A fuzzy programming method for deriving priorities in the analytic hierarchy process

The estimation of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). The priority vector can be derived from these matrices using different techniques, as the most commonly used are the Eigenvector Method (EVM) and the Logarithmic Least Squares Method (LLSM). In this paper a new Fuzzy Programming Method (FPM) is proposed, based on geometrical representation of the prioritisation process. This method transforms the prioritisation problem into a fuzzy programming problem that can easily be solved as a standard linear programme. The FPM is compared with the main existing prioritisation methods in order to evaluate its performance. It is shown that it possesses some attractive properties and could be used as an alternative to the known prioritisation methods, especially when the preferences of the decision-maker are strongly inconsistent.     

Interval estimation of priorities in the AHP

This paper extends and modifies the Analytic Hierarchy Process (AHP) and the Synthetic Hierarchy Method (SHM) of priority estimation to accommodate random data in the pairwise comparison matrices. It employs a Cauchy distribution to describe the pairwise comparison of alternatives in Saaty matrices, and shows how to modify these matrices in order to handle random data. The use of random data yields Saaty matrices that are not reciprocally symmetrical. Several variants of the AHP are then modified (i) to accommodate reciprocally asymmetric matrices, and (ii) to allow each priority estimate to be expressed on an interval of possible values, rather than as a single discrete point. The merits of interval estimation are illustrated by an example.     

On the properties of matrices defining some classes of BVMs

Summary It is known that the matrices defining the discrete problem generated by a k-step Boundary Value Method (BVM) have a quasi-Toeplitz band structure [7]. In particular, when the boundary conditions are skipped, they become Toeplitz matrices. In this paper, by introducing a characterization of positive definiteness for such matrices, we shall prove that the Toeplitz matrices which arise when using the methods in the classes of BVMs known as Generalized BDF and Top Order Methods have such property. Mathematics Subject Classification (2000):65L06, 47B35, 15A48Work supported by G.N.C.S.     

On the quadratic convergence of a generalization of the Jacobi Method to arbitrary matrices

A method of diagonalizing a general matrix is proved to be ultimately quadratically convergent for all normalizable matrices. The method is a slight modification of a method due to P. J. Eberlein, and it brings the general matrix into a normal one by a combination of unitary plane transformations and plane shears (non-unitary). The method is a generalization of the Jacobi Method: in the case of normal matrices it is equivalent to the method given by Goldstine and Horwitz.     

A note on refining eigenelements of symmetric matrices

By means of a Fixed Slope Inexact Newton Method we define an Iterative Refinement Process for approximating eigenvalues and eigenvectors of real symmetric matrices.     

Efficient MFS algorithms in regular polygonal domains

In this work, we apply the Method of Fundamental Solutions (MFS) to harmonic and biharmonic problems in regular polygonal domains. The matrices resulting from the MFS discretization possess a block circulant structure. This structure is exploited to produce efficient Fast Fourier Transform–based Matrix Decomposition Algorithms for the solution of these problems. The proposed algorithms are tested numerically on several examples.      

Solution of the least squares method problem of pairwise comparison matrices

The aim of the paper is to present a new global optimization method for determining all the optima of the Least Squares Method (LSM) problem of pairwise comparison matrices. Such matrices are used, e.g., in the Analytic Hierarchy Process (AHP). Unlike some other distance minimizing methods, LSM is usually hard to solve because of the corresponding nonlinear and non-convex objective function. It is found that the optimization problem can be reduced to solve a system of polynomial equations. Homotopy method is applied which is an efficient technique for solving nonlinear systems. The paper ends by two numerical example having multiple global and local minima. This research was supported in part by the Hungarian Scientific Research Fund, Grant No. OTKA K 60480.     

Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains

In this work, we propose an efficient matrix decomposition algorithm for the Method of Fundamental Solutions when applied to three-dimensional boundary value problems governed by elliptic systems of partial differential equations. In particular, we consider problems arising in linear elasticity in axisymmetric domains. The proposed algorithm exploits the block circulant structure of the coefficient matrices and makes use of fast Fourier transforms. The algorithm is also applied to problems in thermo-elasticity. Several numerical experiments are carried out.     

The heteroscedastic method: Multivariate implementation

Summary Over the past decade, procedures have been developed which allow one (in the univariate case) to make inferences about means even in the presence of unknown and unequal variances. A general method (called The Heteroscedastic Method) allowing this in all statistical problems simultaneously was formulated in 1979 and allowed specifically for the multivariate case (e.g., MANOVA and other multivariate inferences). While in the univariate case The Heteroscedastic Method is readily implemented, in the multivariate case practical implementation was not heretofore possible since a certain problem in construction of matrices required by the method had not been solved. In this paper we solve that problem and give a computer algorithm allowing for use of the solution in The Heteroscedastic Method.     

Panel clustering method and restriction matrices for symmetric Galerkin BEM

In the framework of Symmetric Galerkin Boundary Element Method, different techniques in these last years have been proposed to reduce the computational cost of the Galerkin matrix evaluation: in particular, the Panel Clustering Method [25,26] it is now largely used. On the other side, very recently a theory on restriction matrices has been developed to take computational advantage of possible symmetry properties of the differential or integral problem at hand [4,5]. Here we couple Panel Clustering Method with restriction matrices, presenting the most important algorithms employed and showing several examples, comparisons and numerical results. AMS subject classification 65F30, 65N38     

Implementation of exponential Rosenbrock-type integrators

In this paper, we present a variable step size implementation of exponential Rosenbrock-type methods of orders 2, 3 and 4. These integrators require the evaluation of exponential and related functions of the Jacobian matrix. To this aim, the Real Leja Points Method is used. It is shown that the properties of this method combine well with the particular requirements of Rosenbrock-type integrators. We verify our implementation with some numerical experiments in MATLAB, where we solve semilinear parabolic PDEs in one and two space dimensions. We further present some numerical experiments in FORTRAN, where we compare our method with other methods from literature. We find a great potential of our method for non-normal matrices. Such matrices typically arise in parabolic problems with large advection in combination with moderate diffusion and mildly stiff reactions.     

关于用拉格朗日乘子法求解线性等式约束最小二乘问题

本文讨论用拉格朗日乘子法求解线性等式约束最小二乘问题(简称 LSE 问题)的优点.应用此法能细致地讨论约束条件与变量之间的关系,据此并可证明 LSE 问题与某一个无约束最小二乘问题的等价性.此外,尚可得到参数和拉格朗日乘子的协方差矩阵.最后给出一个数值稳定的解 LSE 问题的算法.     

First Order LSFEM for Fluid‐Structure Problems

The Least‐Squares Finite Element Method (LSFEM) is an interesting alternative to the standard variational principles, which are used to solve partial differential equations. Advantages of the LSFEM are its robustness and the resulting symmetric positive definite matrices, which allow the use of robust iterative solvers like the CG method. In this paper we consider the application of the LSFEM for Fluid‐Structure Interaction (FSI) problems. Our model uses the LSFEM for the discretisation of the instationary incompressible Navier‐Stokes equations, which is coupled with a standard Galerkin FEM model for a linear elastic structure. The results for a simple model problem agree well with results obtained by other authors with different numerical schemes. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inversion of polynomial and rational matrices

The inversion of polynomial and rational matrices is considered. For regular matrices, three algorithms for computing the inverse matrix in a factored form are proposed. For singular matrices, algorithms of constructing pseudoinverse matrices are considered. The algorithms of inversion of rational matrices are based on the minimal factorization which reduces the problem to the inversion of polynomial matrices. A class of special polynomial matrices is regarded whose inverse matrices are also polynomial matrices. Inversion algorithms are applied to the solution of systems with polynomial and rational matrices. Bibliography: 3 titles. Translated by V. N. Kublanovskaya. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 97–109.     

Restriction matrices for numerically exploiting symmetry

In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group of congruences of . This technique will be based upon suitable restriction matrices strictly related to a system of irreducible matrix representation of . Both Abelian and non-Abelian finite groups are considered. In the framework of symmetric Galerkin boundary element method (SGBEM), where the discretization matrices are typically full, to increase the computational gain we couple Panel Clustering Method [30] and Adaptive Cross Approximation algorithm [13] with restriction matrices introduced in this paper, showing some numerical examples. Applications of restriction matrices to SGBEM under the weaker assumption of partial geometrical symmetry, where the boundary has disconnected components, one of which is invariant, are proposed. The paper concludes with several numerical tests to demonstrate the effectiveness of the introduced technique in the numerical resolution of Dirichlet or Neumann invariant BVPs, in their differential or integral formulation.      

Roundoff error analysis of algorithms based on Krylov subspace methods

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices. Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method. These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces of small dimension. The roundoff error analyses show upper bounds for the error affecting the computed approximated solutions.This work was carried out with the financial contribution of the Human Capital and Mobility Programme of the European Union grant ERB4050PL921378.     

Semidefinite programming approach for the quadratic assignment problem with a sparse graph

The matching problem between two adjacency matrices can be formulated as the NP-hard quadratic assignment problem (QAP). Previous work on semidefinite programming (SDP) relaxations to the QAP have produced solutions that are often tight in practice, but such SDPs typically scale badly, involving matrix variables of dimension \(n^2\) where n is the number of nodes. To achieve a speed up, we propose a further relaxation of the SDP involving a number of positive semidefinite matrices of dimension \(\mathcal {O}(n)\) no greater than the number of edges in one of the graphs. The relaxation can be further strengthened by considering cliques in the graph, instead of edges. The dual problem of this novel relaxation has a natural three-block structure that can be solved via a convergent Alternating Direction Method of Multipliers in a distributed manner, where the most expensive step per iteration is computing the eigendecomposition of matrices of dimension \(\mathcal {O}(n)\). The new SDP relaxation produces strong bounds on quadratic assignment problems where one of the graphs is sparse with reduced computational complexity and running times, and can be used in the context of nuclear magnetic resonance spectroscopy to tackle the assignment problem.     

Model order reduction for thermomechanical problems including radiation

Thermal field problems including heat exchange by radiation lead to nonlinear system equations with a high number of inputs and outputs as radiation heat fluxes correspond to the fourth power of the temperature and thermal loads are distributed over the whole surface. In an alternative approach presented here, radiation is defined as a part of the load vector. Thus, the system matrices are constant. Furthermore, loads changing synchronously during operation are grouped into one column of the input matrix and load vector snapshots are used to consider the radiation heat fluxes. Hence, the Krylov Subspace Method can be applied to significantly reduce the system dimension and the computation times allowing transient thermal parameter studies. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eigenvalue perturbation theory of symplectic,orthogonal, and unitary matrices under generic structured rank one perturbations

We study the perturbation theory of structured matrices under structured rank one perturbations, with emphasis on matrices that are unitary, orthogonal, or symplectic with respect to an indefinite inner product. The rank one perturbations are not necessarily of arbitrary small size (in the sense of norm). In the case of sesquilinear forms, results on selfadjoint matrices can be applied to unitary matrices by using the Cayley transformation, but in the case of real or complex symmetric or skew-symmetric bilinear forms additional considerations are necessary. For complex symplectic matrices, it turns out that generically (with respect to the perturbations) the behavior of the Jordan form of the perturbed matrix follows the pattern established earlier for unstructured matrices and their unstructured perturbations, provided the specific properties of the Jordan form of complex symplectic matrices are accounted for. For instance, the number of Jordan blocks of fixed odd size corresponding to the eigenvalue 1 or ?1 have to be even. For complex orthogonal matrices, it is shown that the behavior of the Jordan structures corresponding to the original eigenvalues that are not moved by perturbations follows again the pattern established earlier for unstructured matrices, taking into account the specifics of Jordan forms of complex orthogonal matrices. The proofs are based on general results developed in the paper concerning Jordan forms of structured matrices (which include in particular the classes of orthogonal and symplectic matrices) under structured rank one perturbations. These results are presented and proved in the framework of real as well as of complex matrices.     

The least-squares method for matrices dependent on parameters

Some algorithms are suggested for constructing pseudoinverse matrices and for solving systems with rectangular matrices whose entries depend on a parameter in polynomial and rational ways. The cases of one- and two-parameter matrices are considered. The construction of pseudoinverse matrices are realized on the basis of rank factorization algorithms. In the case of matrices with polynomial occurrence of parameters, these algorithms are the ΔW-1 and ΔW-2 algorithms for one- and two-parameter matrices, respectively. In the case of matrices with rational occurrence of parameters, these algorithms are the irreducible factorization algorithms. This paper is a continuation of the author's studies of the solution of systems with one-parameter matrices and an extension of the results to the case of two-parameter matrices with polynomial and rational entries. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 219, 1994, pp. 176–185. This work was supported by the Russian Foundation of Fundamental Research (grant 94-01-00919). Translated by V. N. Kublanovskaya.