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FEAST for differential eigenvalue problems. (English) Zbl 1439.65088

MSC:
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
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[1] P.-A. Absil, R. Sepulchre, P. Van Dooren, and R. Mahony, Cubically convergent iterations for invariant subspace computation, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 70-96, https://doi.org/10.1137/S0895479803422002. · Zbl 1075.65049
[2] A. J. Akbarfam and A. Mingarelli, Higher order asymptotics of the eigenvalues of Sturm-Liouville problems with a turning point of arbitrary order, Can. Appl. Math. Q., 12 (2004), pp. 275-301. · Zbl 1110.34061
[3] A. J. Akbarfam and A. B. Mingarelli, Higher order asymptotic distribution of the eigenvalues of nondefinite Sturm-Liouville problems with one turning point, J. Comput. Appl. Math., 149 (2002), pp. 423-437. · Zbl 1019.34081
[4] N. Alikakos, P. Bates, and X. Chen, Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Amer. Math. Soc., 351 (1999), pp. 2777-2805. · Zbl 0929.35067
[5] L. M. Anguas, M. I. Bueno, and F. M. Dopico, A comparison of eigenvalue condition numbers for matrix polynomials, Linear Algebra Appl., 564 (2019), pp. 170-200, https://doi.org/https://doi.org/10.1016/j.laa.2018.11.031. · Zbl 1407.15010
[6] F. V. Atkinson and A. B. Mingarelli, Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems, J. Reine Angew. Math., 375 (1987), pp. 380-393. · Zbl 0599.34026
[7] A. Austin, Eigenvalues of Differential Operators by Contour Integral Projection, http://www.chebfun.org/examples/ode-eig/ContourProjEig.html, May 2013.
[8] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, 2000, https://doi.org/10.1137/1.9780898719581. · Zbl 0965.65058
[9] A. Barnett and A. Hassell, Fast computation of high-frequency Dirichlet eigenmodes via spectral flow of the interior Neumann-to-Dirichlet map, Commun. Pure Appl. Math., 67 (2014), pp. 351-407. · Zbl 1288.65160
[10] D. Bindel and M. Zworski, Theory and Computation of Resonances in 1D Scattering, http://www.cs.cornell.edu/ bindel/cims/resonant1d/theo2.html, October 2006.
[11] I. Bogaert, Iteration-free computation of Gauss-Legendre quadrature nodes and weights, SIAM J. Sci. Comput., 36 (2014), pp. A1008-A1026, https://doi.org/10.1137/140954969. · Zbl 1297.65025
[12] F. Chatelin, Spectral Approximation of Linear Operators, SIAM, Philadelphia, 2011, https://doi.org/10.1137/1.9781611970678. · Zbl 1214.01004
[13] L. Chen and H.-P. Ma, Approximate solution of the Sturm-Liouville problems with Legendre-Galerkin-Chebyshev collocation method, Appl. Math. Comput., 206 (2008), pp. 748-754. · Zbl 1157.65431
[14] M. J. Colbrook and A. C. Hansen, On the infinite-dimensional QR algorithm, Numer. Math., 143 (2019), pp. 17-83. · Zbl 07088063
[15] B. Ćurgus, A. Fleige, and A. Kostenko, The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions, Integral Equations Operator Theory, 77 (2013), pp. 533-557. · Zbl 1292.34080
[16] E. B. Davies, Linear Operators and Their Spectra, Cambridge Stud. Adv. Math. 106, Cambridge University Press, Cambridge, 2007.
[17] J. J. Dongarra, B. Straughan, and D. W. Walker, Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems, Appl. Numer. Math., 22 (1996), pp. 399-434. · Zbl 0867.76025
[18] T. A. Driscoll, N. Hale, and L. N. Trefethen, Chebfun Guide, https://www.chebfun.org/docs/guide/, 2014.
[19] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc., 64 (1958), pp. 217-274. · Zbl 0088.32102
[20] S. Filip, A. Javeed, and L. N. Trefethen, Smooth random functions, random ODEs, and Gaussian processes, SIAM Rev., 61 (2019), pp. 185-205, https://doi.org/10.1137/17M1161853. · Zbl 1412.42006
[21] J. Gary, Computing eigenvalues of ordinary differential equations by finite differences, Math. Comp., 19 (1965), pp. 365-379. · Zbl 0131.14302
[22] C.-I. Gheorghiu, Spectral Methods for Non-standard Eigenvalue Problems: Fluid and Structural Mechanics and Beyond, Springer, Cham, 2014. · Zbl 1298.65166
[23] M. A. Gilles and A. Townsend, Continuous analogues of Krylov subspace methods for differential operators, SIAM J. Numer. Anal., 57 (2019), pp. 899-924, https://doi.org/10.1137/18M1177810. · Zbl 1411.65047
[24] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Ser. Math. Sci. 3, Johns Hopkins University Press, Baltimore, 2012.
[25] T. R. Goodman, The numerical solution of eigenvalue problems, Math. Comp., 19 (1965), pp. 462-466. · Zbl 0261.65056
[26] J. Gopalakrishnan, L. Grubišić, and J. Ovall, Filtered Subspace Iteration for Selfadjoint Operators, preprint, https://arxiv.org/abs/1709.06694v1, 2017. · Zbl 1436.65067
[27] J. Gopalakrishnan, L. Grubišić, and J. Ovall, Spectral discretization errors in filtered subspace iteration, Math. Comp., 89 (2020), pp. 203-228. · Zbl 1436.65067
[28] N. Hale and Y. Nakatsukasa, Rayleigh Quotient Iteration for an Operator, http://www.chebfun.org/examples/ode-eig/RayleighQuotient.html, March 2017.
[29] S. M. Han, H. Benaroya, and T. Wei, Dynamics of transversely vibrating beams using four engineering theories, J. Sound Vibration, 225 (1999), pp. 935-988. · Zbl 1235.74075
[30] H. V. Henderson and S. R. Searle, On deriving the inverse of a sum of matrices, SIAM Rev., 23 (1981), pp. 53-60, https://doi.org/10.1137/1023004. · Zbl 0451.15005
[31] T. Kato, Perturbation Theory for Linear Operators, Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin, New York, 1976. · Zbl 0342.47009
[32] J. Kestyn, E. Polizzi, and P. T. P. Tang, FEAST eigensolver for non-Hermitian problems, SIAM J. Sci. Comput., 38 (2016), pp. S772-S799, https://doi.org/10.1137/15M1026572. · Zbl 1352.65119
[33] R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Ration. Mech. Anal., 154 (2000), pp. 3-51. · Zbl 0980.35030
[34] R. S. Laugesen and M. C. Pugh, Properties of steady states for thin film equations, European J. Appl. Math., 11 (2000), pp. 293-351. · Zbl 1041.76008
[35] E. D. Napoli, E. Polizzi, and Y. Saad, Efficient estimation of eigenvalue counts in an interval, Numer. Linear Algebra Appl., 23 (2016), pp. 674-692. · Zbl 1413.65092
[36] S. Olver and A. Townsend, A fast and well-conditioned spectral method, SIAM Rev., 55 (2013), pp. 462-489, https://doi.org/10.1137/120865458. · Zbl 1273.65182
[37] S. Olver and A. Townsend, A practical framework for infinite-dimensional linear algebra, in Proceedings of the 1st Workshop for High Performance Technical Computing in Dynamic Languages, IEEE Press, 2014, pp. 57-62.
[38] S. Olver et al., ApproxFun v,0.10.8 Julia Package, https://github.com/JuliaApproximation/ApproxFun.jl, 2018.
[39] S. A. Orszag, Accurate solution of the Orr–Sommerfeld stability equation, J. Fluid Mech., 50 (1971), pp. 689–703, https://doi.org/10.1017/S0022112071002842. · Zbl 0237.76027
[40] A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. I, Arch. Rational Mech. Anal., 1 (1957), pp. 233-241. · Zbl 0083.12002
[41] R. D. Pantazis and D. B. Szyld, Regions of convergence of the Rayleigh quotient iteration method, Numer. Linear Algebra Appl., 2 (1995), pp. 251-269. · Zbl 0831.65041
[42] B. N. Parlett, The Rayleigh quotient iteration and some generalizations for nonnormal matrices, Math. Comp., 28 (1974), pp. 679-693. · Zbl 0293.65023
[43] E. Polizzi, Density-matrix-based algorithm for solving eigenvalue problems, Phys. Rev. B, 79 (2009), 115112, https://doi.org/10.1103/PhysRevB.79.115112.
[44] M. Reed and B. Simon, Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, San Diego, 1980. · Zbl 0459.46001
[45] W. Rudin, Functional Analysis, Internat. Ser. Pure Appl. Math., 2nd ed., McGraw-Hill, New York, 1991.
[46] Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, 1992. · Zbl 0991.65039
[47] Y. Saad, Analysis of subspace iteration for eigenvalue problems with evolving matrices, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 103-122, https://doi.org/10.1137/141002037.
[48] N. Sanford, K. Kodama, J. D. Carter, and H. Kalisch, Stability of traveling wave solutions to the Whitham equation, Phys. Lett. A, 378 (2014), pp. 2100-2107. · Zbl 1331.35310
[49] J. Shen and L.-L. Wang, Fourierization of the Legendre-Galerkin method and a new space-time spectral method, Appl. Numer. Math., 57 (2007), pp. 710-720. · Zbl 1118.65111
[50] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lect. Anal. II, Princeton University Press, Princeton, NJ, 2003.
[51] G. W. Stewart, Error bounds for approximate invariant subspaces of closed linear operators, SIAM J. Numer. Anal., 8 (1971), pp. 796-808, https://doi.org/10.1137/0708073. · Zbl 0232.47010
[52] G. W. Stewart, Simultaneous iteration for computing invariant subspaces of non-Hermitian matrices, Numer. Math., 25 (1976), pp. 123-136. · Zbl 0328.65025
[53] P. T. P. Tang and E. Polizzi, FEAST as a subspace iteration eigensolver accelerated by approximate spectral projection, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 354-390, https://doi.org/10.1137/13090866X. · Zbl 1303.65018
[54] F. Tisseur, Backward error and condition of polynomial eigenvalue problems, Linear Algebra Appl., 309 (2000), pp. 339-361. · Zbl 0955.65027
[55] A. Townsend and S. Olver, The automatic solution of partial differential equations using a global spectral method, J. Comput. Phys., 299 (2015), pp. 106-123. · Zbl 1352.65579
[56] A. Townsend and L. N. Trefethen, Continuous analogues of matrix factorizations, Proc. A, 471 (2015), 20140585. · Zbl 1372.65095
[57] L. N. Trefethen, Householder triangularization of a quasimatrix, IMA J. Numer. Anal., 30 (2009), pp. 887-897. · Zbl 1202.65049
[58] L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 1997. · Zbl 0874.65013
[59] L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, Princeton, NJ, 2005. · Zbl 1085.15009
[60] L. N. Trefethen and M. R. Trummer, An instability phenomenon in spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023, https://doi.org/10.1137/0724066. · Zbl 0636.65124
[61] W. Wasow, Linear Turning Point Theory, Appl. Math. Sci. 54, Springer-Verlag, New York, 1985. · Zbl 0558.34049
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