Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory

Energy levels, resonanees, vibrations, feature extraetion, faetor analysis - the names vary from discipline to diseipline; however, all involve eigenvalue/eigenveetor eomputations. An engineer or physicist who is modeling a physieal proeess, strueture, or deviee is eonstrained to seleet a model for whieh the subsequently-required eomputations ean be performed. This eonstraint often leads to redueed order or redueed size models whieh may or may not preserve all of the important eharaeteristies of the system being modeled. Ideally, the modeler should not be foreed to make such apriori reduetions. It is our intention to provide here proeedures wh ich will allow the direct and suceessful solution of many large 'symmetrie' eigenvalue problems, so that at least in problems where the computations are of this type there will be no need for model reduetion. Matrix eigenelement eomputations can be c1assified as smalI, medium, or large seale, in terms of their relative degrees of difficulty as measured by the amount of computer storage and time required to eomplete the desired eomputations. A matrix eigenvalue problem is said to be sm all scale if the given matrix has order smaller than 100. Well-documented and reliable FORTRAN pro grams exist for small scale eigenelement computations, see in particular ElS PACK [1976,1977]. Typically those programs explicitly trans form the given matrix into a simpler canonieal form. The eigenelement eomputations are then performed on the canonical form.

Contenu
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 ofComplex 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.