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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



Susan Blackford 2000-11-20