China
Africa
Explore chapters and articles related to this topic
Subspace Algorithms
Published in Philipos C. Loizou, Speech Enhancement, 2013
The system of equations in the second step of the algorithm can be solved using the Levinson recursion in O(K2) operations since R is Toeplitz. Although the Rayleigh quotient iteration algorithm is guaranteed to converge with cubic convergence rate, it may not converge to the desired eigenvalue–eigenvector pair. In order to ensure that the preceding algorithm converges to the desired eigenvalue–eigenvector pair, it is critical that the initial eigenvector u0, or equivalently λ(0), be chosen properly. An efficient method for choosing λ(0) was proposed in [74] based on estimated lower and upper bounds of the desired eigenvalue. These bounds were obtained by performing the LDU factorization of the matrix R − λ(k)I at each iteration. The Rayleigh quotient iteration algorithm requires O(K2) operations to compute each eigenvalue–eigenvector pair.
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.
Hilbert spaces
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
Remark For f ≠ 0, the ratio 〈Lf,f〉‖f‖2 in (19) is sometimes called the Rayleigh quotient. In the finite-dimensional setting, it provides a powerful numerical method, called Rayleigh quotient iteration, for finding eigenvectors and eigenvalues.
An Efficient Eigen-Solver and Some of Its Applications
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2019
M. Rezaiee-Pajand, A. Aftabi Sani, M. S. Kazemiyan
Up to now, various algorithms have been presented for estimation of the natural frequencies and mode shapes of multi-degree-of-freedom structures. These schemes may be categorized into five groups: 1. the vector iteration methods; 2. the transformation methods; 3. the polynomial iteration methods; 4. Lanczos method; 5. the subspace iteration method [1]. It should be added that the inverse iteration, the forward iteration and Rayleigh quotient iteration technique are belonged to the first category. Moreover, Jacobi and Householder are two famous transformation approaches. Another well-known algorithm widely used is named subspace iteration. This strategy was developed by Bathe et al. [1], and it is suitable for the finite element model of the huge structures. By means of this way, any arbitrary number of structures' eigenvalues and eigenvectors can be obtained. The most key advantages of this scheme are its simplicity and efficiency. Meanwhile, its computer program can be easily developed. It is worth emphasizing that various researches have been conducted to improve this famous technique. In what follows, some of these works are reviewed briefly.
Eigenvalue Solvers for Modeling Nuclear Reactors on Leadership Class Machines
Published in Nuclear Science and Engineering, 2018
R. N. Slaybaugh, M. Ramirez-Zweiger, Tara Pandya, Steven Hamilton, T. M. Evans
Three complementary methods have been implemented in the code Denovo2 that accomplish this goal: a multigroup block (MG) Krylov solver, a Rayleigh quotient iteration (RQI) eigenvalue solver, and a multigrid-in-energy (MGE) preconditioner. Each individual method has been generally described before (see Refs. 3 and 4), but this is the first time they have been demonstrated to work together in a complementary way.