Broyden's method for nonlinear eigenproblems
For researchers solving nonlinear eigenvalue problems where matrix-vector products are as expensive as solving linear systems, this provides a derivative-free approach with proven convergence.
The authors adapt Broyden's method for nonlinear eigenvalue problems, achieving local superlinear convergence for simple eigenvalues and enabling computation of multiple eigenvalues via deflation. The method is demonstrated on a machine tool milling PDE problem.
Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE is used to illustrate the approach. The simulations are done in the julia programming language, and are provided as publicly available module for reproducability.