On Smooth LU Decompositions with Applications to Solutions of Nonlinear Eigenvalue Problems

Abstract

We study the smooth LU decomposition of a given analytic functional $\lambda$-matrix $A(\lambda)$ and its block-analogue. Sufficient conditions for the existence of such matrix decompositions are given, some differentiability about certain elements arising from them are proved, and several explicit expressions for derivatives of the specified elements are provided. By using these smooth LU decompositions, we propose two numerical methods for computing multiple nonlinear eigenvalues of $A(\lambda)$, and establish their locally quadratic convergence properties. Several numerical examples are provided to show the feasibility and effectiveness of these new methods.