Iterative solution of fuzzy linear systems

Dehghan, Mehdi; Hashemi, Behnam

doi:10.1016/j.amc.2005.07.033

Cited by 95 publications

(63 citation statements)

References 11 publications

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“…There are several well-known point iterative methods and block numerical iterative methods for FLS such as Jacobi, Gauss-Seidel, SOR, and AOR; see [9][10][11][12][13][14]. As a matter of fact, these methods are generalization of iterative methods for crisp linear systems = .…”

Section: The Mixed Type Splitting Iterative Methods For Fuzzy Linear mentioning

confidence: 99%

“…As a matter of fact, these methods are generalization of iterative methods for crisp linear systems = . For instance, in AOR method for FLS [12] we have…”

Section: The Mixed Type Splitting Iterative Methods For Fuzzy Linear mentioning

confidence: 99%

“…In [1,2] Kandel et al applied the embedding method for fuzzy linear system (hereafter denoted by FLS) and replaced the FLS by a 2 × 2 crisp linear system. This model has been modified later by some other researchers; see [10][11][12][13][14][15][16] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

Najafi

Edalatpanah

Shahabi

2014

Applied Computational Intelligence and Soft Computing

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We consider a class of fuzzy linear systems (FLS) and demonstrate some of the existing methods using the embedding approach for calculating the solution. The main aim in this paper is to design a class of mixed type splitting iterative methods for solving FLS. Furthermore, convergence analysis of the method is proved. Numerical example is illustrated to show the applicability of the methods and to show the efficiency of proposed algorithm.

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Section: The Mixed Type Splitting Iterative Methods For Fuzzy Linear mentioning

confidence: 99%

“…As a matter of fact, these methods are generalization of iterative methods for crisp linear systems = . For instance, in AOR method for FLS [12] we have…”

Section: The Mixed Type Splitting Iterative Methods For Fuzzy Linear mentioning

confidence: 99%

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The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

Najafi

Edalatpanah

Shahabi

2014

Applied Computational Intelligence and Soft Computing

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“…Allahviranloo [11][12][13] uses the iterative Jacobi and Gauss Siedel method, the Adomian method and the Successive over-relaxation method, respectively. Dehghan and Hashemi [14,20] extended several well-known numerical algorithms such as Richardson, Extrapolated Richardson, Jacobi, JOR, Gauss-Seidel, EGS, SOR, AOR, ESOR, SSOR, USSOR, EMA and MSOR to solve system of linear equations. Ma et al [15], starting from the work by Friedman et al [10], analyzed the solution of fuzzy systems of the form A 1 x = A 2 x + b.…”

Section: Related Workmentioning

confidence: 99%

Fuzzy Complex System of Linear EquationsApplied to Circuit Analysis

Rahgooy¹,

Yazdi²,

Monsefi³

2009

IJCEE

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I. INTRODUCTIONWide range of real world applications in many areas including economics, engineering and physics, are using linear system of equations for modeling and solving their respective problems. In many situations the estimation of the system parameters is imprecise and only some vague knowledge about the actual value of parameters is available, therefore to overcome this problem the vagueness of parameters is represented by fuzzy numbers. In many applications some of the parameters are usually represented by complex numbers which have fuzzy nature, thus it is important to develop mathematical models that would appropriately treat fuzzy complex linear systems. Circuits can be modeled in the form of fuzzy complex system of linear equations (FCSLE).A rapid growth of interest in circuit theory and simulation is seen in recent decade. A circuit simulation program (e.g., SPICES [1] and QPMDFSD [2, 3, and 4]) provides very good details in simulation of circuits. These types of software provide various facilities for obtaining best answer which is close to real form. Some contributions have been made in stability theory using functional analysis in the study of nonlinear systems, and computer-aided nonlinear network analysis [5, 6, 7, and 8]. But uncertainty in circuit parameters and environmental conditions lead to develop a new method which takes in to account uncertainty in circuit analysis. In the previous work In the aforementioned works, real coefficients are discussed but in many applications as circuit analysis, while SLE includes complex coefficients and complex variables. Also we encountered situations in circuit solving that needs solving a FSLE model which has been solved by Friedman et al. [10]. Therefore we present a Fuzzy Complex SLE, namely, FCSLE in this paper. This model works well in circuit solving. Major notes in this paper are presentation of FCSLE, and application of FSLE and FCSLE in circuit solving. Organization of this paper is as follows: Fuzzy Complex System of Linear EquationsApplied to Circuit Analysis Taher Rahgooy, Hadi Sadoghi Yazdi, Reza MonsefiInternational Journal of Computer and Electrical Engineering, Vol. 1, No. 5 December, 2009 1793-8163 536 FSLE is explained in section III. Section IV appropriate to the proposed FCSLE and experimental results over circuit analysis are discussed in the section V. Final section includes the conclusion. III. FUZZY SYSTEM OF LINEAR EQUATIONS A. PreliminariesWe represented an arbitrary fuzzy number by an ordered pair of functions which should satisfy the following requirements:1.is a bounded left continuous nondecreasing function over [0,1]. 2.is a bounded left continuous nonincreasing function over [0,1]. 3. Fig. 1 A fuzzy numberFor example, the fuzzy number (1 + r, 4 -2r) is shown in Fig. 1. A crisp number α is simply represented by By appropriate definitions the fuzzy number space becomes a convex cone E 1 which is then embedded isomorphically and isometrically into a Banach space. Definition 1. The n × n linear system of equations Cons...

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“…Numerical methods for SFLE were proposed by Allahviranloo [16,17]. Dehghan and Hashemi [10] applied several iterative methods for solving SFLE. Wang and Zheng [8] studied some block iterative methods to solve SFLE.…”

Section: Introductionmentioning

confidence: 99%

Iterative Methods for Solving System of Fuzzy Sylvester Equations

Rivaz¹,

Abad²

2013

Int. J. of Appl. Math.

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Abstract:In this paper, we present Jacobi and Adomian decomposition iterative methods for finding the approximate solution of system of fuzzy Sylvester equations (SFSE), AX + XB = C, where A and B are two m × m crisp matrices, C is an m × m fuzzy matrix and X is an m × m unknown matrix. It is shown that for a SFSE the Adomian decomposition method is equivalent to the Jacobi iterative method. Finally, the proposed iterative methods are illustrated by solving one example.

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Iterative solution of fuzzy linear systems

Cited by 95 publications

References 11 publications

The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

The Mixed Type Splitting Methods for Solving Fuzzy Linear Systems

Fuzzy Complex System of Linear EquationsApplied to Circuit Analysis

Iterative Methods for Solving System of Fuzzy Sylvester Equations

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