scholarly journals Properweak regular splitting and its application to convergence of alternating iterations

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6563-6573 ◽  
Author(s):  
Debasisha Mishra
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Iterative Method ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Convergence Theory ◽  
System Of Equations ◽  
Comparison Result ◽  
Linear System Of Equations ◽  
Regular Splitting

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

scholarly journals A Higher Order Iterative Method for Computing the Drazin Inverse

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
F. Soleymani ◽  
Predrag S. Stanimirović
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Iterative Method ◽  
Drazin Inverse ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
High Convergence Rate ◽  
Approximate Inverses ◽  
Nonsingular Matrices ◽  
Test Matrices

A method with high convergence rate for finding approximate inverses of nonsingular matrices is suggested and established analytically. An extension of the introduced computational scheme to general square matrices is defined. The extended method could be used for finding the Drazin inverse. The application of the scheme on large sparse test matrices alongside the use in preconditioning of linear system of equations will be presented to clarify the contribution of the paper.

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 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 Perron Complementation on Linear Systems Involving M-Matrices

2021 ◽  
Vol 09 (05) ◽  
Author(s):  
Jianhong Xu ◽  
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Regular Splitting

We propose in this paper a generalized Perron complementation method for uncoupling a consistent linear system which involves an irreducible, either singular or nonsingular, M-matrix. We show that this uncoupling arises naturally from a regular splitting, which also leads to an efficient iterative scheme for solving the linear system.

scholarly journals Optimal design of optical analog solvers of linear systems

2021 ◽  
Vol 2 (6) ◽  
Author(s):  
Kthim Imeri
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Design ◽  
Linear System ◽  
Linear Systems ◽  
Asymptotic Expansions ◽  
Physical Method ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Optical Analog

AbstractIn this paper, given a linear system of equations $$\mathbf {A}\, \mathbf {x}= \mathbf {b}$$ A x = b , we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlying partial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.

scholarly journals An Iterative Method for Symmetric Positive Semidefinite Linear System of Equations

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Davod Khojasteh Salkuyeh
Keyword(s):  
Linear System ◽  
Iterative Method ◽  
Numerical Experiments ◽  
Positive Semidefinite ◽  
Sufficient Condition ◽  
System Of Equations ◽  
Linear System Of Equations

AbstractIn this paper, a new two-step iterative method for solving symmetric positive semidefinite linear system of equations is presented. A sufficient condition for the semiconvergence of the method is also given. Some numerical experiments are presented to show the efficiency of the proposed method.

scholarly journals Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1835-1851
Author(s):  
Vaibhav Shekhar ◽  
Chinmay Giri ◽  
Debasisha Mishra
Keyword(s):  
Linear Systems ◽  
Iterative Scheme ◽  
Iterative Solution ◽  
Differential Algebraic Equations ◽  
Convergence Theory ◽  
Algebraic Equations ◽  
Linear System Of Equations ◽  
Convergence Results ◽  
Comparison Results ◽  
Singular Linear System

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

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