Error Analysis of the Lanczos Algorithm for the Nonsymmetric Eigenvalue Problem

DOI10.2307/2153404zbMath0809.65029OpenAlexW2095282848MaRDI QIDQ4286595

Zhaojun Bai

Publication date: 28 March 1995

Full work available at URL: https://doi.org/10.2307/2153404

zbMATH Keywords

convergenceerror analysisconditioningloss of orthogonalitynon-symmetric matricescancellation errorbi-orthogonal Lanczos algorithm

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Roundoff error (65G50)

Related Items

The Gauss quadrature for general linear functionals, Lanczos algorithm, and minimal partial realization, The second-order biorthogonalization procedure and its application to quadratic eigenvalue problems, Maintaining convergence properties of BiCGStab methods in finite precision arithmetic, Reliable updated residuals in hybrid Bi-CG methods, Arnoldi versus nonsymmetric Lanczos algorithms for solving matrix eigenvalue problems, Abstract perturbed Krylov methods, Unnamed Item, Towards backward perturbation bounds for approximate dual Krylov subspaces, A new quasi-minimal residual method based on a biconjugate \(A\)-orthonormalization procedure and coupled two-term recurrences, An augmented analysis of the perturbed two-sided Lanczos tridiagonalization process, Backward Error Analysis for an Eigenproblem Involving Two Classes of Matrices, A dynamic thick restarted semi-refined ABLE algorithm for computing a few selected eigentriplets of large nonsymmetric matrices, Backward errors for eigenproblem of two kinds of structured matrices, Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem, Extending the eigCG algorithm to nonsymmetric Lanczos for linear systems with multiple right-hand sides, Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems

Uses Software

• LAPACK
• SRRIT
• IRAM
• Algorithm 694

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Rational Krylov sequence methods for eigenvalue computation
• A survey of Lanczos procedures for very large real 'symmetric' eigenvalue problems
• Analysis of the symmetric Lanczos algorithm with reorthogonalization methods
• Lanczos algorithms and field of value rotations for symmetric matrix pencils
• Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
• A new look at the Lanczos algorithm for solving symmetric systems of linear equations
• Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices
• Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
• Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights
• Matrix interpretations and applications of the continued fraction algorithm
• Super-matrix methods
• Numerical solution of large nonsymmetric eigenvalue problems
• A Look-Ahead Lanczos Algorithm for Unsymmetric Matrices
• A Simultaneous Iteration Algorithm for Real Matrices
• Residual Bounds on Approximate Eigensystems of Nonnormal Matrices
• Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations
• Implicit Application of Polynomial Filters in a k-Step Arnoldi Method
• Reduction to Tridiagonal Form and Minimal Realizations
• Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form
• Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix
• The Lanczos Algorithm with Selective Orthogonalization
• Algorithm 776: SRRIT
• Computing selected eigenvalues of sparse unsymmetric matrices using subspace iteration
• Algorithm 694
• An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices
• Laguerre's Method Applied to the Matrix Eigenvalue Problem
• A generalized Bairstow algorithm