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Spectral Transformations R.
Contents
Index
Hermitian Eigenvalue Problems
Subsections
Introduction
Overview of Available Algorithms.
Summary of Choices.
Matrix Preparation.
Orthogonalization.
Eigenvalues Sought.
Storage.
Direct Methods
Single- and Multiple-Vector Iterations
M. Gu
Power Method
Inverse Iteration
Rayleigh Quotient Iteration
Subspace Iteration
Subspace Dimension.
Locking.
Acceleration.
Software Availability
Lanczos Method
A. Ruhe
Algorithm
Convergence Properties
Multiple Eigenvalues.
Spectral Transformation
Reorthogonalization
Full Reorthogonalization.
Selective Reorthogonalization.
Local Reorthogonalization and Detecting Spurious Ritz Values.
Software Availability
Numerical Examples
Results for L-Shaped Membrane.
Results for Medline SVD.
Results for L-Shaped Membrane with Shift-and-Invert.
Implicitly Restarted Lanczos Method
R. Lehoucq and D. Sorensen
Implicit Restart
Shift Selection
Lanczos Method in GEMV Form
Convergence Properties
Computational Costs and Tradeoffs
Deflation and Stopping Rules
Orthogonal Deflating Transformation
Locking or Purging a Single Eigenvalue.
Locking
.
Purging
.
Stability of
.
Implementation of Locking and Purging
Software Availability
Band Lanczos Method
R. Freund
The Need for Deflation
Basic Properties
Algorithm
Variants
Jacobi-Davidson Methods
G. Sleijpen and H. van der Vorst
Basic Theory
Basic Algorithm
Storage and Computational Costs.
Restart and Deflation
Restart Strategy.
Deflation.
Preconditioning.
An Algorithm Template.
Computing Interior Eigenvalues
Software Availability
Numerical Example
Stability and Accuracy Assessments
Z. Bai and R. Li
Residual Vector.
Transfer Residual Error to Backward Error.
Error Bounds for Computed Eigenvalues.
Error Bound for Computed Eigenvectors.
Remarks on Clustered Eigenvalues.
Remarks on Eigenvalue Computations to High Relative Accuracy.
Susan Blackford 2000-11-20
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