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A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver
Author(s) -
Avery P.,
Farhat C.,
Hetmaniuk U.
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2572
Subject(s) - eigenvalues and eigenvectors , factorization , solver , algorithm , mathematics , divide and conquer eigenvalue algorithm , scalar (mathematics) , iterative method , rational function , inertia , eigendecomposition of a matrix , algebra over a field , mathematical optimization , mathematical analysis , pure mathematics , geometry , physics , quantum mechanics , classical mechanics
When computing the solution of a generalized symmetric eigenvalue problem of the form Ku =λ Mu , the Sturm sequence check, also known as the inertia check, is the most popular method for reporting the number of missed eigenvalues within a range [σ L ,σ R ]. This method requires the factorization of the matrices K −σ L M and K −σ R M . 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 −σ M is not desired or feasible and therefore the inertia 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 [σ L ,σ R ]. 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. Copyright © 2009 John Wiley & Sons, Ltd.