A Padé‐based factorization‐free algorithm for identifying the eigenvalues missed by a generalized symmetric eigensolver |
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Authors: | P Avery C Farhat U Hetmaniuk |
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Affiliation: | 1. Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique, 42 Avenue G. Coriolis, 31057 Toulouse Cedex 01, France;2. Department of Mechanical Engineering and Institute for Computational and Mathematical Engineering, Stanford University, Mail Code 3035, Stanford, CA 94305, U.S.A.;3. Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195‐2420, U.S.A. |
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Abstract: | 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 σL,σR]. This method requires the factorization of the matrices K ?σL M and K ?σR 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 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. Copyright © 2009 John Wiley & Sons, Ltd. |
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Keywords: | eigensolver missed eigenvalue Padé Sturm sequence |
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