Regularized Normalization Methods for Solving Linear and Nonlinear Eigenvalue Problems

To solve linear and nonlinear eigenvalue problems, we develop a simple method by directlysolving a nonhomogeneous system obtained by supplementing a normalization condition on theeigen-equation for the uniqueness of the eigenvector. The novelty of the present paper is that wetransform the original homogeneous eigen-equation to a nonhomogeneous eigen-equation by anormalization technique and the introduction of a simple merit function, the minimum of whichleads to a precise eigenvalue. For complex eigenvalue problems, two normalization equations arederived utilizing two different normalization conditions. The golden section search algorithms areemployed to minimize the merit functions to locate real and complex eigenvalues, and simultaneously,we can obtain precise eigenvectors to satisfy the eigen-equation. Two regularized normalizationmethods can accelerate the convergence speed for two extensions of the simple method, and aderivative-free fixed-point Newton iterative scheme is developed to compute real eigenvalues, theconvergence speed of which is ten times faster than the golden section search algorithm. Newtonmethods are developed for solving two systems of nonlinear regularized equations, and the efficiencyand accuracy are significantly improved. Over ten examples demonstrate the high performance of theproposed methods. Among them, the two regularization methods are better than the simple method.