Peaks, plateaus, numerical instabilities in a Galerkin minimal residual pair of methods for solving \(Ax=b\)

DOI10.1016/0168-9274(95)00086-0zbMath0855.65022OpenAlexW1964281402MaRDI QIDQ1917421

Jane K. Cullum

Publication date: 16 February 1997

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0168-9274(95)00086-0

zbMATH Keywords

numerical examplesnumerical instabilitiesbidiagonalizationGalerkin-type methodBLanczosBMinresresidual minimization methodunsymmetric linear equation systems

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Iterative numerical methods for linear systems (65F10)

Related Items (7)

Lanczos-type variants of the COCR method for complex nonsymmetric linear systems ⋮ A framework for generalized conjugate gradient methods -- with special emphasis on contributions by Rüdiger Weiß ⋮ A note on the convergence behavior of GMRES ⋮ A theoretical overview of Krylov subspace methods ⋮ Analysis of peaks and plateaus in a Galerkin/minimal residual pair of methods for solving \(Ax=b\) ⋮ The effect of non-optimal bases on the convergence of Krylov subspace methods ⋮ A note on convergence of quasi-minimal residual smoothing

Uses Software

• LSQR
• CRAIG

Cites Work

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• The convergence behavior of Ritz values in the presence of close eigenvalues
• Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences
• Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem
• The superlinear convergence behaviour of GMRES
• Residual smoothing and peak/plateau behavior in Krylov subspace methods
• A Lanczos Algorithm for Computing Singular Values and Vectors of Large Matrices
• GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
• LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
• Refined Interlacing Properties
• Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix
• Residual Smoothing Techniques for Iterative Methods
• Lanczos Algorithms for Large Symmetric Eigenvalue Computations
• Calculating the Singular Values and Pseudo-Inverse of a Matrix
• Bidiagonalization of Matrices and Solution of Linear Equations
• A Theoretical Comparison of the Arnoldi and GMRES Algorithms

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