scholarly journals NEP

2021 ◽  
Vol 47 (3) ◽  
pp. 1-29
Author(s):  
Carmen Campos ◽  
Jose E. Roman
Keyword(s):  
Large Scale ◽  
Eigenvalue Problems ◽  
Nonlinear Function ◽  
Test Problems ◽  
Scalar Parameter ◽  
Nonlinear Eigenvalue Problems ◽  
Square Roots ◽  
The Past ◽  
Parallel Library ◽  
Nonlinear Eigenvalue

SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

scholarly journals The nonlinear eigenvalue problem

Acta Numerica ◽  
2017 ◽  
Vol 26 ◽  
pp. 1-94 ◽  
Author(s):  
Stefan Güttel ◽  
Françoise Tisseur
Keyword(s):  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Rational Interpolation ◽  
Science And Engineering ◽  
Nonlinear Eigenvalue Problems ◽  
Numerical Examples ◽  
Matlab Code ◽  
The Past ◽  
Mathematical Properties ◽  
Nonlinear Eigenvalue

Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour integration and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

scholarly journals Automatic rational approximation and linearization of nonlinear eigenvalue problems

2021 ◽  
Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Javier Pérez ◽  
Bart Vandereycken
Keyword(s):  
Rational Approximation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Space Representation ◽  
Rational Approximations ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
State Space Representation ◽  
Rational Polynomial ◽  
Nonlinear Eigenvalue

Abstract We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

scholarly journals Erratum to: “Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems”

2017 ◽  
Vol 38 (3) ◽  
pp. 1598-1598
Author(s):  
Karl Meerbergen ◽  
Emre Mengi ◽  
Wim Michiels ◽  
Roel Van Beeumen
Keyword(s):  
Large Scale ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

scholarly journals On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator

2020 ◽  
Vol 26 ◽  
pp. 118
Author(s):  
Roland Glowinski ◽  
Shingyu Leung ◽  
Hao Liu ◽  
Jianliang Qian
Keyword(s):  
Exact Solutions ◽  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Gradient Flow ◽  
Finite Element Approximation ◽  
Test Problems ◽  
Element Approximation ◽  
Nonlinear Eigenvalue Problems ◽  
Monge Ampere ◽  
Nonlinear Eigenvalue

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.

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