A Prediction-Correction Dynamic Method for Large-Scale Generalized Eigenvalue Problems

Xin-Long Luo,Jia-Ru Lin,Wei-Ling Wu

doi:10.1155/2013/845459

Abstract

This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem. First, the smallest generalized eigenvalue problem is converted into an equivalent-constrained optimization problem. Second, according to the Karush-Kuhn-Tucker conditions of this special equality-constrained problem, a special continuous dynamical system of differential-algebraic equations is obtained. Third, based on the implicit Euler method and an analogous trust-region technique, a prediction-correction method is constructed to follow this system of differential-algebraic equations to compute its steady-state solution. Consequently, the smallest generalized eigenvalue of the original problem is obtained. The local superlinear convergence property for this new algorithm is also established. Finally, in comparison with other methods, some promising numerical experiments are presented.

Highlights

In this paper, we consider the smallest generalized eigenvalue problem, which is often encountered in engineering applications such as automatic control, dynamical analysis of structure, electronic structure calculations, and quantum chemistry
This paper gives a new prediction-correction method based on the dynamical system of differential-algebraic equations for the smallest generalized eigenvalue problem
For this old and active problem, recently, Gao et al gave an interesting continuous projected method [3, 4]. This article follows this line and gives a new prediction-correction method based on the dynamical system of differential-algebraic equations (DAEs) for this problem

Summary

Introduction

We consider the smallest generalized eigenvalue problem, which is often encountered in engineering applications such as automatic control, dynamical analysis of structure, electronic structure calculations, and quantum chemistry (see [1, 2] and references therein). For this old and active problem, recently, Gao et al gave an interesting continuous projected method [3, 4]. Throughout this article, ‖ ⋅ ‖ denotes the Euclidean vector norm and the corresponding induced matrix norm

Continuous Dynamical Model

A Prediction-Correction Method

Local Convergence Analysis

Numerical Experiments

Conclusions