**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 | 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
**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
**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