Uzawa Algorithms for Fully Fuzzy Linear Systems

Abstract

Recently, 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.

The Inverse and General Inverse of Trapezoidal Fuzzy Numbers with Modified Elementary Row Operations

“…This method is an extension of the triangular fuzzy number concept, which is developed into a trapezoidal fuzzy number, as shown in [1][2][3][4][5][6][7][8][9][10][11][12]. For further expansion, many authors also use approaches in the form of intervals, such as in [1,11,[13][14][15][16][17][18][19][20][21][22][23][24], but the general notation used for trapezoidal fuzzy numbers is ∼ u = (a, b, α, β). This form can also be changed into the parametric form ∼ u(r) = [u(r), u(r)].…”

“…Every fuzzy number and fuzzy matrix should have an inverse; if it does not have an inverse, this will have an impact on the solving of the trapezoidal fuzzy number linear system. Various authors, such as [1][2][3][4][5][6][7][8][9][10][11][12][13][14]20,[22][23][24][25][26][27][28][29][30]34,[37][38][39][40][41], have found alternative methods to solve it, which involve modifying various existing concepts in real matrices. This method was not able to show the existence of…”

“…As a result of the differences in arithmetic operations used by each author, different results are produced; this is shown in [9], which discusses the same problem addressed in [7,8] but obtains different results. The same conditions also occur when different methods and solutions are applied to trapezoidal fuzzy numbers, as discussed in [1][2][3][4][5][6][7][10][11][12]14,22,25,26,28,30,[41][42][43]. More specifically, as shown in [41], the standard fuzzy arithmetic operation performed by many authors on the basis of the extension principle does not take into account all of the available information, and the results obtained are inaccurate or, in some cases, wrong.…”

“…Although the solutions obtained using this method seem reasonable, they sometimes require long procedures and are computationally inefficient. However, the solution offered by [41] is an approach that is essentially no different from the approach given by [22,23]. In applications related to medical diagnoses [44][45][46], the steps commonly used by various authors are as follows.…”