A linear eigenvalue algorithm for the nonlinear eigenvalue problem

DOI10.1007/s00211-012-0453-0zbMath1256.65043OpenAlexW2078578549MaRDI QIDQ453331

Karl Meerbergen, Elias Jarlebring, Wim Michiels

Publication date: 19 September 2012

Published in: Numerische Mathematik (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s00211-012-0453-0

zbMATH Keywords

Chebyshev polynomialsnumerical examplesnonlinear eigenvalue problemKrylov subspace methodArnoldi algorithmdelay eigenvalue problemoperator eigenvalue problem

Mathematics Subject Classification ID

Numerical computation of eigenvalues and eigenvectors of matrices (65F15) Numerical solution of nonlinear eigenvalue and eigenvector problems (65H17)

Related Items (32)

On spurious solutions encountered in Helmholtz scattering resonance computations in \(\mathbb{R}^d\) with applications to nano-photonics and acoustics ⋮ Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structures ⋮ A Structure Preserving Shift-Invert Infinite Arnoldi Algorithm for a Class of Delay Eigenvalue Problems with Hamiltonian Symmetry ⋮ Spectra, signless Laplacian and Laplacian spectra of complementary prisms of graphs ⋮ Preconditioned Eigensolvers for Large-Scale Nonlinear Hermitian Eigenproblems with Variational Characterizations. II. Interior Eigenvalues ⋮ A Block Preconditioned Harmonic Projection Method for Large-Scale Nonlinear Eigenvalue Problems ⋮ A mathematical and numerical investigation of the hemodynamical origins of oscillations in microvascular networks ⋮ Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map ⋮ Robust Rational Approximations of Nonlinear Eigenvalue Problems ⋮ Infinite GMRES for Parameterized Linear Systems ⋮ A Padé approximate linearization algorithm for solving the quadratic eigenvalue problem with low-rank damping ⋮ Contour Integration for Eigenvector Nonlinearities ⋮ The nonlinear eigenvalue problem ⋮ A rank-exploiting infinite Arnoldi algorithm for nonlinear eigenvalue problems ⋮ A smallest singular value method for nonlinear eigenvalue problems ⋮ The infinite Lanczos method for symmetric nonlinear eigenvalue problems ⋮ The Infinite Bi-Lanczos Method for Nonlinear Eigenvalue Problems ⋮ A panoramic sketch about the robust stability of time-delay systems and its applications ⋮ A successive quadratic approximations method for nonlinear eigenvalue problems ⋮ On restarting the tensor infinite Arnoldi method ⋮ Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property ⋮ Sylvester-based preconditioning for the waveguide eigenvalue problem ⋮ Chebyshev interpolation for nonlinear eigenvalue problems ⋮ Compact Two-Sided Krylov Methods for Nonlinear Eigenvalue Problems ⋮ A Newton-Type Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling ⋮ Krylov Approximation of Linear ODEs with Polynomial Parameterization ⋮ Localization Theorems for Nonlinear Eigenvalue Problems ⋮ Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation ⋮ PDE eigenvalue iterations with applications in two-dimensional photonic crystals ⋮ Nonlinearizing Two-parameter Eigenvalue Problems ⋮ Compact Rational Krylov Methods for Nonlinear Eigenvalue Problems ⋮ The Waveguide Eigenvalue Problem and the Tensor Infinite Arnoldi Method

Uses Software

• Matlab
• eigs
• NLEVP
• ARPACK
• Chebfun

Cites Work

• Unnamed Item
• A block Newton method for nonlinear eigenvalue problems
• Structured polynomial eigenproblems related to time-delay systems
• Nonlinear Rayleigh-Ritz iterative method for solving large scale nonlinear eigenvalue problems
• Implicit QR algorithms for palindromic and even eigenvalue problems
• Introduction to functional differential equations
• An Arnoldi method for nonlinear eigenvalue problems
• Jacobi-Davidson type methods for generalized eigenproblems and polynomial eigenproblems
• Pseudospectral approximation of eigenvalues of derivative operators with nonlocal boundary conditions
• A numerical method for nonlinear eigenvalue problems using contour integrals
• Solving Rational Eigenvalue Problems via Linearization
• Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form
• NLEVP
• Harmonic and refined Rayleigh–Ritz for the polynomial eigenvalue problem
• The Quadratic Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
• Residual Inverse Iteration for the Nonlinear Eigenvalue Problem
• Inverse Iteration, Ill-Conditioned Equations and Newton’s Method
• ARPACK Users' Guide
• Spectral Methods in MATLAB
• Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods
• A Krylov Method for the Delay Eigenvalue Problem
• SOAR: A Second-order Arnoldi Method for the Solution of the Quadratic Eigenvalue Problem
• Algorithms for the Nonlinear Eigenvalue Problem
• Nichtlineare Behandlung von Eigenwertaufgaben
• The principle of minimized iterations in the solution of the matrix eigenvalue problem

This page was built for publication: A linear eigenvalue algorithm for the nonlinear eigenvalue problem