Matrix polynomial methods for eigenvector nonlinear problems

This talk addresses two topics: (a) matrix functions in the sense of Higham, and (b) nonlinear eigenvalue problems with eigenvector nonlinearities, such as those arising in electronic structure calculations. The focus is on a recent class of methods for solving the nonlinear eigenvalue problem by evaluating matrix functions of large, dense matrices, particularly in settings where we aim to compute a large invariant subspace. In these situations, the matrix-matrix product is the dominating computational component, and the evaluation should be done with as few matrix-matrix multiplications as possible. We present an approach for achieving this by characterizing the univariate semi-algebraic set of polynomials whose corresponding matrix polynomials can be evaluated with a fixed (low) number of matrix-matrix multiplications.