Solvability criteria for systems of fuzzy relation equations

By solving systems of fuzzy relation equations, qualitative process models can be obtained. To give more information on the solving procedure and to help constructing models, solvability criteria for-systems of fuzzy relation equations are necessary. In this article such criteria will be developed. Both methods are considered. In addition to some ideas on general , the is evaluated in detail. Criteria of practical use will be developed. These criteria will limit the variety of premise intersections to guarantee solvability. Nevertheless, they will still allow to model the significant behaviour of the processes.     

An algorithm for solving Fredholm integro-differential equations

The aim of this paper is to present an efficient numerical procedure for solving linear second order Fredholm integro-differential equations. The scheme is based on B-spline collocation and cubature formulas. The analysis is accompanied by numerical examples. The results demonstrate reliability and efficiency of the proposed algorithm.      

An algorithm for solving linear Volterra integro-differential equations

An efficient numerical procedure for solving linear second order Volterra integro-differential equations is presented herein. The scheme is based on B-spline collocation and cubature formulas. Analysis is accompanied by numerical examples. Results confirm reliability and efficiency of the proposed algorithm.     

A relaxed cutting plane algorithm for solving fuzzy inequality systems

This paper studies a system of infinitely many fuzzy inequalities with concavemembership functions. By using the tolerance approach, we show that solving such system can be reduced to a semi-infinite programming problem. A relaxed cutting plane algorithm is proposed. In each iteration, we solve a finite convex optimization problem and add one or two more constraints. The proposed algorithm chooses a point at which the infinite constraints are violated to a degree rather than at which the violation is maximized. The iterative process ends when an optimal solution is identified. A convergence proof, under some mild conditions, is given. An efficient implementation based on the "method of centres" with "entropic regularization" techniques is also included. Some computational results confirm the efficiency of the proposed method and show its potential for solving large scale problems.     

A general projection algorithm for solving systems of linear equations

In this paper, we give a general projection algorithm for implementing some known extrapolation methods such as the MPE, the RRE, the MMPE and others. We apply this algorithm to vectors generated linearly and derive new algorithms for solving systems of linear equations. We will show that these algorithms allow us to obtain known projection methods such as the Orthodir or the GCR.     

A Monte Carlo algorithm for solving systems of non-linear equations

A Monte Carlo method for solving systems of non-linear equations is presented and discussed. The method does not require the differentiation of left side functions of the system and provides the solution with arbitrary accuracy, although it may be applied only to systems with not too large a number of equations. Illustrative examples are given.     

An effective variational iteration algorithm for solving Riccati differential equations

The piecewise variational iteration method (VIM) for solving Riccati differential equations (RDEs) provides a solution as a sequence of iterates. Therefore, its application to RDEs leads to the calculation of terms that are not needed and more time is consumed in repeated calculations for series solutions. In order to overcome these shortcomings, we propose an easy-to-use piecewise-truncated VIM algorithm for solving the RDEs. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the classical fourth-order Runge–Kutta method (RK4) verify that the new method is very effective and convenient for solving Riccati differential equations.     

An iterative method for solving systems of linear algebraic equations

An iterative solution process for systems of linear algebraic equations is proposed. It converges starting from any initial approximation and theoretically does not require preliminary transformation of the input data.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 64–68, 1985.     

An efficient method for solving difference systems for elliptic differential equations

A method for solving systems of linear algebraic equations arising in connection with the approximation of boundary value problems for elliptic partial differential equations is proposed. This method belongs to the class of conjugate directions method applied to a preliminary transformed system of equations. A model example is used to explain the idea underlying this method and to investigate it. Results of numerical experiments that confirm the method’s efficiency are discussed.     

Optimization of the quasi-Monte Carlo algorithm for solving systems of linear algebraic equations

An interesting conclusion about error reduction of the modified quasi-Monte Carlo method for solving systems of linear algebraic equations is suggested. The Monte Carlo method is compared with the quasi-Monte Carlo method and its modification. The optimal choice of the parameters of the Markov chain for the modified Monte Carlo method applied to solving systems of linear equations is substantiated.     

A CLASS OF FACTORIZATION UPDATE ALGORITHM FOR SOLVING SYSTEMS OF SPARSE NONLINEAR EQUATIONS

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A bidiagonalization algorithm for solving large and sparse ill-posed systems of linear equations

An iterative method based on Lanczos bidiagonalization is developed for computing regularized solutions of large and sparse linear systems, which arise from discretizations of ill-posed problems in partial differential or integral equations. Determination of the regularization parameter and termination criteria are discussed. Comments are given on the computational implementation of the algorithm.Dedicated to Peter Naur on the occasion of his 60th birthday     

An identification algorithm in fuzzy relational systems

The paper presents an effective identification method in fuzzy relational systems. We propose an algorithm for constructing models on the basis of fuzzy and nonfuzzy data with the aid of fuzzy discretization and clustering techniques. The usefulness of the method provided is demonstrated by means of two numerical examples. Also a possible way of generating a linguistic decision-making algorithm is discussed.     

Hamiltonian algorithm for solving singularly perturbed equations

Translated from Programmnoe Oborudovanie i Voprosy Prinyatiya Reshenii, pp. 224–237, 1989.     

Globally convergent algorithm for solving nonlinear equations

A general iterative method is proposed for finding the maximal rootx _max of a one-variable equation in a given interval. The method generates a monotone-decreasing sequence of points converging tox _max or demonstrates the nonexistence of a real root. It is globally convergent. A concrete realization of the general algorithm is also given and is shown to be locally quadratically convergent. Computational experience obtained for eight test problems indicates that the new method is comparable to known methods claiming global convergence.     

Application of fuzzy differential transform method for solving fuzzy Volterra integral equations

This paper deals with the solutions of fuzzy Volterra integral equations with separable kernel by using fuzzy differential transform method (FDTM). If the equation considered has a solution in terms of the series expansion of known functions, this powerful method catches the exact solution. To this end, we have obtained several new results to solve mentioned problem when FDTM has been applied. In order to show this capability and robustness, some fuzzy Volterra integral equations are solved in detail as numerical examples.