Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

Abstract

This study sets up two new merit functions, which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems. For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less, where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector. 1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues. Simultaneously, the real and complex eigenvectors can be computed very accurately. A simpler approach to the nonlinear eigenvalue problems is proposed, which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly. The real eigenvalues can be computed by the fictitious time integration method (FTIM), which saves computational costs compared to the one-dimensional golden section search algorithm (1D GSSA). The simpler method is also combined with the Newton iteration method, which is convergent very fast. All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.