Index

ABLE

An Adaptively Blocked Lanczos

accuracy assessment

of GHEP

Stability and Accuracy Assessments

of GNHEP

Stability and Accuracy Assessments

of HEP

Stability and Accuracy Assessments

of NHEP

Stability and Accuracy Assessments

of SVD

Introduction

adaptive blocking

An Adaptively Blocked Lanczos

Arnoldi factorization

Arnoldi Procedure in GEMV

Arnoldi method

basic

Arnoldi Method   Y. Saad

block

Block Arnoldi Method  

implicitly restarted

Implicitly Restarted Arnoldi Method

Arnoldi procedure

Basic Algorithm | Arnoldi Procedure in GEMV

Arnoldi vector

Basic Algorithm | Arnoldi Procedure in GEMV

ARPACK

for GHEP

Software Availability.

for HEP

Software Availability

for NHEP

Software Availability

for SVD

Software Availability

backward error analysis

Numerical Stability and Conditioning

of GHEP

Transfer Residual Error to | Transfer Residual Error to

of GNHEP

Transfer Residual Errors to

of HEP

Transfer Residual Error to

of NHEP

Transfer Residual Errors to

balancing

accuracy of eigenvalues

Accuracy of Eigenvalues Computed

dense matrix

Direct Balancing

direct, dense matrix

Direct Balancing

direct, sparse matrix

Direct Balancing

iterative

Krylov Balancing Algorithms

bidiagonalization

for dense matrices

Direct Methods

for sparse matrices

Golub-Kahan-Lanczos Bidiagonalization Procedure.

biorthogonality

full

An Adaptively Blocked Lanczos

loss of

Algorithm

semi-

An Adaptively Blocked Lanczos

bisection method

for GHEP

Direct Methods

for HEP

Direct Methods

for SVD

Direct Methods

BLAS

BLAS

for structured matrices

CDS Matrix-Vector Product

sparse

Sparse BLAS

breakdown

of band Lanczos method

Basic Properties

of block Lanczos method

Basic Algorithm

of complex symmetric Lanczos method

Properties of the Algorithm

of Lanczos method

Algorithm

of symmetric indefinite Lanczos method

Algorithm

Cayley transform

Matrix Transformations

for QEP

QEP with Cayley Transform.

characteristic polynomial

Eigenvalues and Eigenvectors | Eigenvalues and Eigenvectors | Eigenvalues and Eigenvectors | Generalized Non-Hermitian Eigenproblems  

chordal metric

Conditioning | Some Combination of $A$ | Stability and Accuracy Assessments

companion matrix

Related Eigenproblems

block

Related Eigenproblems | Higher Order Polynomial Eigenvalue

complex symmetric eigenproblem

Lanczos Method for Complex

condition number

Introduction | Numerical Stability and Conditioning

of eigenvalue, GNHEP

Error Bound for Computed

of eigenvalue, NHEP

Error Bound for Computed

conditioning

of GHEP

Conditioning

of GNHEP

Conditioning | Conditioning.

of HEP

Conditioning | Conditioning

of SVD

Conditioning

congruence transformation

Equivalences (Congruences)

conjugate gradient method

covariant

Under the Hood

convergence properties

Davidson method

Davidson Method

of inexact Cayley, for GNHEP

Arnoldi Method with Inexact

of inverse iteration, for HEP

Inverse Iteration

of inverse iteration, for NHEP

Inverse Iteration

of IRAM

Convergence Properties

of IRLM

Convergence Properties

of Lanczos method, for GHEP

Convergence Properties.

of Lanczos method, for HEP

Convergence Properties

of Lanczos method, for NHEP

Convergence Properties

of power method, for HEP

Power Method

of power method, for NHEP

Power Method

of preconditioned power method

Preconditioned Shifted Power Method

of preconditioned subspace iteration

Preconditioned Simultaneous Iterations

of subspace iteration, for HEP

Subspace Dimension.

covariant

conjugate gradient method

Under the Hood

differentiation

Covariant Differentiation

Newton's method

Under the Hood

Stiefel-Grassmann gradient

Inner Products, Gradients, and

Crawford number

Some Combination of $A$

CS (cosine/sine) decomposition

Related Singular Value Problems

Davidson method

Basic Theory

definite (matrix) pencil

Generalized Hermitian Eigenproblems  

deflating subspace

Deflating Subspaces

deflation

band Lanczos method, for HEP

The Need for Deflation

band Lanczos method, for NHEP

Deflation

block Arnoldi method

Deflation.

IRAM

Deflation and Stopping Rules

IRLM

Deflation and Stopping Rules

Jacobi-Davidson method, for GNHEP

Deflation and Restart

Jacobi-Davidson method, for HEP

Deflation.

Jacobi-Davidson method, for NHEP

Schur Form and Restart

subspace iteration, for NHEP

Subspace Iteration

diagonal form

of GNHEP

Eigendecompositions

of NHEP

Eigendecompositions

differentiation

covariant

Covariant Differentiation

direct methods

for GHEP

Direct Methods

for GNHEP

Direct Methods

for HEP

Direct Methods

for NHEP

Direct Methods

for SVD

Direct Methods

divide-and-conquer method

for GHEP

Direct Methods

for HEP

Direct Methods

for SVD

Direct Methods

DQDS algorithm

Direct Methods

eigendecompositions

of GHEP

Eigendecompositions

of GNHEP

Eigendecompositions | Eigendecompositions.

of HEP

Eigendecompositions

of NHEP

Eigendecompositions

eigenspace

Introduction

equivalence transformations

Equivalences | Equivalences | Equivalences.

error bound

of eigenvalue, GHEP

Error Bounds for Computed | Error Bound for Computed

of eigenvalue, GNHEP

Error Bound for Computed

of eigenvalue, HEP

Error Bounds for Computed

of eigenvalue, NHEP

Error Bound for Computed

of eigenvalues, singular pencil

More on GUPTRI and

of eigenvector, GHEP

Error Bounds for Computed | Error Bound for Computed

of eigenvector, GNHEP

Error Bound for Computed

of eigenvector, HEP

Error Bound for Computed

of eigenvector, NHEP

Error Bound for Computed

Galerkin condition

Orthogonal Projection Methods.

gap

Conditioning | Error Bounds for Computed | Error Bounds for Computed | Introduction | Criterion for Determining the

generalized Schur-staircase form

Eigendecompositions.

generalized Hermitian eigenproblem (GHEP)

Generalized Hermitian Eigenproblems   | Generalized Hermitian Eigenvalue Problems

generalized non-Hermitian eigenproblem (GNHEP)

Generalized Non-Hermitian Eigenproblems   | Generalized Non-Hermitian Eigenvalue Problems

generalized Schur decomposition

Eigendecompositions

generalized Schur-staircase form

Generalized Schur-Staircase Form

Golub-Kahan-Lanczos method

Golub-Kahan-Lanczos Method

gradient

Stiefel-Grassmann

Inner Products, Gradients, and

Gram-Schmidt process

classical

Arnoldi Procedure in GEMV

modified

Basic Algorithm

two-sided

Algorithm | Algorithm

GUPTRI

Singular Matrix Pencils   | Singular Matrix Pencils  

harmonic Ritz value

Harmonic Ritz Values. | Computing Interior Eigenvalues

Hermitian definite pencil

Introduction | Some Combination of $A$

Hermitian eigenproblem (HEP)

Hermitian Eigenproblems   J. | Hermitian Eigenvalue Problems

Hessenberg matrix

Basic Algorithm

Hessenberg reduction

Direct Methods

ill-conditioning

Introduction

of singular pencil

Ill-Conditioning

implicitly restarted Arnoldi method (IRAM)

Implicit Restart

implicitly restarted Lanczos method (IRLM)

Implicit Restart

inexact method

Lanczos method

Preconditioned Lanczos Method

invariant subspace

Invariant Subspaces

inverse iteration

for GHEP

Inverse Iteration.

for HEP

Direct Methods | Inverse Iteration

for NHEP

Inverse Iteration

for SVD

Direct Methods

Jacobi method

for HEP

Direct Methods

for SVD

Direct Methods

Jacobi-Davidson method

Cayley transform

Jacobi-Davidson Method with Cayley

for GHEP

Jacobi-Davidson Methods  G. Sleijpen and

for GNHEP

Jacobi-Davidson Method  G. Sleijpen and

for HEP

Jacobi-Davidson Methods   G. Sleijpen

for NHEP

Jacobi-Davidson Methods   G. Sleijpen

for QEP

Jacobi-Davidson Method

Jordan (canonical) form

Eigendecompositions

Jordan structure

nearest

Nearest-Jordan Structure

nearest, sg_min example

A Sample Jordan Block

Jordan-Schur form

Eigendecompositions

Kronecker (canonical) form

Eigendecompositions. | Kronecker Canonical Form

Krylov subspace

Basic Ideas Y. Saad

Lanczos factorization

for HEP

Implicitly Restarted Lanczos Method

for NHEP

Algorithm

Lanczos method

band, for HEP

Band Lanczos Method  

band, for NHEP

Band Lanczos Method  

basic, for HEP

Lanczos Method   A.

basic, for NHEP

Lanczos Method   Z. Bai

block, for NHEP

Block Lanczos Methods  

for complex symmetric eigenproblem

Lanczos Method for Complex

for GHEP

Lanczos Methods   A.

for SVD

Golub-Kahan-Lanczos Method

implicitly restarted

Implicitly Restarted Lanczos Method

in GEMV form

Lanczos Method in GEMV

preconditioned

Preconditioned Lanczos Method

symmetric indefinite, for GNHEP

Symmetric Indefinite Lanczos Method

Lanczos vector

Implicitly Restarted Lanczos Method | Algorithm

LAPACK

for GHEP

Direct Methods

for GNHEP

Direct Methods

for HEP

Direct Methods

for NHEP

Direct Methods

for SVD

Direct Methods

linear least squares problem

Related Problems   J.

linear solver

direct

A Brief Survey of

for band matrices

Direct Solvers for Band

for dense matrices

Direct Solvers for Dense

for sparse matrices

Direct Solvers for Sparse

for structured matrices

Direct Solvers for Structured

iterative

A Brief Survey of

linearization

of PEP

Higher Order Polynomial Eigenvalue

of QEP

Transformation to Linear Form

locking

IRAM

Deflation and Stopping Rules

IRLM

Deflation and Stopping Rules

subspace iteration

Locking.

look-ahead technique

Algorithm | Algorithm | Software Availability | Software Availability

manifolds

Manifolds

MATLAB

for GHEP

Direct Methods

for GNHEP

Direct Methods

for HEP

Direct Methods

for NHEP

Direct Methods

for SVD

Direct Methods

nearest Jordan structure

Nearest-Jordan Structure

by sg_min

Nearest-Jordan Structure

sg_min example

A Sample Jordan Block

Newton's method

covariant

Under the Hood

non-Hermitian eigenproblem (NHEP)

Non-Hermitian Eigenproblems  J. Demmel | Non-Hermitian Eigenvalue Problems

nonlinear eigenproblem (NLEP)

Nonlinear Eigenproblems   J. | Nonlinear Eigenvalue Problems

numerical rank

Criterion for Determining the

numerical stability

Numerical Stability and Conditioning

of IRAM, for NHEP

Numerical Stability

of Lanczos method, for NHEP

Lanczos Method   Z. Bai

of QR algorithm

Direct Methods

numerical stability assessment

of GHEP

Stability and Accuracy Assessments

of GNHEP

Stability and Accuracy Assessments

of HEP

Stability and Accuracy Assessments

of NHEP

Stability and Accuracy Assessments

oblique projection

Oblique Projection Methods.

optimization

sg_min

MATLAB Templates

orthogonal projection

Orthogonal Projection Methods.

orthogonality

full

Algorithm

local

Algorithm

loss of

Algorithm

of Arnoldi vectors

Variants | Arnoldi Procedure in GEMV

selective

Algorithm

orthogonality constraints

Nonlinear Eigenvalue Problems with

parallelism

Parallelism   J. Dongarra and to Solvers.

inner products

Inner Products.

matrix-vector products

Matrix-Vector Products. | Matrix-Vector Products.

solver

Solvers.

vector updates

Vector Updates.

Petrov-Galerkin condition

Basic Ideas Y. Saad | Oblique Projection Methods.

polynomial acceleration

Acceleration.

polynomial eigenproblem (PEP)

Higher Order Polynomial Eigenvalue

power method

for GHEP

Power Method.

for HEP

Power Method

for NHEP

Power Method

preconditioned

subspace iteration

Preconditioned Simultaneous Iterations

preconditioned eigensolvers

Introduction

preconditioning

left-

Basic Algorithm

right-

Basic Algorithm

Procrustes problem

The Procrustes Problem

by sg_min

The Procrustes Problem

sg_min example

A Sample Procrustes Problem

projection method

Basic Ideas Y. Saad

approximate problem

Orthogonal Projection Methods.

oblique

Basic Ideas Y. Saad

orthogonal

Basic Ideas Y. Saad

refined

Refined Projection Methods.

projection process

Basic Ideas Y. Saad

purging

in IRAM

Deflation and Stopping Rules

in IRLM

Deflation and Stopping Rules

QR algorithm

for GHEP

Direct Methods

for HEP

Direct Methods

for NHEP

Direct Methods

for SVD

Direct Methods

quadratic eigenproblem (QEP)

Quadratic Eigenvalue Problems Z. Bai,

quadratic Rayleigh quotient

Introduction

quotient SVD (QSVD)

Related Singular Value Problems | Related Problems   J.

QZ algorithm

Direct Methods

rational Krylov method

Rational Krylov Subspace Method

Rayleigh quotient

Rayleigh Quotient Iteration

Stiefel-Grassmann optimization

Trace Minimization

by sg_min

Trace Minimization

sg_min example

A Sample Trace Minimization

Rayleigh quotient iteration (RQI)

Rayleigh Quotient Iteration

for GHEP

Rayleigh Quotient Iteration.

Rayleigh-Ritz procedure

Orthogonal Projection Methods.

reduced-order modeling

Application to Reduced-Order Modeling

reducing subspace

Reducing Subspaces.

refined projection method

Refined Projection Methods.

regular matrix pencil

Regular Versus Singular Problems

relatively robust representation algorithm

for HEP

Direct Methods

reorthogonalization

of Lanczos method, for GHEP

Algorithm.

of Lanczos method, for HEP

Reorthogonalization

residual vector

of GHEP

Residual Vector. | Residual Vector.

of GNHEP

Residual Vectors.

of HEP

Residual Vector.

of NHEP

Residual Vectors.

of QEP

Introduction

of SVD

Introduction

restart

of Arnoldi method, explicit

Explicit Restarts

of Arnoldi method, implicit

Implicit Restart

of block Arnoldi method

Restarting a Block Arnoldi

of Jacobi-Davidson method, for GNHEP

Deflation and Restart

of Jacobi-Davidson method, for HEP

Restart Strategy.

of Jacobi-Davidson method, for NHEP

Schur Form and Restart

of Lanczos method, implicit

Implicit Restart

Ritz

harmonic, value

Harmonic Ritz Values.

value

Orthogonal Projection Methods.

vector

Orthogonal Projection Methods.

roots of polynomial

Related Eigenproblems

ScaLAPACK

for HEP

Direct Methods

for NHEP

Direct Methods

for SVD

Direct Methods

Schur decomposition (form)

Eigendecompositions

generalized

Eigendecompositions

generalized, partial

Basic Theory

partial

Implicitly Restarted Arnoldi Method | Computing Interior Eigenvalues

simultaneous

Simultaneous Schur Decomposition Problem

simultaneous, sg_min example

A Sample Simultaneous Schur

sg_min

modifying

Modifying the Templates

nearest Jordan structure problem

Nearest-Jordan Structure

Procrustes problem

The Procrustes Problem

Rayleigh quotient

Trace Minimization

simultaneous Schur decomposition

Simultaneous Schur Decomposition Problem

shift selection

for IRLM

Shift Selection

shift-and-invert

Spectral Transformations   R. | Matrix Transformations

for GNHEP

Shift-and-Invert.

for QEP

Shift-and-Invert QEP.

inexact

Inexact Shift-and-Invert

IRAM, for NHEP

Eigenvector Computation with Spectral

Lanczos method, for GHEP

Lanczos Algorithm with SI.

Lanczos method, for HEP

Spectral Transformation

Lanczos method, for NHEP

Convergence Properties

rational Krylov

Rational Krylov Subspace Method

symmetric indefinite Lanczos

Algorithm

similarity transformation

Equivalences (Similarities) | Equivalences (Similarities)

simultaneous Schur decomposition

by sg_min

Simultaneous Schur Decomposition Problem

singular (matrix) pencil

Singular Case | Regular Versus Singular Problems

singular subspace

Singular Subspaces

singular value decomposition (SVD)

Singular Values and Singular | Introduction

compact

Decompositions | Introduction

generalized

Related Singular Value Problems | Related Problems   J.

more generalized

Related Problems   J.

partial

Decompositions

thin

Decompositions | Introduction

truncated

Decompositions | Introduction

sparse matrix storage

Sparse Matrix Storage Formats to Skyline Storage

BCRS

Block Compressed Row Storage to Block Compressed Row Storage

CCS

Compressed Column Storage to Compressed Column Storage

CDS

Compressed Diagonal Storage

CRS

Compressed Row Storage to Compressed Row Storage

JDS

Jagged Diagonal Storage to Jagged Diagonal Storage

SKS

Skyline Storage to Skyline Storage

spectral transformation

Spectral Transformations   R.

for QEP

Spectral Transformations for QEP

Lanczos method, for HEP

Spectral Transformation

Stiefel-Grassmann optimization

Nonlinear Eigenvalue Problems with

sg_min

MATLAB Templates

subspace iteration

for HEP

Subspace Iteration

for NHEP

Subspace Iteration

preconditioned

Preconditioned Simultaneous Iterations

subspace of approximants

Basic Ideas Y. Saad

symmetric definite pencil

Introduction

preconditioned solver

Introduction

symmetric indefinite matrix pencil

Symmetric Indefinite Lanczos Method

transfer function

Application to Reduced-Order Modeling

transformation to standard problem

GHEP

Transformation to Standard Problem

GNHEP

Transformation to Standard Problems

Weierstrass form

Eigendecompositions

Weierstrass-Schur form

Eigendecompositions