Based on the recently developed abs algorithm for solving linear Diophantine equations, we present a special abs algorithm for solving such equations which is effective in computation and storage, not requiring the co...
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Based on the recently developed abs algorithm for solving linear Diophantine equations, we present a special abs algorithm for solving such equations which is effective in computation and storage, not requiring the computation of the greatest common divisor. A class of equations always solvable in integers is identified. Using this result, we discuss the ILP problem with upper and lower bounds on the variables.
We propose an approach for computing the general compromised solution of an LR fuzzy linear system by use of a ranking function when the coefficient matrix is a crisp m x n matrix. The solution is so that mean values ...
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We propose an approach for computing the general compromised solution of an LR fuzzy linear system by use of a ranking function when the coefficient matrix is a crisp m x n matrix. The solution is so that mean values of a compromised solution satisfies the corresponding crisp linear system. We show that if the corresponding crisp system is incompatible, then the fuzzy linear system lacks any solution. Otherwise, we solve a constrained least squares problem to compute a compromised solution. If the optimal value of the constrained least squares problem is zero, then we obtain the LR solution, namely the exact solution, of the system with respect to a ranking function. On the other hand, if the optimal value of the constrained least squares problem is nonzero, then no exact solution exists and thus we introduce and compute approximate (or weak) solutions of the system with respect to a ranking function. Also, when the ranking function is a member of a certain class of ranking functions, we propose a class of algorithms, based on abs class of algorithms, to compute the general compromised solution. (C) 2010 Elsevier Inc. All rights reserved.
In this paper we present results of testing codes for KKT linear systems based upon the abs algorithms versus classical codes (from packages absPACK and LAPACK). Classes of problems are found where codes from absPACK ...
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In this paper we present results of testing codes for KKT linear systems based upon the abs algorithms versus classical codes (from packages absPACK and LAPACK). Classes of problems are found where codes from absPACK perform significantly better in terms of speed and accuracy.
Using a strict bound of Spedicato to the condition number of bordered positive-definite matrices, we show that the scaling parameter in the abs class for linear systems can always be chosen so that the bound of a cert...
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Using a strict bound of Spedicato to the condition number of bordered positive-definite matrices, we show that the scaling parameter in the abs class for linear systems can always be chosen so that the bound of a certain update matrix is globally minimized. Moreover, if the scaling parameter is so chosen at every iteration, then the condition number itself is globally minimized. The resulting class of optimally conditioned algorithms contains as a special case the class of optimally stable algorithms in the sense of Broyden.
A method, called the (I.) abs-MPVT algorithm, for solving a system comprising linear equations and linear inequalities is presented. This method is characterized by solving the system of linear equations first via the...
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A method, called the (I.) abs-MPVT algorithm, for solving a system comprising linear equations and linear inequalities is presented. This method is characterized by solving the system of linear equations first via the abs algorithms and then solving an unconstrained minimization obtained by substituting the abs general form of solutions into the system of linear inequalities. For the unconstrained minimization problem it can be solved by a (modified) parallel algorithm. The convergence of this method is also given. (C) 2007 Elsevier Ltd. All rights reserved.
In this paper the abs algorithm with singular initial matrix is considered and used to construct an active set algorithm for solving the linear programming problem. We prove that this active set algorithm is reduced t...
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In this paper the abs algorithm with singular initial matrix is considered and used to construct an active set algorithm for solving the linear programming problem. We prove that this active set algorithm is reduced to the simplex algorithm when it starts from a vertex or it meets a vertex.
The method, called the Multi-Stage abs algorithm, for solving the over-determined linear inequalities system and the system combined with the over-determined linear inequalities and the equations is presented. This me...
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The method, called the Multi-Stage abs algorithm, for solving the over-determined linear inequalities system and the system combined with the over-determined linear inequalities and the equations is presented. This method is characterized by translating inequalities system to an equations system with slack variables. The explicit solution with the slack variables of the equations system are given by the implicit LU algorithm, then the slack variables can be given by the abs algorithm. Finally, the upper multiplications of the algorithm are given.
The general scheme of an algorithm, called an abs-FRE algorithm, for solving systems of fuzzy relation equations (systems of FRE) via the abs algorithms is presented. As special cases, two particular algorithms for ob...
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The general scheme of an algorithm, called an abs-FRE algorithm, for solving systems of fuzzy relation equations (systems of FRE) via the abs algorithms is presented. As special cases, two particular algorithms for obtaining the greatest and minimal solutions of systems of FRE are described. Several new operations used in this scheme are given, for instance, operators underscored logical OR sign and underscored logical AND sign, called quasi-inverses of operators [logical OR] and [logical AND] , respectively, etc.
The abs class for linear and nonlinear systems has been recently introduced by Abaffy, Broyden, Galantai and Spedicato. Here we consider various ways of applying these algorithms to the determination of the minimal eu...
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The abs class for linear and nonlinear systems has been recently introduced by Abaffy, Broyden, Galantai and Spedicato. Here we consider various ways of applying these algorithms to the determination of the minimal euclidean norm solution of over-determined linear systems in the least squares sense. Extensive numerical experiments show that the proposed algorithms are efficient and that one of them usually gives better accuracy than standard implementations of the QR orthogonalization algorithm with Householder reflections.
A modified abs algorithm for solving a class of singular nonlinear systems, F(x) = 0, F is a member of the set of Rn, constructed by combining the discreted abs algorithm and a method of Hoy and Schwetlick (1990), is ...
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A modified abs algorithm for solving a class of singular nonlinear systems, F(x) = 0, F is a member of the set of Rn, constructed by combining the discreted abs algorithm and a method of Hoy and Schwetlick (1990), is presented. The second differential operation of F at a point is not required to be calculated directly in this algorithm. Q-quadratic convergence of this algorithm is given.
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