# MultiParEig

Version 2.8.0.0 (328 KB) by
Toolbox for multiparameter and singular eigenvalue problems
Updated 29 Aug 2024

This is a joined work with Andrej Muhič, who wrote part of the code, in particular the staircase algorithm for a singular multiparameter eigenvalue problem. If you use the toolbox to solve a singular MEP, please cite: A. Muhič, B. Plestenjak: On the quadratic two-parameter eigenvalue problem and its linearization, Linear Algebra Appl. 432 (2010) 2529-2542.
Toolbox contains numerical methods for multiparameter eigenvalue problems (MEPs), rectangular MEPs and singular generalized eigenvalue problems (GEPs)
A matrix two-parameter eigenvalue problem (2EP) has the form
A1*x = lambda*B1*x + mu*C1*x,
A2*y = lambda*B2*y + mu*C2*y,
and we are looking for an eigenvalue (lambda,mu) and nonzero eigenvectors x,y. A 2EP is related to a pair of GEPs
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). The 2EP is nonsingular when Delta0 is nonsingular. This can be generalized to 3EP and MEP.
In many applications a PDE has to be solved on a 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 MEP, some 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 2EP, where a good method of choice is the Chebyshev collocation.
Functions in the toolbox can:
- compute Delta matrices for a MEP
- solve a nonsingular or singular MEP with arbitrary number of parameters (the limitation is the size of the corresponding Delta matrices),
- compute few eigenpairs of a 2EP using implicitly restarted Arnoldi or Krylov-Schur method,
- compute few eigenpairs of a 2EP or 3EP using the Jacobi-Davidson 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 into a 2EP or 3EP,
- find finite regular eigenvalues of a singular pencil using rank-completin perturbations,
- most of the methods support multiprecision using Advanpix Multiprecision Computing Toolbox.
Main functions in the toolbox
2EP:
- twopareig: solve a 2EP (set options to solve a singular 2EP)
- twopareigs: few eigenpairs using implicitly restarted Arnoldi or Krylov-Schur method
- twopareigs_si: subspace iteration with Arnoldi expansion
- twopareigs_jd: Jacobi-Davidson method
- trqi: tensor Rayleigh quotient iteration
- twopar_delta: Delta matrices
Singular eigenvalue problem:
- double_eig: lambda such that A+lambda*B has a multiple eigenvalue
- singgep: finite regular eigenvalues of a singular GEP
3EP:
- threepareig: solve a 3EP (set options to solve a singular 3EP)
- threepareigs: few eigenpairs using implicitly restarted Arnoldi method
- threepareigs_si: subspace iteration with Arnoldi expansion
- threepareigs_jd: Jacobi-Davidson method
- trqi_3p: tensor Rayleigh quotient iteration
- threepar_delta: Delta matrices
MEP:
- multipareig: solve a MEP (set options to solve a singular MEP)
- trqi_np: tensor Rayleigh quotient iteration
- multipar_delta: Delta matrices
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
- quad_twopareig: eigenpairs of a Q2EP
- linearize_quadtwopar: linearize Q2EP as a linear two-parameter matrix pencil
Rectangular MEP:
- rect_multipareig: solve a linear rectangular multiparameter eigenvalue problem
- arma11, arma21, lti2: returns stationary points for the ARMA(1,1), ARMA(2,1), LTI(2) models
See folder Examples with many demos. In particular:
- folder BdeMep contains numerical examples 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.
- folder SingGep contains numerical examples from: M.E. Hochstenbach, C. Mehl, B. Plestenjak: Solving singular generalized eigenvalue problems by a rank-completing perturbation, SIAM J. Matrix Anal. Appl. 40 (2019) 1022-1046 and M.E. Hochstenbach, C. Mehl, B. Plestenjak: Solving singular generalized eigenvalue problems part II: projection and augmentation, arXiv 2208.01359
- folder Subspace3 contains numerical examples from: M.E. Hochstenbach, K. Meerbergen, E. Mengi, B. Plestenjak: Subspace methods for 3-parameter eigenvalue problems, Numer. Linear Algebra Appl. 26 (2019) e2240
- folder RectangularMEP contains numerical examles from M.E.Hochstenbach, T.Kosir, B.Plestenjak: On the solution of rectangular multiparameter eigenvalue problems, arXiv 2212.01867
See Contents.m for references for the methods and please cite an appropriate reference if you use the toolbox in your paper.

### Cite As

Bor Plestenjak (2024). MultiParEig (https://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig), MATLAB Central File Exchange. Retrieved .

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Inspired by: DMSUITE, lapack

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#### Examples/Subspace3

Version Published Release Notes
2.8.0.0

Eigenvalues are computed by default using new randomized solver for joint eigenvalues based on Rayleigh quotients and eigenvectors of a random linear combination, more efficient solver more rectangular eigenvalue problems

2.7.0.0

Support for rectangular multiparameter eig. problems. Solutions of ARMA(1,1), ARMA(2,1) and LTI(2) models using RMEPs. See M.E.Hochstenbach, T.Košir, B.Plestenjak: On the solution of rectangular multiparameter eigenvalue problems, arXiv 2212.01867

2.6.0.0

Improved method singgep for finite eigenvalues of a singular GEP with new numerical examples, see M.E. Hochstenbach, C. Mehl, B. Plestenjak: Solving singular generalized eigenvalue problems part II: projection and augmentation, arXiv 2208.01359

2.5.0.0

New method singgep for finite regular eigenvalues of a singular GEP with several numerical examples.

M.E. Hochstenbach, C. Mehl, B. Plestenjak: Solving singular generalized eigenvalue problems by a rank-completing perturbation, arXiv:1805:07657

2.4.0.0

Improved subspace methods for 3-parameter problems and numerical examples from M.E. Hochstenbach, K. Meerbergen, E. Mengi, B. Plestenjak, Subspace methods for 3-parameter eigenvalue problems, arXiv:1802:07386. All methods now support multiprecision.

2.3.0.0

Support for multiprecision using Advanpix Multiprecision Computing Toolbox. Default in twopareigs is twopareigs_ks, that now works for complex matrices as well. Methods from DMSUITE are replaced by methods that support multiprecision. Improved TRQI.

2.2.0.0

Updated Jacobi-Davidson for 2EP and 3EP, option of faster (and less accurate) rank revealing QR instead of SVD for singular MEPs, new aerolastic flutter examples, minor corrections and updates, output of rank choices and gaps for singular MEPs

2.1.0.0

Updated description of the toolbox with an acknowledgement to Andrej Muhič and a list of references. Updated references in some files.

2.0.0.0

New methods for two- and three-parameter eigenvalue problems with large matrices. Support for singular problems with three or more parameters. Tensor Rayleigh quotient iteration generalized to three or more parameters. Construction of Delta matrices.

1.0.0.0