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Index
Contents
Acknowledgements
Nederlandse samenvatting
Outline of the thesis
Introduction to semiseparable matrices
Semiseparable and related matrices, definitions and properties
Definitions for semiseparable and related matrices
The nullity theorem
Inverse of structured rank matrices
Generalizations of the nullity theorem
The generator definition, investigated
Common misunderstandings about generator representable semiseparable matrices
A theoretical problem, with numerical consequences
Connection between the generator representable semiseparable and semiseparable matrices
Conclusions
The representation of semiseparable matrices
The definition of a representation
The generator representation
The representation based on the diagonal and subdiagonal
A new representation for symmetric semiseparable matrices
Other types of representations
Some examples
Retrieving the representation from a semiseparable matrix
Some algorithms connected to the representation
Swapping the representation
A fast matrix vector multiplication
The determinant of a semiseparable matrix in order
flops.
Conclusions
An overview of semiseparable matrices.
Oscillation matrices
Introduction
Definition and examples
The inverse of a one-pair matrix
The example of the one-pair matrix
Some other interesting applications
The connection with eigenvalues and eigenvectors
Semiseparable matrices as covariance matrices
The multinomial distribution
Some other matrices
Discretization of integral equations
Orthogonal rational functions
An historical overview of the literature
Conclusions
The reduction of matrices to semiseparable matrices
Reduction algorithms to semiseparable form
An orthogonal similarity reduction to semiseparable form
A similarity reduction applied to a nonsymmetric matrix
Reduction to lower (upper) semiseparable form
Conclusions
Convergence properties of the reductions
The Arnoldi(Lanczos)-Ritz values
Ritz values and Arnoldi(Lanczos)-Ritz values
Necessary conditions to obtain the Arnoldi(Lanczos)-Ritz values as eigenvalues in the already reduced block of the matrix
Sufficient conditions to obtain the convergence behavior
Some general remarks
Convergence properties of the reduction to semiseparable form
Lanczos-Ritz values in a similarity reduction to semiseparable form
subspace iteration in the similarity reduction
subspace iteration and the Lanczos convergence behavior
Convergence properties of the reduction to Hessenberg-like
Transformation to upper triangular semiseparable form
Lanczos-Ritz values
subspace iteration
Conclusions
Implementation of the algorithm and numerical experiments
Reduction to symmetric semiseparable
Similarity reduction to Hessenberg-like
The reduction to upper triangular semiseparable form
Numerical experiments
Experiment 1
Experiment 2
Experiment 3
Experiment 4
Numerical experiments
Numerical experiments: Deflation possibilities
Conclusions
QR-algorithms
Theoretical results for
-algorithms
The
-factorization of Hessenberg-like plus diagonal matrices
Unreduced Hessenberg-like matrices
A
-step maintains the Hessenberg-like structure
The reduction to unreduced Hessenberg-like form
The implicit
-theorem
Conclusions
Implicit
-algorithms for semiseparable matrices
An implicit
-algorithm for symmetric semiseparable matrices
Unreduced symmetric semiseparable matrix
The shift
An implicit
-step on a symmetric semiseparable matrix
Proof of the correctness of the implicit approach
An implicit
-algorithm for Hessenberg-like matrices
The shift
An implicit
-step on the Hessenberg-like matrix
Chasing the disturbance in the matrix
An implicit
-algorithm for computing the singular values
The standard
-method for the calculation of the singular values
Unreduced upper triangular semiseparable matrices and the shift
The new method via upper triangular semiseparable matrices
Chasing the bulge
One iteration of the new method applied to upper triangular semiseparable matrices.
Conclusions
Implementations and numerical experiments of
-algorithms
Implementation of the
-factorization
The
-algorithm for symmetric semiseparable matrices
The reduction to unreduced form
The
-algorithm for symmetric semiseparable matrices
Deflation after a step of the
-algorithm
Computing the eigenvectors
Computing all the eigenvectors
Selected eigenvectors
The eigenvectors of an arbitrary symmetric matrix
The implementation of the
-algorithms
Tests on the symmetric eigenvalue solver
The block experiment
Stewart's devil's stairs
Problem matrices
Cutting off the last eigenvalue
The
-algorithm for the computation of the singular values
Conclusions
Software package for semiseparable matrices
General tools
Retrieving the Givens-vector representation
Constructing the matrices from the representation
Various tools
Reduction Algorithms
-tools
-solver
Reduction to unreduced form
An implicit
-step without shift
An implicit
-step with shift
The eigen(singular)value decomposition
Eigenvalue and singular value tools
Conclusions and future research
Bibliography
Index
Curriculum Vitae
About this document ...
Subsections
Acknowledgements
Nederlandse samenvatting
Outline of the thesis
Raf Vandebril 2004-05-03