SIAM Journal on Matrix Analysis and Applications 1996, 17, 789 Calvetti, D. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations Springer, 1997 Lehoucq, R. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen, D. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. Practically, only a small subset of eigenvalues is required. Typically in these problems, the dimension of Hilbert space is huge. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Knowledge of eigenvalues is essential in quantum molecular science. Their study is mainly motivated by their high importance in a wide range of applications. Based on the direct algebraic methods an iterative algorithm is developed for efficiently calculating the eigenvalues and eigenvectors of the perturbed geometry from the eigenvalues and eigenvectors of the unperturbed geometry.ĮvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.Įigenvalues and eigenvectors are an essential theme in numerical linear algebra. Two common approaches for the calculation of eigenpair derivatives, namely modal superposition method and direct algebraic methods, are discussed in this paper. The perturbed eigenpairs can be approximated using eigenpair derivatives. Directly solving the eigenvalue problem for each perturbation is computationally costly. Geometrical perturbations may arise by manufacturing tolerances, harsh operating conditions or during shape optimization. Geometrical perturbations of the structure under concern lead to a new generalized eigenvalue problems with different system matrices. They may arise in electromagnetic fields from the discretization of the Helmholtz equation by for example the finite element method (FEM). Generalized eigenvalue problems are standard problems in computational sciences. Gorgizadeh, Shahnam Flisgen, Thomas van Rienen, Ursula Eigenmode computation of cavities with perturbed geometry using matrix perturbation methods applied on generalized eigenvalue problems
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