We present a method for solving nonlinear eigenvalue problems using rational approximation. The method uses the AAA method by Nakatsukasa, Sète, and Trefethen to approximate the nonlinear eigenvalue problem by a rational eigenvalue problem and is embedded in the state space representation of a rational polynomial by Su and Bai. The advantage of the method, compared to related techniques such as NLEIGS and infinite Arnoldi, is the efficient computation by an automatic procedure. In addition, a set-valued approach is developed that allows building a low degree rational approximation of a nonlinear eigenvalue problem. The method perfectly fits the framework of the Compact rational Krylov methods (CORK and TS-CORK), allowing to efficiently solve large scale nonlinear eigenvalue problems. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

We describe a generalization of the compact rational Krylov (CORK) methods for polynomial and rational eigenvalue problems that usually but not necessarily come from polynomial or rational approximations of genuinely nonlinear eigenvalue problems. CORK is a family of one-sided methods that reformulates the polynomial or rational eigenproblem as a generalized eigenvalue problem. By exploiting the Kronecker structure of the associated pencil, it constructs a right Krylov subspace in compact form and thereby avoids the high memory and orthogonalization costs that are usually associated with linearizations of high degree matrix polynomials. CORK approximates eigenvalues and their corresponding right eigenvectors but is not suitable in its current form for the computation of left eigenvectors. Our generalization of the CORK method is based on a class of Kronecker structured pencils that include as special cases the CORK pencils, the transposes of CORK pencils, and the symmetrically structured linearizations by Robol, Vandebril and Van Dooren, 2016. This class of structured pencils facilitates the development of a general framework for the computation of both right and left-sided Krylov subspaces in compact form, and hence allows the development of two-sided compact rational Krylov methods for nonlinear eigenvalue problems. The latter are particularly efficient when the standard inner product is replaced by a cheaper to compute quasi inner product. We show experimentally that convergence results similar to CORK can be obtained for a certain quasi inner product.

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