8 research outputs found
An Improved Method for Computing Eigenpair Derivatives of Damped System
The calculation of eigenpair derivatives plays an important role in vibroengineering. This paper presents an improved algorithm for the eigenvector derivative of the damped systems by dividing it into a particular solution and general solution of the corresponding homogeneous equation. Compared with the existing methods, the proposed algorithm can significantly reduce the condition number of the equation for particular solution. Therefore, the relative errors of the calculated solutions are notably cut down. The results on two numerical examples show that such strategy is effective in reducing the condition numbers for both distinct and repeated eigenvalues
Rational Krylov and ADI iteration for infinite size quasi-Toeplitz matrix equations
We consider a class of linear matrix equations involving semi-infinite
matrices which have a quasi-Toeplitz structure. These equations arise in
different settings, mostly connected with PDEs or the study of Markov chains
such as random walks on bidimensional lattices. We present the theory
justifying the existence in an appropriate Banach algebra which is
computationally treatable, and we propose several methods for their solutions.
We show how to adapt the ADI iteration to this particular infinite dimensional
setting, and how to construct rational Krylov methods. Convergence theory is
discussed, and numerical experiments validate the proposed approaches
A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems
We consider the nonlinear eigenvalue problem: M (λ)x = 0, where M (λ) is a large parameter-dependent matrix. In several applications, M (λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldiâs method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169â195]. In this paper the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems.status: publishe
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A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems
We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169â195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems. Copyright © 2016 John Wiley & Sons, Ltd
A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems
We consider the nonlinear eigenvalue problem: M(λ) x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp.169-195]. In this paper the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems.nrpages: 22status: publishe
A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems
We consider the nonlinear eigenvalue problem M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi-based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi's method applied to an operator whose reciprocal eigenvalues are solutions to the nonlinear eigenvalue problem. The iterates in the algorithm are functions represented in a particular structured vector-valued polynomial basis similar to the construction in the infinite Arnoldi method [Jarlebring, Michiels, and Meerbergen, Numer. Math., 122 (2012), pp. 169â195]. In this paper, the low-rank structure is exploited by applying an additional operator and by using a more compact representation of the functions. This reduces the computational cost associated with orthogonalization, as well as the required memory resources. The structure exploitation also provides a natural way in carrying out implicit restarting and locking without the need to impose structure in every restart. The efficiency and properties of the algorithm are illustrated with two large-scale problems. Copyright © 2016 John Wiley & Sons, Ltd