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Dynamics of Beams and Plates
Published in Irving H. Shames, Clive L. Dym, Energy and Finite Element Methods in Structural Mechanics, 2018
Irving H. Shames, Clive L. Dym
We shall now employ the Ritz approach of Chapter 3 in conjunction with the Rayleigh quotient to form the very valuable Rayleigh-Ritz method. Instead of employing a single function ϕ for the Rayleigh quotient, we shall now select a set of n linearly independent functions‡ϕi, each satisfying the boundary conditions of the problem, to represent Wapp as a parameter-laden sum as follows () Wapp=A1ϕ1+⋯+Anϕn
Subspace Algorithms
Published in Philipos C. Loizou, Speech Enhancement, 2013
Several iterative algorithms were also proposed for computing the smallest eigenvalue–eigenvector pair of symmetric Toeplitz matrices. These algorithms made use of the fact that the inverse of a symmetric Toeplitz matrix can be efficiently computed using the Levinson recursion in O(K2) operations. Algorithms based on the Levinson recursion were proposed in [75,76] for estimating the eigenvalues of Toeplitz matrices. The Rayleigh quotient iteration algorithm was used in [74] to compute the smallest eigenvalue–eigenvector pair of a symmetric Toeplitz matrix. This algorithm is an extension of the inverse iteration algorithm [8, p. 383] commonly used for computing selected eigenvector/eigenvalue pairs based on the Rayleigh quotient formula. The Rayleigh quotient iteration algorithm is described in the following text:
Numerical Methods for Eigenvalue Problems
Published in Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas, Eigenvalue Problems in Power Systems, 2020
Federico Milano, Ioannis Dassios, Muyang Liu, Georgios Tzounas
Despite the ability of the inverse power method to compute any eigenpair (λi,ui ), a bad selection of the spectral shift σ may lead to very slow convergence, or convergence to a different eigenvector than the one desired. The Rayleigh quotient iteration method is an extension of the inverse power method, that updates the applied spectral shift at each step of the iteration.
Splitting methods for the Eigenvalue Complementarity Problem
Published in Optimization Methods and Software, 2019
Alfredo N. Iusem, Joaquim J. Júdice, Valentina Sessa, Paula Sarabando
In this paper, we discuss splitting methods for the numerical solution of the symmetric and nonsymmetric EiCP. This kind of methods has been applied for a long time for solving Linear Algebra problems, like systems of linear equations (see [27] for an early reference) or Linear Complementarity Problems (LCP; see [6]). On the other hand, to the best of our knowledge, it is the first time that this approach is used for solving the symmetric and nonsymmetric EiCP. For the case of linear equations Ax=b with given and , we choose a non-singular matrix D, set E=A−D, and assuming that is the kth iterate generated by the method, we solve the linear system with , whose unique solution gives the next iterate . The matrix D is chosen so that a linear system with the matrix D is much easier to solve than Ax=b, and the sequence generated in this fashion is expected to converge to a solution of the original linear system. We suitably extend this approach to EiCP, and study four variants. In the first one, called A1, we take A=C−D, where D is a symmetric PD (SPD) matrix and the method consists of solving, at each step, an LCP with matrix D. When A and B are symmetric, the direction of the method at iteration k, namely , turns out to be an ascent direction for the Rayleigh quotient, and, hence, it is possible to add a line search ensuring increase of the quotient, which improves the convergence properties of the algorithm. This version is called A2.