*Title*: **The NLFEAST Algorithm for Large-Scale Nonlinear Eigenvalue Problems.**

*Abstract*: Eigenvalue problems in which the coefficient matrices depend nonlinearly on the eigenvalues arise in a variety
of applications in science and engineering, e.g., dynamic analysis of structures or computational nanoelectronics, to mention just a few.
This talk will discuss how the Cauchy integral-based approaches offer an attractive framework to develop highly efficient and flexible techniques for solving large-scale nonlinear eigenvalue problems.

We will first introduce the nonlinear counterpart of the well-established linear FEAST algorithm. Like its linear predecessor, the nonlinear FEAST (NLFEAST) algorithm can be used to solve nonlinear
eigenvalue problems for the eigenpairs corresponding to eigenvalues that lie in a user-specified region in the complex plane, thereby allowing for the calculation of large number of eigenpairs in parallel.
To develop a nonlinear FEAST algorithm that enables the iterative refinement of a subspace of a fixed dimension, we propose to use a modified form of the contour integral resulting
from the relationship between the NLFEAST and the residual inverse iteration by Neumaier for the nonlinear eigenvalue problems.
Finally, we will use several real-world examples to illustrate the method. This is a joint work with B. Gavin, E. Polizzi and Y. Saad.

CAM Seminar Spring 2019