A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver

Abstract

SUMMARYWhen computing the solution of a generalized symmetric eigenvalue problem of the form Ku = λMu, the Sturm sequence check is the most popular method for reporting the number of missed eigenvalues within a range [σL, σR]. This method requires the factorization of the matrices K − σLM and K − σRM. When the size of the problem is reasonable and the matrices K and M are assembled, these factorizations are possible. When the eigensolver is equipped with an iterative solver, which is nowadays the preferred choice for large-scale problems, the factorization of K − σM is not desired or feasible and therefore the Sturm sequence check cannot be performed. To this effect, the purpose of this paper is to present a factorization-free algorithm for detecting and identifying the eigenvalues that were missed by an eigensolver equipped with an iterative linear equation solver within an interval of interest [σL, σR]. This algorithm constructs a scalar, rational, transfer function whose poles are exactly the eigenvalues of the symmetric pencil (K, M), approximates it by a Padé expansion, and computes the poles of this approximation to detect and identify the missed eigenvalues. The proposed algorithm is illustrated with an academic numerical example. Its potential for real engineering applications is also demonstrated with a large-scale structural vibrations problem.

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