Elementary fuzzy calculus

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.

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“…Gauss-Seidel algorithm uses the decompositions (26) and (27) to transform the linear equations (7) and (8) into iterative methods of the forms…”

Section: Uzawa-gauss-seidel (Ugs) Methodsmentioning

confidence: 99%

Uzawa Algorithms for Fully Fuzzy Linear Systems

Zareamoghaddam

Chronopoulos

Kadijani

et al. 2016

IJCIS

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“…Goetschel and Voxman in [37] proved that if the fuzzy function ϕ (x) is continuous in the metric D, its definite integral exists and also,…”

Section: Preliminariesmentioning

confidence: 99%

“…The topic of fuzzy integration was first discussed in [61]. Particularly, it has been studied by Kaleva [41], Goetschel and Voxman [37], Nanda [47], Ralescu and Adams [49], Wang [59], Bede and Gal [16] and others [32,33]. The fuzzy Riemann integral and its numerical integration was investigated by Wu in [60].…”

Section: Introductionmentioning

confidence: 99%

Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

Noeiaghdam¹,

Araghi²

2017

Journal of Fuzzy Set Valued Analysis

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The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By using this method, the optimal number of points in the fuzzy numerical integration rules and the optimal approximate solution are obtained. Also, the accuracy of the fuzzy quadrature rules are discussed. An algorithm is given to illustrate the implementation of the method. In this case, the termination criterion is considered as the Hausdorff distance between two sequential results to be an informatical zero. Two sample fuzzy integrals are evaluated based on the proposed algorithm to show the importance and advantage of using the stochastic arithmetic in place of the floating-point arithmetic.

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“…FN jest to podzbiór rozmyty ∈ ℱ(ℝ) o ograniczonym domknięciu nośnika spełniający dodatkowo warunek [Goetschel, Voxman, 1986]:…”

Section: Rysunek 1 Behawioralna Wartość Bieżąca (Bpv)unclassified

O pewnych modyfikacjach teorii skierowanych liczb rozmytych

Piasecki

2017

Optimum

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StreszczenieSkierowane liczby rozmyte zostały zdefiniowane w doskonały i intuicyjny sposób przez Witolda Kosińskiego. Z tej przyczyny skierowane liczby rozmyte coraz częściej określa się mianem liczb Kosińskie-go. W pierwszej części tej pracy zaproponowano w pełni sformalizowaną definicję liczby Kosińskiego. Definicję tę następnie uogólniono do przypadku skierowanej liczby rozmytej z nieciągłą funkcją przynależności. Istotną wadą arytmetyki zaproponowanej przez Kosińskiego był brak zamknięcia przestrzeni skierowanych liczb rozmytych ze względu na podstawowe działania arytmetyczne, takie jak: dodawanie, odejmowanie, mnożenie i dzielenie. Głównym celem prezentowanej pracy jest taka modyfikacja działań arytmetycznych, aby przestrzeń liczb Kosińskiego była zamknięta z racji zmodyfikowanych działań arytmetycznych. Słowa kluczowe: skierowane liczby rozmyte, arytmetyka ON CERTAIN MODIFICATIONS OF ORDERED FUZZY NUMBERS THEORY SummaryOrdered fuzzy numbers have been defined in an excellent, intuitive way by Witold Kosiński. For this reason, they are increasingly referred to as Kosiński's numbers. A fully formalized definition of a Kosiński's number is proposed in the first part of this work. This definition is generalized so as to fit an ordered fuzzy number with an upper semi-continuous membership function. A significant drawback of Kosiński's arithmetic is that the space of ordered fuzzy numbers is not closed under addition, subtraction, multiplication, or division. The main aim of this paper is to modify the arithmetic in such a way that the space of ordered fuzzy numbers is closed under the modified arithmetic operations.

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Elementary fuzzy calculus

Cited by 1,129 publications

References 4 publications

Uzawa Algorithms for Fully Fuzzy Linear Systems

Uzawa Algorithms for Fully Fuzzy Linear Systems

Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

O pewnych modyfikacjach teorii skierowanych liczb rozmytych

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