# Convergence of the Dominant Pole Algorithm and Rayleigh Quotient Iteration

@article{Rommes2008ConvergenceOT, title={Convergence of the Dominant Pole Algorithm and Rayleigh Quotient Iteration}, author={Joost Rommes and Gerard L. G. Sleijpen}, journal={SIAM J. Matrix Anal. Appl.}, year={2008}, volume={30}, pages={346-363} }

The dominant poles of a transfer function are specific eigenvalues of the state space matrix of the corresponding dynamical system. In this paper, two methods for the computation of the dominant poles of a large scale transfer function are studied: two-sided Rayleigh quotient iteration (RQI) and the dominant pole algorithm (DPA). First, a local convergence analysis of DPA will be given, and the local convergence neighborhoods of the dominant poles will be characterized for both methods. Second…

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## References

SHOWING 1-10 OF 25 REFERENCES

Regions of convergence of the Rayleigh quotient iteration method

- MathematicsNumer. Linear Algebra Appl.
- 1995

The regions of the unit sphere which include all possible initial vectors converging to a specific eigenvector are studied and the generalized eigenvalue problem Ax = λBx is considered and it is shown that the regions do not change when the matrix is shifted or multiplied by a scalar.

The Rayleigh Quotient Iteration and Some Generalizations for Nonnormal Matrices

- Mathematics
- 1974

The Rayleigh Quotient Iteration (RQI) was developed for real symmetric matrices. Its rapid local convergence is due to the stationarity of the Rayleigh Quotient at an eigenvector. Its excellent…

Criteria for Combining Inverse and Rayleigh Quotient Iteration

- Mathematics
- 1988

A method is presented to find selected eigenvalue-eigenvector pairs of the generalized problem $Ax = \lambda Bx$. Here A and B are real symmetric matrices and B is positive definite. It is further…

Efficient computation of transfer function dominant poles using subspace acceleration

- Computer ScienceIEEE Transactions on Power Systems
- 2006

A new algorithm to compute the dominant poles of a high-order scalar transfer function that is more robust than existing methods in finding both real and complex dominant poles and faster because of subspace acceleration.

Efficient Computation of Multivariable Transfer Function Dominant Poles Using Subspace Acceleration

- Computer ScienceIEEE Transactions on Power Systems
- 2006

A new algorithm to compute the dominant poles of a high-order multiple-input multiple-output (MIMO) transfer function, called the Subspace Accelerated MIMO Dominant Pole Algorithm (SAMDP), which can be used to produce good modal equivalents automatically.

Localization criteria and containment for Rayleigh quotient iteration

- Mathematics
- 1989

Rayleigh quotient iteration can often yield an eigenvalue-eigenvector pair of a positive-definite Hermitian problem in a very short time. The primary hindrance associated with its use as a regular…

Computing dominant poles of power system transfer functions

- Mathematics
- 1996

This paper describes the first algorithm to efficiently compute the dominant poles of any specified high order transfer function. As the method is closely related to Rayleigh iteration (generalized…

Methods for eigenvalue problems with applications in model order reduction

- Computer Science
- 2007

The algorithms described in this thesis are efficient and effective methods for the computation of specific dominant eigenvalues that can be used to construct reduced-order models in the form of modal approximations, but also to improve reduced- order models computed by Krylov subspace based techniques.

A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems

- MathematicsSIAM Rev.
- 1996

A new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors is proposed that has improved convergence properties and that may be used for general matrices.