scholarly journals Iterative Methods for Solving a System of Linear Equations in a Bipolar Fuzzy Environment

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 728 ◽  
Author(s):  
Muhammad Akram ◽  
Ghulam Muhammad ◽  
Ali N. A. Koam ◽  
Nawab Hussain
Keyword(s):  
Iterative Methods ◽  
Linear Equations ◽  
Jacobi Method ◽  
System Of Linear Equations ◽  
Linear System Of Equations ◽  
Sor Method ◽  
Fuzzy Environment ◽  
Fuzzy Linear System ◽  
Richardson Method ◽  
Successive Over Relaxation

We develop the solution procedures to solve the bipolar fuzzy linear system of equations (BFLSEs) with some iterative methods namely Richardson method, extrapolated Richardson (ER) method, Jacobi method, Jacobi over-relaxation (JOR) method, Gauss–Seidel (GS) method, extrapolated Gauss-Seidel (EGS) method and successive over-relaxation (SOR) method. Moreover, we discuss the properties of convergence of these iterative methods. By showing the validity of these methods, an example having exact solution is described. The numerical computation shows that the SOR method with ω = 1 . 25 is more accurate as compared to the other iterative methods.

Splitting iterative methods for fuzzy system of linear equations

2009 ◽  
Vol 20 (3) ◽  
pp. 326-335 ◽  
Author(s):  
Jun-Feng Yin ◽  
Ke Wang
Keyword(s):  
Iterative Methods ◽  
Fuzzy System ◽  
Linear Equations ◽  
System Of Linear Equations

scholarly journals A Novel Technique to Solve the Fuzzy System of Equations

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 850
Author(s):  
Nasser Mikaeilvand ◽  
Zahra Noeiaghdam ◽  
Samad Noeiaghdam ◽  
Juan J. Nieto
Keyword(s):  
Linear System ◽  
Fuzzy Number ◽  
Fuzzy System ◽  
Linear Equations ◽  
Second Step ◽  
Negative Solution ◽  
System Of Linear Equations ◽  
Vector Solution ◽  
Novel Technique ◽  
Fuzzy Linear System

The aim of this research is to apply a novel technique based on the embedding method to solve the n × n fuzzy system of linear equations (FSLEs). By using this method, the strong fuzzy number solutions of FSLEs can be obtained in two steps. In the first step, if the created n × n crisp linear system has a non-negative solution, the fuzzy linear system will have a fuzzy number vector solution that will be found in the second step by solving another created n × n crisp linear system. Several theorems have been proved to show that the number of operations by the presented method are less than the number of operations by Friedman and Ezzati’s methods. To show the advantages of this scheme, two applicable algorithms and flowcharts are presented and several numerical examples are solved by applying them. Furthermore, some graphs of the obtained results are demonstrated that show the solutions are fuzzy number vectors.

Certain efficient iterative methods for bipolar fuzzy system of linear equations

2020 ◽  
Vol 39 (3) ◽  
pp. 3971-3985 ◽  
Author(s):  
Muhammad Saqib ◽  
Muhammad Akram ◽  
Shahida Bashir
Keyword(s):  
Approximate Solution ◽  
Linear System ◽  
Iterative Methods ◽  
Fuzzy Set ◽  
Fuzzy System ◽  
Linear Equations ◽  
Performance Validity ◽  
System Of Linear Equations ◽  
Numerical Examples ◽  
Iterative Schemes

A bipolar fuzzy set model is an extension of fuzzy set model. We develop new iterative methods: generalized Jacobi, generalized Gauss-Seidel, refined Jacobi, refined Gauss-seidel, refined generalized Jacobi and refined generalized Gauss-seidel methods, for solving bipolar fuzzy system of linear equations(BFSLEs). We decompose n ×  n BFSLEs into 4n ×  4n symmetric crisp linear system. We present some results that give the convergence of proposed iterative methods. We solve some BFSLEs to check the validity, efficiency and stability of our proposed iterative schemes. Further, we compute Hausdorff distance between the exact solutions and approximate solution of our proposed schemes. The numerical examples show that some proposed methods converge for the BFSLEs, but Jacobi and Gauss-seidel iterative methods diverge for BFSLEs. Finally, comparison tables show the performance, validity and efficiency of our proposed iterative methods for BFSLEs.

scholarly journals Convergence of relaxed chaotic parallel iterative methods

1994 ◽  
Vol 50 (1) ◽  
pp. 167-176 ◽  
Author(s):  
Peter E. Kloeden ◽  
Dong-Jin Yuan
Keyword(s):  
Iterative Methods ◽  
Large Scale ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
System Of Linear Equations ◽  
Parallel Iterative Methods ◽  
Parallel Iterative

Sufficient conditions involving uniform multisplittings are established for the convergence of relaxed and AOR versions of asynchronous or chaotic parallel iterative methods for solving a large scale nonsingular system of linear equations Ax = b.

FUZZY CENTRE BASED SOLUTION OF FUZZY COMPLEX LINEAR SYSTEM OF EQUATIONS

2013 ◽  
Vol 21 (04) ◽  
pp. 629-642 ◽  
Author(s):  
DIPTIRANJAN BEHERA ◽  
S. CHAKRAVERTY
Keyword(s):  
Linear Time ◽  
Linear Equations ◽  
Electric Circuit ◽  
System Of Linear Equations ◽  
New Approach ◽  
Linear System Of Equations ◽  
Linear Time Invariant ◽  
Complex Linear ◽  
Time Invariant ◽  
Good Agreement

A new approach to solve Fuzzy Complex System of Linear Equations (FCSLE) based on fuzzy complex centre procedure is presented here. Few theorems related to the investigation are stated and proved. Finally the presented procedure is used to analyze an example problem of linear time invariant electric circuit with complex crisp coefficient and fuzzy complex sources. The results obtained are also compared with the known solutions and are found to be in good agreement.

scholarly journals Analysis of the Production Task Model With Fuzzy Information About Direct Cost Factors and the Final Product Demand

2019 ◽  
Vol 09 (4) ◽  
pp. 32-45 ◽  
Author(s):  
A.V. Panteleev ◽  
V.S. Saveleva
Keyword(s):  
Strong Solution ◽  
Direct Cost ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Production Task ◽  
Fuzzy Information ◽  
System Of Linear Equations ◽  
Fuzzy Matrix ◽  
Linear System Of Equations ◽  
The Matrix

The article discusses the study of a mathematical model of execution of the production task in the presence of fuzzy information about the matrixes of direct costs and final demand. By solving a problem with fuzzy information we mean the solution of a linear system of equations with a fuzzy matrix and a fuzzy right-hand side described by fuzzy triangular numbers in a form of deviations from the mean. In this task of search of inter-sectoral balance the LU-decomposition method for the matrix of direct cost which is further used for solving the system of linear equations is applied. A software implementation of a numerical method for finding a strong solution of a fuzzy system of linear equations consisting of two successive stages is described. At the first stage, the necessary and sufficient conditions for the existence of a strong solution are verified. At the second stage, the solution of the system is found, which is written in the form of a fuzzy matrix. The influence of the fuzzy numbers parameters on the final result was studied.

scholarly journals Refinement of Iterative Methods for the Solution of System of Linear Equations Ax=b

2014 ◽  
Vol 10 (3) ◽  
pp. 70-73
Author(s):  
Anamul Haque Laskar ◽  
◽  
Samira Behera
Keyword(s):  
Iterative Methods ◽  
Linear Equations ◽  
System Of Linear Equations

Modified Iterative Methods for Solving Fully Fuzzy Linear Systems

Fuzzy Systems ◽  
2017 ◽  
pp. 55-73
Author(s):  
S. A. Edalatpanah
Keyword(s):  
Linear Systems ◽  
Iterative Methods ◽  
Fuzzy System ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Numerical Procedure ◽  
Numerical Results ◽  
System Of Linear Equations ◽  
Iterative Schemes ◽  
Fuzzy Linear Systems

In the present chapter, we give an overview of computational iterative schemes for fuzzy system of linear equations. We also consider fully fuzzy linear systems (FFLS) and demonstrate a class of the existing iterative methods using the splitting approach for calculating the solution. Furthermore, the main aim in this work is to design a numerical procedure for improving this algorithm. Some numerical experiments are illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm, we report the numerical results of large-scaled fuzzy problems.

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