A note on harmonic Ritz values and their reciprocals

DOI10.1002/nla.653zbMath1240.65125OpenAlexW1741406338MaRDI QIDQ3090783

Christof Vömel

Publication date: 2 September 2011

Published in: Numerical Linear Algebra with Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1002/nla.653

zbMATH Keywords

Rayleigh quotientmatrix pencilspreconditioned conjugate gradientslarge sparse matrixRitz valueharmonic Ritz valuesymmetric eigenproblemgeneralized symmetric positive definite eigenprobleminterior eigenpairmatrix-free computationsparse iterative eigensolver

Mathematics Subject Classification ID

Computational methods for sparse matrices (65F50) Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Preconditioners for iterative methods (65F08)

Related Items

Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices, The Convergence of Harmonic Ritz Vectors and Harmonic Ritz Values, Revisited, A homogeneous Rayleigh quotient with applications in gradient methods

Uses Software

• eigs
• JDQR
• JDQZ
• IRAM

Cites Work

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• Generalizations of harmonic and refined Rayleigh-Ritz
• The use of bulk states to accelerate the band edge state calculation of a semiconductor quantum dot
• State-of-the-art eigensolvers for electronic structure calculations of large scale nano-systems
• Quasi-kernel polynomials and their use in non-Hermitian matrix iterations
• Improved algorithms for the lowest few eigenvalues and associated eigenvectors of large matrices
• The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices
• Efficient expansion of subspaces in the Jacobi-Davidson method for standard and generalized eigenproblems
• Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems
• New iterative methods for solution of the eigenproblem
• Harmonic and refined extraction methods for the singular value problem, with applications in least squares problems
• Matrix Algorithms
• Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems Under Limited Memory. Part II: Seeking Many Eigenvalues
• The Jacobi-Davidson method
• Generalizations of Davidson’s Method for Computing Eigenvalues of Sparse Symmetric Matrices
• Implicit Application of Polynomial Filters in a k-Step Arnoldi Method
• The Rayleigh Quotient Iteration and Some Generalizations for Nonnormal Matrices
• The Davidson Method
• Templates for the Solution of Algebraic Eigenvalue Problems
• The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors
• Nearly Optimal Preconditioned Methods for Hermitian Eigenproblems under Limited Memory. Part I: Seeking One Eigenvalue
• The Rotation of Eigenvectors by a Perturbation. III