Lanczos algorithms for large symmetric eigenvalue computations

TLDR: This chapter discusses Lanczos Procedures with no Reorthogonalization for Real Symmetric Problems, and an Identification Test, 'Good' versus' spurious' Eigenvalues.

Abstract: 0 Preliminaries: Notation and Definitions.- 0.1 Notation.- 0.2 Special Types of Matrices.- 0.3 Spectral Quantities.- 0.4 Types of Matrix Transformations.- 0.5 Subspaces, Projections, and Ritz Vectors.- 0.6 Miscellaneous Definitions.- 1 Real' symmetric' Problems.- 1.1 Real Symmetric Matrices.- 1.2 Perturbation Theory.- 1.3 Residual Estimates of Errors.- 1.4 Eigenvalue Interlacing and Sturm Sequencing.- 1.5 Hermitian Matrices.- 1.6 Real Symmetric Generalized Eigenvalue Problems.- 1.7 Singular Value Problems.- 1.8 Sparse Matrices.- 1.9 Reorderings and Factorization of Matrices.- 2 Lanczos Procedures, Real Symmetric Problems.- 2.1 Definition, Basic Lanczos Procedure.- 2.2 Basic Lanczos Recursion, Exact Arithmetic.- 2.3 Basic Lanczos Recursion, Finite Precision Arithmetic.- 2.4 Types of Practical Lanczos Procedures.- 2.5 Recent Research on Lanczos Procedures.- 3 Tridiagonal Matrices.- 3.1 Introduction.- 3.2 Adjoint and Eigenvector Formulas.- 3.3 Complex Symmetric or Hermitian Tridiagonal.- 3.4 Eigenvectors, Using Inverse Iteration.- 3.5 Eigenvalues, Using Sturm Sequencing.- 4 Lanczos Procedures with no Reorthogonalization for Real Symmetric Problems.- 4.1 Introduction.- 4.2 An Equivalence, Exact Arithmetic.- 4.3 An Equivalence, Finite Precision Arithmetic.- 4.4 The Lanczos Phenomenon.- 4.5 An Identification Test, 'Good' versus' spurious' Eigenvalues.- 4.6. Example, Tracking Spurious Eigenvalues.- 4.7 Lanczos Procedures, Eigenvalues.- 4.8 Lanczos Procedures, Eigenvectors.- 4.9 Lanczos Procedure, Hermitian, Generalized Symmetric.- 5 Real Rectangular Matrices.- 5.1 Introduction.- 5.2 Relationships With Eigenvalues.- 5.3 Applications.- 5.4 Lanczos Procedure, Singular Values and Vectors.- 6 Nondefective Complex Symmetric Matrices.- 6.1 Introduction.- 6.2 Properties of Complex Symmetric Matrices.- 6.3 Lanczos Procedure, Nondefective Matrices.- 6.4 QL Algorithm, Complex Symmetric Tridiagonal Matrices.- 7 Block Lanczos Procedures, Real Symmetric Matrices.- 7.1 Introduction.- 7.2 Iterative Single-vector, Optimization Interpretation.- 7.3 Iterative Block, Optimization Interpretation.- 7.4 Iterative Block, A Practical Implementation.- 7.5 A Hybrid Lanczos Procedure.- References.- Author and Subject Indices.