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Keywords:

linearization, matrix pencil, rational Krylov, nonlinear eigenvalue problem, Science & Technology, Physical Sciences, Mathematics, Applied, Mathematics, MATRIX POLYNOMIALS, ARNOLDI METHOD, LINEARIZATIONS, INTERPOLATION, ALGORITHM, 0102 Applied Mathematics, 0103 Numerical and Computational Mathematics, Numerical & Computational Mathematics, 4901 Applied mathematics

Abstract:

We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.