solved by the restarted generalized minimal residual method. Eigenvalue distribution of the preconditioned matrix is also discussed. Finally some numerical experiments are given to show the effectiveness and robustness of the new preconditioner. Numerical results show that the modified GSS method is superior to the classical GSS method.Comment: 21 pages, submitte

Interpolated variational iteration method for initial value problems

In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced which converges to the solution under some suitable conditions. In the nonlinear case, after a few iterations the terms of the sequence become complicated, and therefore, computing a highly accurate solution would be difficult or even impossible. In this paper, for one-dimensional initial value problems, we propose a new approach which is based on approximating each term of the sequence by a piecewise linear function. Moreover, the convergence of the method is proved. Three illustrative examples are given to show the superiority of the proposed method over the classical variational iteration method.Comment: 17 pages, 8 figures in Applied Mathematical Modelling, 201

A new iterative method for solving a class of two-by-two block complex linear systems

We present a stationary iteration method, namely Alternating Symmetric positive definite and Scaled symmetric positive semidefinite Splitting (ASSS), for solving the system of linear equations obtained by using finite element discretization of a distributed optimal control problem together with time-periodic parabolic equations. An upper bound for the spectral radius of the iteration method is given which is always less than 1. So convergence of the ASSS iteration method is guaranteed. The induced ASSS preconditioner is applied to accelerate the convergence speed of the GMRES method for solving the system. Numerical results are presented to demonstrate the effectiveness of both the ASSS iteration method and the ASSS preconditioner.Comment: 17 Pages, Revised, Submitte

Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. They are now mostly used as preconditioners for the popular iterative solvers. In this paper a generalization of these methods are proposed and their convergence properties are studied. Some numerical experiments are given to show the efficiency of the new methods

On the generalized shift-splitting preconditioner for saddle point problems

In this paper, the generalized shift-splitting preconditioner is implemented for saddle point problems with symmetric positive definite (1,1)-block and symmetric positive semidefinite (2,2)-block. The proposed preconditioner is extracted form a stationary iterative method which is unconditionally convergent. Moreover, a relaxed version of the proposed preconditioner is presented and some properties of the eigenvalues distribution of the corresponding preconditioned matrix are studied. Finally, some numerical experiments on test problems arisen from finite element discretization of the Stokes problem are given to show the effectiveness of the preconditioners.Comment: 7 pages, 1 figure and 2 tables, Applied Mathematics Letters, 201

A preconditioner based on the shift-splitting method for generalized saddle point problems

In this paper, we propose a preconditioner based on the shift-splitting method for generalized saddle point problems with nonsymmetric positive definite (1,1)-block and symmetric positive semidefinite $(2,2)$-block. The proposed preconditioner is obtained from an basic iterative method which is unconditionally convergent. We also present a relaxed version of the proposed method. Some numerical experiments are presented to show the effectiveness of the method.Comment: 4 pages, submitte

A New Preconditioner for the GeneRank Problem

Identifying key genes involved in a particular disease is a very important problem which is considered in biomedical research. GeneRank model is based on the PageRank algorithm that preserves many of its mathematical properties. The model brings together gene expression information with a network structure and ranks genes based on the results of microarray experiments combined with gene expression information, for example from gene annotations (GO). In the present study, we present a new preconditioned conjugate gradient algorithm to solve GeneRank problem and study its properties. Some numerical experiments are given to show the effectiveness of the suggested preconditioner.Comment: 10 pages, 3 figure