Index

$A(i,j)$

Definitions for semiseparable and | Connection between the generator | Connection between the generator

Abstract of the thesis

thesis

Acknowledgements

Acknowledgements to Acknowledgements

Arnoldi-Ritz values

The Arnoldi(Lanczos)-Ritz values | Convergence properties of the

in an orthogonal reduction to Hessenberg-like form

Convergence properties of the

BHL

Constructing the matrices from

Binomial coefficients

An historical overview of

Binomial distribution

The multinomial distribution

Biology

An historical overview of

Block experiment

The block experiment

Block off-diagonal rank

Inverse of structured rank

Block off-diagonal structure

Inverse of structured rank

BLS

Constructing the matrices from

Boundary value problems

An historical overview of | An historical overview of | An historical overview of

BRHL

Retrieving the Givens-vector representation

BRLS

Retrieving the Givens-vector representation

BRSS

Retrieving the Givens-vector representation

BRUS

Retrieving the Givens-vector representation

BSS

Constructing the matrices from

BUS

Constructing the matrices from

Cardinality of a set

The nullity theorem

Cauchy matrix

Definition and examples

Charged particles

An historical overview of

Chase the disturbance

Chasing the disturbance in

Chasing the bulge

Chasing the bulge

CHL

Reduction Algorithms

Cholesky decomposition

An historical overview of

Cholesky decomposition of structured rank matrices

Generalizations of the nullity

CLS

Reduction Algorithms

Complexity of

reduction to Hessenberg-like form

A similarity reduction applied

reduction to semiseparable form

An orthogonal similarity reduction

reduction to upper triangular semiseparable form

Reduction to lower (upper)

Continuant matrices

An historical overview of

Continuum

Introduction

Convergence properties

Arnoldi-Ritz values

Convergence properties of the

interaction between the subspace iteration and the Lanczos-Ritz values

subspace iteration and the

Lanczos-Ritz values

Lanczos-Ritz values in a | Lanczos-Ritz values

subspace iteration

subspace iteration in the | Convergence properties of the | subspace iteration

Convergence properties of

subspace iteration

subspace iteration in the

the reduction algorithms

Convergence properties of the to subspace iteration

the reduction to a similar Hessenberg-like matrix

Convergence properties of the

the reduction to a similar semiseparable matrix

Convergence properties of the

the reduction to an upper triangular semiseparable matrix

Transformation to upper triangular

Counteridentity matrix

Numerical experiments

Covariance matrices

Semiseparable matrices as covariance to Some other matrices

examples

Some other matrices to Some other matrices

CSS

Reduction Algorithms

CUS

Reduction Algorithms

Decay rates

An historical overview of | An historical overview of | An historical overview of | An historical overview of | An historical overview of

Decompositions

Cholesky

An historical overview of

$LU$

Generalizations of the nullity

$QR$

Generalizations of the nullity

Deflation criteria

Deflation after a step to Deflation after a step | Cutting off the last

Determinant of a semiseparable matrix

The determinant of a to The determinant of a

DETHL

Various tools

DETLS

Various tools

DETSS

Various tools

DETUS

Various tools

Diagonal subdiagonal representation

The representation based on to The representation based on

Difference between sets

The nullity theorem

Difference equation

An historical overview of

Differentiation

An historical overview of

Discretization of integral equations

Discretization of integral equations to Discretization of integral equations

Distance between subspaces

subspace iteration in the

Distribution

binomial

The multinomial distribution

exponential

Some other matrices

multinomial

The multinomial distribution

Divide and conquer method

An historical overview of

Dominant tridiagonal matrices

An historical overview of

Double implicit shift

The shift $\kappa$

Dutch summary

Nederlandse samenvatting to Nederlandse samenvatting

Eigenvectors

Computing the eigenvectors | Computing all the eigenvectors to The eigenvectors of an

EIGHL

The eigen(singular)value decomposition

EIGSS

The eigen(singular)value decomposition

Essentially the same matrices

The implicit $Q$-theorem

Exchange matrix

Numerical experiments

Exponential distribution

Some other matrices

Extended diagonal position

Inverse of structured rank

Finite boundary value problems

An historical overview of

Gap

subspace iteration and the

Gauss-Markov processes

An historical overview of

Generalized Hessenberg matrices

Definitions for semiseparable and

Generalized semiseparable matrices

An historical overview of

Generator representable semiseparable matrices

The generator definition, investigated to Connection between the generator

Generator representation

The generator representation to The generator representation

Givens-vector representation

A new representation for to A new representation for

Graded matrices

Numerical experiments | Problem matrices

Green's kernel

Discretization of integral equations | An historical overview of

$\mathcal{H}$-matrices

An historical overview of

Harmonic oscillation

Introduction

Hessenberg-like

Definitions for semiseparable and

matrices retains the structure under a $QR$-step

A $QR$-step maintains the

HLEIG

Eigenvalue and singular value

Implementation of

$QR$-algorithms

Implementations and numerical experiments

eigenvector calculation

Computing all the eigenvectors to The eigenvectors of an

$QR$-algorithms

Implementation of the $QR$-factorization to Implementation of the $QR$-factorization | The $QR$-algorithm for symmetric to The $QR$-algorithm for symmetric | The $QR$-algorithm for symmetric to The $QR$-algorithm for symmetric | The implementation of the | The implementation of the to The implementation of the

reduction algorithms

Implementation of the algorithm to The reduction to upper

reduction to Hessenberg-like form

Similarity reduction to Hessenberg-like to Similarity reduction to Hessenberg-like

reduction to semiseparable form

Reduction to symmetric semiseparable to Reduction to symmetric semiseparable

reduction to unreduced form

The reduction to unreduced to The reduction to unreduced

reduction to upper triangular semiseparable form

The reduction to upper to The reduction to upper

Influence coefficients

Introduction

Influence function of a string

The example of the

Influence matrix

Introduction

Inner product

Orthogonal rational functions

Integral equations

Discretization of integral equations | An historical overview of

Integration

An historical overview of

Interaction between Lanczos and subspace convergence behavior

Numerical experiments to Experiment 4

Inverse eigenvalue problem

Orthogonal rational functions to Orthogonal rational functions | An historical overview of

Inverse iteration

Selected eigenvectors | Selected eigenvectors

Inverse of

block off-diagonal rank

Inverse of structured rank

lower bidiagonal matrices

Inverse of structured rank

lower triangular rank

Inverse of structured rank

lower triangular semiseparable matrices

Inverse of structured rank

off-diagonal rank

Inverse of structured rank

one-pair matrices

The inverse of a to The inverse of a

$\{p\}$-generalized Hessenberg matrices

Inverse of structured rank | Inverse of structured rank | Inverse of structured rank | Inverse of structured rank | Inverse of structured rank | Inverse of structured rank

rank $k$ plus diagonal matrices

Inverse of structured rank

rank $k$ plus block diagonal matrices

Inverse of structured rank

semiseparable matrices

Inverse of structured rank

semiseparable plus block diagonal matrices

Inverse of structured rank

semiseparable plus diagonal matrices

Inverse of structured rank

structured rank matrices

The nullity theorem | Inverse of structured rank to Inverse of structured rank

tridiagonal matrices

Inverse of structured rank

$Z$-matrices

An historical overview of

Jacobi matrix

Oscillation matrices | Definition and examples

Kernel

Green's

Discretization of integral equations | An historical overview of

oscillation

Introduction

semiseparable

An historical overview of

separable

An historical overview of

Krylov subspace

Ritz values and Arnoldi(Lanczos)-Ritz

$LU$-decomposition of

of structured rank matrices

Generalizations of the nullity

semiseparable matrices

Generalizations of the nullity

Lanczos-Ritz values

The Arnoldi(Lanczos)-Ritz values | Lanczos-Ritz values in a | Lanczos-Ritz values

in an orthogonal reduction to semiseparable form

Lanczos-Ritz values in a

in an orthogonal reduction to upper triangular semiseparable form

Lanczos-Ritz values

interaction with subspace iteration

subspace iteration and the

Largest eigenvalues

subspace iteration in the

Look ahead algorithm

An historical overview of

Matrices

Cauchy

Definition and examples

continuant

An historical overview of

counteridentity

Numerical experiments

covariance

Semiseparable matrices as covariance to Some other matrices

diagonal of type $r$

An historical overview of

dominant tridiagonal

An historical overview of

essentially the same

The implicit $Q$-theorem

exchange

Numerical experiments

graded

Numerical experiments | Problem matrices

$\mathcal{H}$-matrices

An historical overview of

influence

Introduction

Jacobi

Oscillation matrices | Definition and examples

minor

Introduction

nongraded

Numerical experiments | Problem matrices

one-pair

Oscillation matrices | Definition and examples | The inverse of a

oscillation

Oscillation matrices to The connection with eigenvalues

patterned

Semiseparable matrices as covariance

pentadiagonal

An historical overview of

single-pair

Oscillation matrices

Toeplitz

An historical overview of to An historical overview of

totally nonnegative

Definition and examples

totally positive

Definition and examples

``une matrice factorisable''

An historical overview of

Vandermonde

Definition and examples

weakly semiseparable

An historical overview of

$Z$-matrices

An historical overview of

zero-tailed

The implicit $Q$-theorem

Minor of a matrix

Introduction

Minus operator for sets

The nullity theorem

Moving objects

subspace iteration in the

MULHL

Various tools

MULLS

Various tools

MULSS

Various tools

Multinomial distribution

The multinomial distribution

Multiplication of a semiseparable matrix and a vector

A fast matrix vector to A fast matrix vector

MULUSS

Various tools

Necessary conditions to obtain the Lanczos-Ritz values

Necessary conditions to obtain

Nederlandse samenvatting

Nederlandse samenvatting to Nederlandse samenvatting

Nested subspace iteration

subspace iteration in the

Nongraded matrices

Numerical experiments | Problem matrices

Nonsingularity of an unreduced Hessenberg-like matrix

Unreduced Hessenberg-like matrices

Nullity of a matrix

The nullity theorem

Nullity theorem

The nullity theorem

generalizations

Generalizations of the nullity to Generalizations of the nullity

$LU$-decomposition

Generalizations of the nullity

$QR$-decomposition

Generalizations of the nullity

Numerical experiments

a $QR$-step on an upper triangular semiseparable matrix

The $QR$-algorithm for the

a $QR$-step on an upper triangular semiseparable matrix

to The $QR$-algorithm for the

accuracy

The $QR$-algorithm for the

accuracy of the reduction algorithms

Numerical experiments to Numerical experiments

block experiment

The block experiment

deflation criteria

Cutting off the last

deflation possibilities of the reduction algorithms

Numerical experiments: Deflation possibilities to Numerical experiments: Deflation possibilities

interaction between Lanczos and subspace convergence behavior

no title to Experiment 4

number of $QR$-steps

The $QR$-algorithm for the

on the $QR$-algorithms

Tests on the symmetric

on the $QR$-algorithms

to The $QR$-algorithm for the

on the symmetric eigenvalue solver

Tests on the symmetric to Cutting off the last

problem matrices

Problem matrices

reduction algorithms

no title to Numerical experiments: Deflation possibilities

Stewart's devil's stairs

Stewart's devil's stairs

Off-diagonal rank

Inverse of structured rank

Off-diagonal structure

Inverse of structured rank

One-pair matrix

Oscillation matrices | Definition and examples | The inverse of a

Operator theory

An historical overview of

Optical flow

subspace iteration in the

Orthogonal rational functions

Orthogonal rational functions to Orthogonal rational functions

Orthogonal reduction to

lower (upper) triangular semiseparable
form

Reduction to lower (upper) | Reduction to lower (upper)

Orthogonal similarity reduction

The Arnoldi(Lanczos)-Ritz values

to Hessenberg-like form

A similarity reduction applied

to symmetric semiseparable form

An orthogonal similarity reduction

Oscillation kernel

Introduction

Oscillation matrices

Oscillation matrices to The connection with eigenvalues

definition

Definition and examples

eigenvectors and eigenvalues

The connection with eigenvalues

examples

Definition and examples

properties

Definition and examples

Oscillation properties

Introduction

Outline

Outline of the thesis to Outline of the thesis

$\{p\}$-generator representable matrices

Definitions for semiseparable and | Definitions for semiseparable and | Definitions for semiseparable and | Definitions for semiseparable and | Definitions for semiseparable and | Definitions for semiseparable and | The generator definition, investigated | The generator definition, investigated

Particles

An historical overview of

Partition of a set

Inverse of structured rank

Patterned matrices

Semiseparable matrices as covariance

Pentadiagonal matrices

An historical overview of

Perfect shift

A $QR$-step maintains the

Pointwise closure of the class of semiseparable matrices

Connection between the generator

Pointwise limit

A theoretical problem, with

$QR$-step

The $QR$-factorization of Hessenberg-like to The $QR$-factorization of Hessenberg-like | The implicit $Q$-theorem

deflation

Deflation after a step to Deflation after a step | Computing the eigenvectors

destroys Hessenberg structure

A $QR$-step maintains the to A $QR$-step maintains the

destroys the Hessenberg-like structure

A $QR$-step maintains the

eigenvectors

Computing all the eigenvectors to The eigenvectors of an

essential uniqueness

Unreduced Hessenberg-like matrices

for bidiagonal matrices

The standard $QR$-method for to The standard $QR$-method for

for semiseparable matrices

Implicit $QR$-algorithms for semiseparable to One iteration of the

for upper triangular semiseparable matrices

An implicit $QR$-algorithm for to One iteration of the

Francis

The shift $\kappa$ | Conclusions and future research

Hessenberg-like matrices

An implicit $QR$-algorithm for to An implicit $QR$-step on to Chasing the disturbance in to Chasing the disturbance in

if $R$ is invertible

A $QR$-step maintains the

implicit $Q$-theorem for Hessenberg-like matrices

The implicit $Q$-theorem

implicit $Q$-theorem for Hessenberg-like matrices

to The implicit $Q$-theorem

implicit $Q$-theorem for tridiagonal

The standard $QR$-method for

implicit on a Hessenberg-like matrix

An implicit $QR$-algorithm for

implicit on a symmetric semiseparable matrix

An implicit $QR$-step on to An implicit $QR$-step on

implicit on an upper triangular semiseparable matrix

The new method via to One iteration of the

maintaining the structure

A $QR$-step maintains the to A $QR$-step maintains the

maintains the structure

A $QR$-step maintains the to A $QR$-step maintains the

of semiseparable matrices

Generalizations of the nullity

of structured rank matrices

Generalizations of the nullity

on a generator representable semiseparable matrix

A $QR$-step maintains the

on a Hessenberg-like matrix

A $QR$-step maintains the

on a Hessenberg-like plus diagonal matrix

A $QR$-step maintains the

semiseparable matrices

An implicit $QR$-algorithm for to Proof of the correctness

theoretical results

Theoretical results for $QR$-algorithms to The implicit $Q$-theorem

with shift on a matrix with weakly lower triangular rank $1$

A $QR$-step maintains the

without shift on a Hessenberg-like matrix

A $QR$-step maintains the

QRHL

An implicit $QR$-step without

QRSHL

An implicit $QR$-step with

QRSS

An implicit $QR$-step without

QRSSS

An implicit $QR$-step with

QRSUS

An implicit $QR$-step with

QRUS

An implicit $QR$-step without

Quasiseparable matrices

Other types of representations | An historical overview of

$\range(A)$

Sufficient conditions to obtain

Ranks of matrices

block off-diagonal rank

Inverse of structured rank

off-diagonal rank

Inverse of structured rank

weakly lower triangular block

Inverse of structured rank

weakly upper triangular block

Inverse of structured rank

Rational interpolation

An historical overview of

Rayleigh shift

The shift $\kappa$

Rectangular distribution

Some other matrices

Recurrence relation

Orthogonal rational functions

Recursively semiseparable

Discretization of integral equations

REDHL

Reduction to unreduced form

REDSS

Reduction to unreduced form

Reduction algorithms

Reduction algorithms to semiseparable to Reduction to lower (upper)

Reduction to

convergence properties

Convergence properties of the to subspace iteration

lower (upper) triangular semiseparable
form

Reduction to lower (upper) | Reduction to lower (upper)

similar Hessenberg-like form

A similarity reduction applied

similar semiseparable form

An orthogonal similarity reduction

unreduced form

The reduction to unreduced to The reduction to unreduced

unreduced symmetric semiseparable form

Unreduced symmetric semiseparable matrix

unreduced upper triangular semiseparable form

Unreduced upper triangular semiseparable

REDUS

Reduction to unreduced form

Representation

The definition of a

Givens-vector

A new representation for to A new representation for

other types

Other types of representations to Other types of representations

representation map

The definition of a

swapping of the Givens-vector representation

Swapping the representation to Swapping the representation

unsymmetric semiseparable matrices

An historical overview of

with diagonal and subdiagonal

The representation based on to The representation based on

with generators

The generator representation to The generator representation

Representation for unsymmetric semiseparable matrices

An historical overview of

Representation map

The definition of a

Representation of

tridiagonal matrices

The definition of a

Retrieving the Givens-vector representation

Retrieving the representation from to Retrieving the representation from

Ritz values

The Arnoldi(Lanczos)-Ritz values

necessary conditions

Necessary conditions to obtain to Necessary conditions to obtain

sufficient conditions

Sufficient conditions to obtain to Some general remarks

Routines

BHL

Constructing the matrices from

BLS

Constructing the matrices from

BRHL

Retrieving the Givens-vector representation

BRLS

Retrieving the Givens-vector representation

BRSS

Retrieving the Givens-vector representation

BRUS

Retrieving the Givens-vector representation

BSS

Constructing the matrices from

BUS

Constructing the matrices from

CHL

Reduction Algorithms

CLS

Reduction Algorithms

CSS

Reduction Algorithms

CUS

Reduction Algorithms

DETHL

Various tools

DETLS

Various tools

DETSS

Various tools

DETUS

Various tools

EIGHL

The eigen(singular)value decomposition

EIGSS

The eigen(singular)value decomposition

HLEIG

Eigenvalue and singular value

MULHL

Various tools

MULLS

Various tools

MULSS

Various tools

MULUS

Various tools

QRHL

An implicit $QR$-step without

QRSHL

An implicit $QR$-step with

QRSS

An implicit $QR$-step without

QRSSS

An implicit $QR$-step with

QRSUS

An implicit $QR$-step with

QRUS

An implicit $QR$-step without

REDHL

Reduction to unreduced form

REDSS

Reduction to unreduced form

REDUS

Reduction to unreduced form

SOLVHLD

$QR$-solver

SOLVSSD

$QR$-solver

SR

Various tools

SSEIG

Eigenvalue and singular value

SVDUS

The eigen(singular)value decomposition

USSVD

Eigenvalue and singular value

Scattering theory

An historical overview of

Semiseparable matrices

Hessenberg-like

Definitions for semiseparable and

Semiseparable and related matrices

Semiseparable and related matrices, to Conclusions

Semiseparable kernel

An historical overview of

Semiseparable matrices

Definitions for semiseparable and | Definitions for semiseparable and to Definitions for semiseparable and

as discretization matrices

Discretization of integral equations

covariance matrices

Semiseparable matrices as covariance

generalized

An historical overview of

generator representable

The generator definition, investigated

Givens-vector representation

A new representation for

historical overview

An historical overview of to An historical overview of

$LU$-decomposition

Generalizations of the nullity

lower $\{p\}$-Hessenberg-like

Definitions for semiseparable and

one-pair matrices

Oscillation matrices

oscillation matrices

Oscillation matrices

overview

An overview of semiseparable to An historical overview of

$\{p\}$-generator representable

Definitions for semiseparable and | Definitions for semiseparable and | Definitions for semiseparable and | The generator definition, investigated | The generator definition, investigated

problem matrices

Some examples

$QR$-decomposition

Generalizations of the nullity

quasiseparable

Other types of representations | An historical overview of

recursively

Discretization of integral equations

representation with diagonal and subdiagonal

The representation based on

semiseparability rank $p$

Definitions for semiseparable and

semiseparable

Definitions for semiseparable and

semiseparable plus diagonal

Definitions for semiseparable and

sequentially

Other types of representations | Discretization of integral equations | An historical overview of

unreduced

Unreduced Hessenberg-like matrices

upper $\{p\}$-Hessenberg-like

Definitions for semiseparable and

weakly semiseparable

An historical overview of

Semiseparable plus diagonal matrices

Definitions for semiseparable and

Separable kernel

An historical overview of

Sequentially semiseparable

Other types of representations | Discretization of integral equations | An historical overview of

Sets

a partition of

Inverse of structured rank

cardinality

The nullity theorem

difference

The nullity theorem

minus

The nullity theorem

Shift

The shift $\kappa$ | The shift $\kappa$

double implicit shift

The shift $\kappa$

perfect shift

A $QR$-step maintains the

Rayleigh shift

The shift $\kappa$

Wilkinson shift

The shift $\kappa$

Signal processing

An historical overview of

Similarity reduction to

Hessenberg-like form

A similarity reduction applied

semiseparable form

An orthogonal similarity reduction

Single-pair matrix

Oscillation matrices

Singular values

of bidiagonal matrices

The standard $QR$-method for to The standard $QR$-method for

of upper triangular semiseparable matrices

An implicit $QR$-algorithm for to One iteration of the

Software package

Software package for semiseparable to Eigenvalue and singular value

SOLVHLD

$QR$-solver

SOLVSSD

$QR$-solver

SR

Various tools

SSEIG

Eigenvalue and singular value

Stationary time series

Some other matrices

Statistics

An historical overview of

Stewart's devil's stairs

Stewart's devil's stairs

Strictly lower triangular structure

see weak-
ly lower triangular structure

Strictly upper triangular structure

see weak-
ly upper triangular structure

Structure

Definitions for semiseparable and

block off-diagonal structure

Inverse of structured rank

off-diagonal structure

Inverse of structured rank

$p$-upper triangular

Definitions for semiseparable and

strictly lower triangular

see weakly lower triangular

strictly upper triangular

see weakly upper triangular

subdiagonal

see weakly lower triangular

superdiagonal

see weakly upper triangular

upper triangular

Definitions for semiseparable and

weakly lower triangular block structure

Inverse of structured rank

weakly upper triangular

Definitions for semiseparable and

weakly upper triangular block structure

Inverse of structured rank

Subdiagonal structure

see weakly lower triangular structure

Subspace iteration

subspace iteration in the | subspace iteration

in an orthogonal reduction to Hessenberg-like form

Convergence properties of the | subspace iteration

in an orthogonal reduction to semiseparable form

subspace iteration in the

interaction with the Lanczos-Ritz values

subspace iteration and the

Sufficient conditions for the Lanczos-Ritz values

Sufficient conditions to obtain

Superdiagonal structure

see weakly upper triangular structure

SVDUS

The eigen(singular)value decomposition

Time varying linear systems

An historical overview of

Toeplitz matrices

An historical overview of to An historical overview of

Totally nonnegative matrices

Definition and examples

Totally positve matrices

Definition and examples

Trapezoidal rule

Discretization of integral equations

Tridiagonal matrices

the representation

The definition of a

$u(i:j)$

Connection between the generator

Unreduced

for matrices coming from the reduction algorithms of Part II

The reduction to unreduced

Hessenberg matrices

Unreduced Hessenberg-like matrices

Hessenberg-like matrices

Unreduced Hessenberg-like matrices to Unreduced Hessenberg-like matrices

Hessenberg-like matrix

Unreduced Hessenberg-like matrices

number

The implicit $Q$-theorem

reduction to

The reduction to unreduced to The reduction to unreduced

sum of rank $1$ plus strictly upper triangular

The implicit $Q$-theorem

symmetric semiseparable matrices

Unreduced symmetric semiseparable matrix

upper triangular semiseparable matrices

Unreduced upper triangular semiseparable

Upper triangular structure

Definitions for semiseparable and

USSVD

Eigenvalue and singular value

Vandermonde matrix

Definition and examples

Weakly lower triangular block rank

Inverse of structured rank

Weakly lower triangular block structure

Inverse of structured rank

Weakly semiseparable matrices

An historical overview of

Weakly upper triangular block rank

Inverse of structured rank

Weakly upper triangular block structure

Inverse of structured rank

Weakly upper triangular structure

Definitions for semiseparable and

Wilkinson shift

The shift $\kappa$

$Z$-matrices

An historical overview of

Zero-tailed matrix

The implicit $Q$-theorem