Robust rational approximations of nonlinear eigenvalue problems

We develop algorithms that construct robust (i.e., reliable for a given tolerance, and scaling independent) rational approximants of matrix-valued functions on a given subset of the complex plane. We consider matrix-valued functions provided in both split form (i.e., as a sum of scalar functions times constant coefficient matrices) and in black box form. We develop a new error analysis and use it for the construction of stopping criteria, one for each form. Our criterion for split forms adds weights chosen relative to the importance of each scalar function, leading to the weighted AAA algorithm, a variant of the set-valued AAA algorithm that can guarantee to return a rational approximant with a user-chosen accuracy. We propose two-phase approaches for black box matrix-valued functions that construct a surrogate AAA approximation in phase one and refine it in phase two, leading to the surrogate AAA algorithm with exact search and the surrogate AAA algorithm with cyclic Leja--Bagby refinement. The stopping criterion for black box matrix-valued functions is updated at each step of phase two to include information from the previous step. When convergence occurs, our two-phase approaches return rational approximants with a user-chosen accuracy. We select problems from the NLEVP collection that represent a variety of matrix-valued functions of different sizes and properties and use them to benchmark our algorithms