Tridiagonal matrix

About: Tridiagonal matrix is a research topic. Over the lifetime, 3653 publications have been published within this topic receiving 62596 citations.

Lanczos algorithms for large symmetric eigenvalue computations

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.

A Parallel Algorithm for the Efficient Solution of a General Class of Recurrence Equations

Abstract: An mth-order recurrence problem is defined as the computation of the series x 1 , x 2 , ..., X N , where x i = f i (x i-1 , ..., x i-m ) for some function f i . This paper uses a technique called recursive doubling in an algorithm for solving a large class of recurrence problems on parallel computers such as the Iliac IV. Recursive doubling involves the splitting of the computation of a function into two equally complex subfunctions whose evaluation can be performed simultaneously in two separate processors. Successive splitting of each of these subfunctions spreads the computation over more processors. This algorithm can be applied to any recurrence equation of the form x i = f(b i , g(a i , x i-1 )) where f and g are functions that satisfy certain distributive and associative-like properties. Although this recurrence is first order, all linear mth-order recurrence equations can be cast into this form. Suitable applications include linear recurrence equations, polynomial evaluation, several nonlinear problems, the determination of the maximum or minimum of N numbers, and the solution of tridiagonal linear equations. The resulting algorithm computes the entire series x 1 , ..., x N in time proportional to [log 2 N] on a computer with N-fold parallelism. On a serial computer, computation time is proportional to N.

A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices

Abstract: A class of matrices ( $\Cal H$ -matrices) is introduced which have the following properties (i) They are sparse in the sense that only few data are needed for their representation (ii) The matrix-vector multiplication is of almost linear complexity (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the $\Cal H$ -matrix format are again of almost linear complexity (iv) The same statement holds for the inverse of an $\Cal H$ -matrix This paper is the first of a series and is devoted to the first introduction of the $\Cal H$ -matrix concept Two concret formats are described The first one is the simplest possible Nevertheless, it allows the exact inversion of tridiagonal matrices The second one is able to approximate discrete integral operators

LINPACK Users' Guide

Abstract: General matrices Band matrices Positive definite matrices Positive definite band matrices Symmetric Indefinite Matrices Triangular matrices Tridiagonal matrices The Cholesky decomposition The QR decomposition Updating QR and Cholesky decompositions The singular value decomposition References Basic linear algebra subprograms Timing data Program listings BLA Listings.