Robust Rayleigh quotient minimization and generalized eigenvalue problems

Robust Rayleigh quotient minimization and nonlinear eigenvalue problems

In this paper, we study the robust Rayleigh quotient optimization problems that arise when optimizing in worst-case the Rayleigh quotient of data matrices subject to uncertainties. We propose to solve such problem by exploiting its characterization as a nonlinear eigenvalue problem with eigenvector nonlinearity. With this approach, we can show that a commonly used iterative method can be diverg...

A Comparison of Parallel Preconditioners for the Sparse Generalized Eigenvalue Problems by Rayleigh-Quotient Minimization

Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems

Four classes of eigenvalue problems that admit similar min-max principles and the Cauchy interlacing inequalities as the symmetric eigenvalue problem famously does are investigated. These min-max principles pave ways for efficient numerical solutions for extreme eigenpairs by optimizing the so-called Rayleigh quotient functions. In fact, scientists and engineers have already been doing that for...

Generalized Rayleigh Quotient Shift Strategy in QR Algorithm for Eigenvalue Problems

QR algorithm for eigenproblems is often applied with single or double shift strategies. To save computation effort, double implicit shift technique is employed. In Watkins and Elsner[1], a higher-order shift strategy, generalized Rayleigh quotient shift strategy, is introduced. In this paper, we give a generalization of the double implicit shift technique for the higher-order strategy, and comp...

Rayleigh Quotient Iteration and Simplified Jacobi-davidson with Preconditioned Iterative Solves for Generalised Eigenvalue Problems

The computation of a right eigenvector and corresponding finite eigenvalue of a large sparse generalised eigenproblem Ax = λMx using preconditioned Rayleigh quotient iteration and the simplified JacobiDavidson method is considered. Both methods are inner-outer iterative methods and we consider GMRES and FOM as iterative algorithms for the (inexact) solution of the inner systems that arise. The ...