Numerical methods and algorithms. Experimental.zip

# MultiParEig # This is a joined work with Andrej Muhič, who wrote part of the code, among other things he contributed the staircase algorithm that is used to solve a singular multiparameter eigenvalue problem. If you use this toolbox to solve a singular multiparameter eigenvalue problem, please cite the reference: A. Muhič, B. Plestenjak: On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl. 432 (2010) 2529-2542. **A MATLAB toolbox for multiparameter eigenvalue problems** Version 2.3 Toolbox contains numerical methods for multiparameter eigenvalue problems. Other contributors: * Pavel Holoborodko, Advanpix LLC, http://www.advanpix.com/ Methods chebdif_mp and lagdif_mp are based on chebdif and lagdif from Matlab toolbox [DMSUITE](http://www.mathworks.com/matlabcentral/fileexchange/29-dmsuite) by J.A.C. Weideman. # Short description of the problem In a matrix two-parameter eigenvalue problem, which has the form A1 x = lambda B1 x + mu C1 x, A2 y = lambda B2 y + mu C2 y, we are looking for an eigenvalue (lambda,mu) and nonzero eigenvectors x,y such that the above system is satisfied. A two-parameter eigenvalue problem is related to a pair of generalized eigenvalue problems Delta1 z = lambda Delta0 z, Delta2 z = mu Delta0 z, where Delta0, Delta1 and Delta2 are operator determinants Delta0 = kron(C2, B1) - kron(B2, C1) Delta1 = kron(C2, A1) - kron(A2, C1) Delta2 = kron(A2, B1) - kron(B2, A1) and z = kron(x,y). We say that the problem is nonsingular when Delta0 is nonsingular. The above can straightforward be generalized to three or more parameters. In many applications a partial differential equation has to be solved on some domain that allows the use of the method of separation of variables. In several coordinate systems separation of variables applied to the Helmholtz, Laplace, or Schrödinger equation leads to a multiparameter eigenvalue problem, some important cases are Mathieu's system, Lamé's system, and a system of spheroidal wave functions. A generic two-parameter boundary value eigenvalue problem has the form p1(x1) y1''(x1) + q1(x1) y1'(x1) + r1(x2) y1(x1) = lambda s1(x1) y1(x1) + mu s2(x2) y1(x1), p2(x2) y2''(x2) + q2(x2) y2'(x2) + r2(x2) y2(x2) = lambda s2(x2) y2(x2) + mu s2(x2) y2(x2), where x1 in [a1,b1] and x2 in [a2,b2] together with the boundary conditions. Such system can be discretized into a matrix two-parameter eigenvalue problem, where a good method of choice is the Chebyshev collocation. ### Functions in the toolbox can be used to: ### * compute Delta matrices for a multiparameter eigenvalue problem * solve a nonsingular or singular multiparameter eigenvalue problem with arbitrary number of parameters (the limit is the overall size of the corresponding Delta matrices), * compute few eigenvalues and eigenvectors of a two-parameter eigenvalue problem using implicitly restarted Arnoldi or Krylov-Schur method, * compute few eigenvalues and eigenvectors of a two- or three-parameter eigenvalue problem using the Jacobi-Davidson method or the subspace iteration method * refine an eigenpair using the tensor Rayleigh quotient iteration * discretize a two- or three-parameter boundary value eigenvalue problem with the Chebyshev collocation (package Dmsuite is required) into a matrix two- or three-parameter eigenvalue problem, * solve a quadratic two-parameter eigenvalue problem. * most of the methods support multiprecision using [Advanpix Multiprecision Computing Toolbox](http://www.advanpix.com/) ### Dependence on other toolboxes: ### * in Matlab older than 2014a method *twopareigs* run faster if package [lapack](http://www.mathworks.com/matlabcentral/fileexchange/16777-lapack) is installed * multiprecision examples require [Advanpix Multiprecision Computing Toolbox](http://www.advanpix.com/) ## Main methods ## ### Two-parameter eigenvalue problems (2EP): ### * *twopareig*: solve a 2EP (set options to solve a singular 2EP) * *twopareigs*: few eigenvalues and eigenvectors of a 2EP using implicitly restarted Arnoldi method or Krylov-Schur method * *twopareigs_si*: subspace iteration with Arnoldi expansion for a 2EP * *twopareigs_jd*: Jacobi-Davidson method for a 2EP * *trqi*: tensor Rayleigh quotient iteration for a 2EP * *twopar_delta*: Delta matrices for a 2EP ### Three-parameter eigenvalue problems (3EP): ### * *threepareig*: solve a 3EP (set options to solve a singular 3EP) * *threepareigs*: few eigenvalues and eigenvectors of a 3EP using implicitly restarted Arnoldi method * *threepareigs_si*: subspace iteration with Arnoldi expansion for a 3EP * *threepareigs_jd*: Jacobi-Davidson method for a 3EP * *trqi_3p*: tensor Rayleigh quotient iteration for a 3EP * *threepar_delta*: Delta matrices for a 3EP ### Multi-parameter eigenvalue problems (MEP): ### * *multipareig*: solve a MEP (set options to solve a singular MEP) * *trqi_np*: tensor Rayleigh quotient iteration for a MEP * *multipar_delta*: Delta matrices for a MEP ### Two and three-parameter boundary differential equations: ### * *bde2mep*: discretizes two-parameter BDE as a two-parameter matrix pencil using the Chebyshev collocation * *bde3mep*: discretizes three-parameter BDE as a three-parameter matrix pencil using the Chebyshev collocation ### Quadratic two-parameter eigenvalue problem: ### * *quad_twopareig*: eigenvalues and eigenvectors of a quadratic two-parameter eigenvalue problem * *linearize_quadtwopar*: linearize quadratic two-parameter matrix pencil into a linear two-parameter matrix pencil ### Other applications: ### * *double_eig*: values of parameter lambda such that A + lambda*B has a multiple eigenvalue See directory Examples with many examples in the following subdirectories: * MEP : demos for basic solvers for multiparameter eigenvalue problems * BdeMep : demo functions that recreate numerical results from B. Plestenjak, C.I. Gheorghiu, M.E. Hochstenbach: Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems, J. Comp. Phys. 298 (2015) 585-601 * Multiprecision : demos that use the multiprecision to obtain more accurate results * SingularMep : examples that use the staircase algorithm to solve singular multiparameter eigenvalue problems * JD : examples that use the Jacobi-Davidson method ## References ## This is a list of references for some of the implemented algorithms. Please cite an appropriate reference if you use the toolbox in your paper. * twopareig (nonsingular), twopareigs_jd: M.E. Hochstenbach, T. Košir, B. Plestenjak: A Jacobi-Davidson type method for the two-parameter eigenvalue problem, SIAM J. Matrix Anal. Appl. 26 (2005) 477-497 * twopareig (singular), quad_twopareig, linearize_quadtwopar: A. Muhič, B. Plestenjak: On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl. 432 (2010) 2529-2542 * twopareigs, twopareigs_si: K. Meerbergen, B. Plestenjak: A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants, Numer. Linear Algebra Appl. 22 (2015) 1131-1146 * bde2mep, bde3mep: B. Plestenjak, C.I. Gheorghiu, M.E. Hochstenbach: Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems, J. Comput. Phys. 298 (2015) 585-601 * twopareigs_jd (harmonic Ritz values): M.E. Hochstenbach, B. Plestenjak: Harmonic Rayleigh-Ritz for the multiparameter eigenvalue problem, Electron. Trans. Numer. Anal. 29 (2008) 81-96. * double_eig: A. Muhič, B. Plestenjak: A method for computing all values lambda such that A + lambda*B has a multiple eigenvalue, Linear Algebra Appl. 440 (2014) 345-359 MultiParEig toolbox B. Plestenjak, University of Ljubljana FreeBSD License, see LICENSE.txt