Abstract
Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “\(m=\infty \)” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature.
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Aurentz, J.L., Trefethen, L.N.: Block operators and spectral discretizations. SIAM Rev. 59, 423–446 (2017)
Austin, A.P., Krishnamoorthy, M., Leyffer, S., Mrenna, S., Müller, J., Schulz, H.: Practical algorithms for multivariate rational approximation. Comput. Phys. Commun. 261, 107663 (2021)
Barnett, A.H., Betcke, T.: Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains. J. Comp. Phys. 14, 7003–7026 (2008)
Battles, Z., Trefethen, L.N.: An extension of MATLAB to continuous functions and operators. SIAM J. Sci. Comput. 25, 1743–1770 (2004)
Betcke, T., Trefethen, L.N.: Computed eigenmodes of planar regions. Contemp. Math. 412, 297–314 (2006)
Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer (2009)
Boutry, G., Elad, M., Golub, G.H., Milanfar, P.: The generalized eigenvalue problem for nonsquare pencils using a minimal perturbation approach. SIAM J. Matrix Anal. Appl. 27, 582–601 (2005)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods, 2nd edn. Dover (2001)
Brubeck, P.D., Nakatsukasa, Y., Trefethen, L.N.: Vandermonde with Arnoldi. SIAM Rev. 63, 405–415 (2021)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge U Press (2003)
Driscoll, T.A., Hale, N.: Rectangular spectral collocation. IMA J. Numer. Anal. 36, 108–132 (2016)
Driscoll, T.A., Hale, N., Trefethen, L.N.: Chebfun Guide. Pafnuty Press, Oxford (2014). see also www.chebfun.org
Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comp. Math. 9, 69–95 (1998)
Fasshauer, G.: Meshfree Approximation Methods with MATLAB. World Scientific (2007)
Gander, M.J., Wanner, G.: From Euler, Ritz, and Galerkin to modern computing. SIAM Rev. 54, 627–666 (2012)
Gopal, A., Trefethen, L.N.: Solving Laplace problems with corner singularities via rational functions. SIAM J. Numer. Anal. 57, 2074–2094 (2019)
Hashemi, B., Nakatsukasa, Y.: Least-squares spectral methods for ODE eigenvalue problems. SIAM J. Sci. Comput. 44, A3244–A3264 (2022)
Hokanson, J.M.: Multivariate rational approxmation using a stabilized Sanathanan–Koerner iteration arXiv:2009.10803v1 (2020)
Ito, S., Murota, K.: An algorithm for the generalized eigenvalue problem for nonsquare matrix pencils by minimal perturbation approach. SIAM J. Matrix Anal. Appl. 37, 409–419 (2016)
Jiang, B.: The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetism. Springer (1998)
Kamath, A., Manzhos, S.: Inverse multiquadratic functions as the basis for the rectangular collocation method to solve the vibrational Schrödinger equation. Mathematics 6, 253 (2018)
Manzhos, S., Yamashita, K., Carrington Jr., T.: On the advantages of a rectangular matrix collocation equation for computing vibrational spectra from small basis sets. Chem. Phys. Lett. 511, 434–439 (2011)
Millar, R.F.: Singularities and the Rayleigh hypothesis for soutions to the Helmholtz equation. IMA J. Appl. Math. 37, 155–171 (1986)
Monk, P., Wang, D. -Q.: A least-squares method for the Helmholtz equation. Comput. Methods Appl. Mech. Engrg. 175, 121–136 (1999)
Nakatsuji, H., Nakashima, H., Kurokawa, Y., Ishikawa, A.: Solving the Schrödinger equation of atoms and molecules without analytical integration based on the free iterative-complement-interaction wave function. Phys. Rev. Lett. 99, 240402 (2007)
Nakatsukasa, Y., Trefethen, L.N.: Reciprocal-log approximation and planar PDE solvers. SIAM J. Numer. Anal. 59, 2801–2822 (2021)
Platte, R.B.: How fast do radial basis function interpolants of analytic functions converge IMA J. Numer. Anal. 31, 1578–1597 (2011)
Platte, R.B., Driscoll, T.A.: Computing eigenmodes of elliptic operators using radial basis functions. Computers and Math. with Applics. 48, 561–576 (2004)
Platte, R.B., Driscoll, T.A.: Eigenvalue stability of radial basis function discretizations for time-dependent problems. Comput. Math. Applics. 51, 1251–1268 (2006)
Saad, Y.: Numerical Methods for Large Eigenvalue Problems, revised edn. SIAM (2011)
Shen, J., Tang, T., Wang, L. -L.: Spectral Methods: Algorithms, Analysis and Applications. Springer (2011)
Trefethen, L.N.: Householder triangularization of a quasimatrix. IMA J. Numer. Anal. 30, 887–897 (2010)
Trefethen, L.N.: Series solution of Laplace problems. ANZIAM J. 60, 1–26 (2018)
Trefethen, L.N., Nakatsukasa, Y., Weideman, J.A.C.: Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math. 147, 227–254 (2021)
Wright, T.G., Trefethen, L.N.: Pseudospectra of rectangular matrices. IMA J. Numer. Anal. 22, 501–519 (2002)
Zhu, K.W.: Multivariate Least-Squares Approximations in Irregular Domains via Vandermonde with Arnoldi. MSc diss. University of Oxford (2021)
Acknowledgements
We are grateful for helpful suggestions from Alex Barnett, Timo Betcke, Toby Driscoll, Mark Embree, Greg Fasshauer, Abi Gopal, Dave Hewett, Norm Levenberg, Rodrigo Platte, Euan Spence, and Alex Townsend—and also from two conscientious referees.
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Communicated by: Tobin Driscoll
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Hashemi, B., Nakatsukasa, Y. & Trefethen, L.N. Rectangular eigenvalue problems. Adv Comput Math 48, 80 (2022). https://doi.org/10.1007/s10444-022-09994-8
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DOI: https://doi.org/10.1007/s10444-022-09994-8
Keywords
- Eigenvalue problems
- Quasimatrix
- Spectral methods
- Method of fundamental solutions
- Lightning solver
- Vandermonde with Arnoldi
- Helmholtz equation
- Fourier extension