Uzawa Algorithms for Fully Fuzzy Linear Systems

Z Zareamoghaddam,H Zareamoghaddam,A.T Chronopoulos,M Nouri Kadijani

doi:10.1080/18756891.2016.1237194

Abstract

AbstractRecently, there have been many studies on solving different kinds of fuzzy equations. In this paper, the solution of a trapezoidal fully fuzzy linear system (FFLS) is studied. Uzawa approach, which is a popular iterative technique for saddle point problems, is considered for solving such FFLSs. In our Uzawa approach, it is possible to compute the solution of a fuzzy system using various relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR as well as Krylov subspace methods such as GMRES, QMR and BiCGSTAB. Krylov subspace iterative methods are known to converge for a larger class of matrices than relaxation iterative methods and they exhibit higher convergence rates. Thus, they are more widely used in practical problems. Numerical experiments are to illustrate the performance of our suggested methods.

Highlights

IntroductionResearchers in numerical computing have developed relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR
Linear system of equations Ax = b, where A and b are respectively a crisp square matrix and vector, mostly comes from the discretization of some practical partial differential equations
conjugate gradient (CG) is applicable for symmetric positive definite matrices and it is proven to converge in a finite number of steps and in general at higher convergence rates than the relaxation iterative methods

Summary

Introduction

Researchers in numerical computing have developed relaxation iterative methods such as Richardson, Jacobi, Gauss-Seidel, SOR, SSOR Some of these methods are applicable to only to diagonally dominant and others for symmetric positive definite matrices. Friedman et al Ref. 10 studied a general form of fuzzy linear system of equations (FLSE) with a crisp coefficient matrix and an arbitrary right hand side fuzzy vector. We propose practical implementations, which solve the solution of a trapezoidal FFLSs based on Uzawa approach. The extrapolation scalar τ is a real number which is named as Uzawa parameter This papers aim is to compute the solution of a trapezoidal FFLS by the above Uzawa instruction based on several different appropriate iterative methods. Our proposed algorithms are illustrated by solving some numerical examples in Sections 4 and 5

Preliminaries

Uzawa algorithm

Uzawa Iterative methods for FFLSs

Uzawa algorithms for large sparse FFLSs

Numerical Experiments

Conclusion