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Real and Integer Extended Rank Reduction Formulas and Matrix Decompositions: A Review 3rd

Real and Integer Extended Rank Reduction Formulas and Matrix...

3rd International Conference on Numerical Analysis and Optimization - Theory, Methods, Applications and Technology Transfer ((NAO)

作者： Mahdavi-Amiri, Nezam Golpar-Raboky, Effat Sharif Univ Technol Fac Math Sci Tehran Iran Univ Qom Dept Math Qom Iran

ISBN: (纸本)9783319176895;9783319176888

We have recently developed an extended rank reducing process for rank reduction of a matrix leading to various matrix decompositions containing the Abaffy-Broyden-Spedicato (abs) and Wedderburn processes. Notably, the extended process contains both the Wedderburn biconjugation process and the scaled extended abs class of algorithms. The process provides a general finite iterative approach for constructing factorizations of a matrix and its transpose under a common framework of a general decomposition having various useful structures such as triangular, orthogonal, diagonal, banded and Hessenberg and many others. One main new result is the derivation of an extended rank reducing process for an integer matrix leading to the so-called Smith normal form. For this process, to solve the arising quadratic Diophantine equations, we have proposed two algorithms. Here, we report some numerical results on randomly generated test problems showing a better performance of one algorithm, based on a recent abs algorithm, in controlling the size of the solution. We also report results obtained by our algorithm on the Smith normal form having a more balanced distribution of the intermediate values as compared to the ones obtained by Maple.

关键词： Linear systems Matrix decomposition Wedderburn rank reduction abs algorithms Smith normal form Quadratic Diophantine equation

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New solutions of LR fuzzy linear systems using ranking functions and abs algorithms

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APPLIED MATHEMATICAL MODELLING 2010年第11期34卷 3363-3375页

作者： Ghanbari, Reza Mahdavi-Amiri, Nezam Sharif Univ Technol Fac Math Sci Tehran 113659415 Iran Ferdowsi Univ Mashhad Dept Math Mashhad Iran

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.

关键词： Fuzzy linear systems Fuzzy LR numbers Ranking functions abs algorithms Parametric solutions

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Approximate Nonnegative Symmetric Solution of Fully Fuzzy Systems Using Median Interval Defuzzification

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FUZZY INFORMATION AND ENGINEERING 2014年第3期6卷 331-358页

作者： Ezzati, R. Khezerloo, S. Mandavi-Amiri, N. Valizadeh, Z. Islamic Azad Univ Coll Basic Sci Karaj Branch Dept Math Alborz Iran Islamic Azad Univ South Tehran Branch Dept Math Tehran Iran Sharif Univ Technol Fac Math Sci Tehran *** Iran Roudehen Islamic Azad Univ Fac Basic Sci Dept Math Tehran Iran

We propose an approach for computing an approximate nonnegative symmetric solution of some fully fuzzy linear system of equations, where the components of the coefficient matrix and the right hand side vector are nonnegative fuzzy numbers, considering equality of the median intervals of the left and right hand sides of the system. We convert the m x n fully fuzzy linear system to two m x n real linear systems, one being related to the cores and the other being concerned with spreads of the solution. We propose an approach for solving the real systems using the modified Huang method of the Abaffy-Broyden-Spedicato (abs) class of algorithms. An appropriate constrained least squares problem is solved when the solution does not satisfy nonnegative fuzziness conditions, that is, when the obtained solution vector for the core system includes a negative component, or the solution of the spread system has at least one negative component, or there exists an index for which the component of the spread is greater than the corresponding component of the core. As a special case, we discuss fuzzy systems with the components of the coefficient matrix as real crisp numbers. We finally present two computational algorithms and illustrate their effectiveness by solving some randomly generated consistent as well as inconsistent systems.

关键词： Fully fuzzy systems Fuzzy numbers abs algorithms Median interval defuzzification

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A abs algorithm for solving singular nonlinear system with space transformation

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2009年第1-2期30卷 335-348页

作者： Ge Rendong Xia Zunquan Wang Jinzhi Dalian Natl Univ Sch Sci Dalian 116600 Peoples R China Dalian Univ Technol Dept Appl Math Dalian 116024 Peoples R China

A modified abs algorithm for solving a class of singular nonlinear systems, F (x) = 0, where x, F is an element of R-n, is presented. This method is constructed by combining the discreted Brown algorithm with the space transformation method. The second order information of F (x) at a point is not required calculating, which is different from the tensor method and the Hoy's method. The Q-quadratic convergence of this algorithm and some numerical examples are given as well.

关键词： System of nonlinear equations Singular system of linear equations abs algorithms Jacobian matrix

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abs algorithms FOR DIOPHANTINE LINEAR EQUATIONS AND INTEGER LP PROBLEMS

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2005年第1-2期17卷 93-107页

作者： Zou, Mei-Feng Xia, Zun-Quan Dalian Univ Technol Dalian Peoples R China Dalian Univ Technol Dept Appl Math Dalian 116024 Peoples R China

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.

关键词： abs algorithms Abaffian Diophantine equations Egervarian implicit LU algorithm implicit LX algorithm

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A method for solving the system of linear equations and linear inequalities

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MATHEMATICAL AND COMPUTER MODELLING 2007年第5-6期46卷 823-836页

作者： Pang, Li-Ping Spedicato, Emilio Xia, Zun-Quan Wang, Wei Dalian Univ Technol Dept Appl Math CORA Dalian 116024 Peoples R China Dalian Univ Technol Dept Phys Dalian 116024 Peoples R China Univ Bergamo Dept Math Stat Informat & Applicat I-24128 Bergamo Italy Liaoning Normal Univ Dept Math Dalian 116029 Peoples R China

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.

关键词： abs algorithms abaffian matrix system of linear equations system of linear inequalities parallel algorithm convex programming minimax nonlinear programming

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AN TWO PHASE abs METHOD FOR SOLVING OVER DETERMINED SYSTEMS OF LINEAR INEQUALITIES

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2006年第1-2期21卷 259-267页

作者： Guo, Qiang Liu, Jian-Guo Dalian Nationalities Univ Sch Sci Dalian 116600 Peoples R China Dalian Univ Technol Inst Syst Engn Dalian 116024 Peoples R China

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.

关键词： abs algorithms Abaffian matrix system of linear inequalities system of linear equations LU algorithm

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An abs-FRE algorithm for solving systems of fuzzy relation equations

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Journal of Applied Mathematics and Computing 2004年第1-2期15卷 285-297页

作者： Xia, Zun-Quan Guo, Fang-Fang Department of Applied Mathematics Laboratory 2 Dalian University of Technology Dalian 116024 China

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.

关键词： abs algorithms Fuzzy relation equations Fuzzy systems

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Computational experiments with abs algorithms for KKT linear systems

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OPTIMIZATION METHODS & SOFTWARE 2001年第1-4期16卷 85-99页

作者： Bodon, E Del Popolo, A Luksan, L Spedicato, E Univ Bergamo Dept Math I-24129 Bergamo Italy Acad Sci Czech Republ Inst Comp Sci Prague 18204 8 Czech Republic

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 ... 详细信息

关键词： abs algorithms KKT systems

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An abs algorithm for solving singular nonlinear systems with rank defects

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Journal of Applied Mathematics and Computing 2003年第1-2期12卷 1-20页

作者： Ge, Rendong Xia, Zun-Quan CORA Department of Applied Mathematics Dalian University of Technology Dalian 116024 China Department of Basic Science Nationality Institute of Dalian Dalian 116024 China

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.

关键词： abs algorithms Jacobian matrix Singular system of linear equations System of nonlinear equations

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