For tridiagonals \(T\) replace \(T\) with \(LDL\)
type: Item
instance of: scholarly article
title: For tridiagonals \(T\) replace \(T\) with \(LDL\)
zbMATH Open document ID: 0970.65032
DOI: 10.1016/S0377-0427(00)00394-0
author: Beresford N. Parlett
published in: Journal of Computational and Applied Mathematics
publication date: 2001-10-07T00:00:00Z
review text: The author discusses two of the ideas needed to compute eigenvectors that are orthogonal without making use of the Gram-Schmidt procedure when some of the eigenvalues are tightly clustered. In the first of the new schemes, the radical new goal is to compute an approximate eigenvector for a given approximate eigenvalue with a relative residual property. In the second scheme, due to the relative gaps in the spectrum the origin is shifted and the triangular factorization is used. In the development, the recently discovered differential stationary QD algorithms are used. Examples of both ideas, using four by four systems, are given.
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 1546543
zbMATH Keywords: LDU factorization
zbMATH Keywords: orthogonal eigenvectores
zbMATH Keywords: clustered eigenvalues
zbMATH Keywords: numerical examples
zbMATH Keywords: eigenvalue
zbMATH Keywords: triangular factorization
zbMATH Keywords: QD algorithm
reviewed by: R. P. Tewarson
describes a project that uses: LAPACK
MaRDI profile type: Publication
cites work: Q4841244
cites work: Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices
cites work: An analysis of the HR algorithm for computing the eigenvalues of a matrix
cites work: Q4348513
cites work: An Efficient Implementation of the Nonsymmetric Lanczos Algorithm
cites work: Q4327451
cites work: Relative Perturbation Techniques for Singular Value Problems
cites work: Accurate singular values and differential qd algorithms
cites work: Fernando's solution to Wilkinson's problem: An application of double factorization
cites work: Orthogonal Eigenvectors and Relative Gaps
cites work: Relatively robust representations of symmetric tridiagonals
cites work: Q4348515
cites work: An implementation of the dqds algorithm (positive case)
cites work: Der Quotienten-Differenzen-Algorithmus
cites work: Q5735511
cites work: Q3996941
cites work: The DQR algorithm, basic theory, convergence, and conditional stability
Recommended article: Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple
Recommended article: Q4502690
Recommended article: Some tridiagonal determinants
Recommended article: Tridiagonal-Diagonal Reduction of Symmetric Indefinite Pairs
Recommended article: Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair
Recommended article: Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
Recommended article: Lidstone approximation on the triangle
Recommended article: Tridiagonal matrices
Recommended article: Triangles and triple products of Laplace eigenfunctions
Recommended article: Tridiagonal Substitution Hamiltonians
description: scientific article; zbMATH DE number 1546543