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On the convergence of Hill's method
Author(s):
Christopher
W.
Curtis;
Bernard
Deconinck.
Journal:
Math. Comp.
79
(2010),
169-187.
MSC (2000):
Primary 34L16, 65L07
Posted:
July 6, 2009
MathSciNet review:
2552222
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Abstract:
Hill's method is a means to numerically approximate spectra of linear differential operators with periodic coefficients. In this paper, we address different issues related to the convergence of Hill's method. We show the method does not produce any spurious approximations, and that for self-adjoint operators, the method converges in a restricted sense. Furthermore, assuming convergence of an eigenvalue, we prove convergence of the associated eigenfunction approximation in the  -norm. These results are not restricted to selfadjoint operators. Finally, for certain selfadjoint operators, we prove that the rate of convergence of Hill's method to the least eigenvalue is faster than any polynomial power.
References:
- 4
- 1.
- K. Atkinson, The numerical solution of integral equations of the second kind, Cambridge University Press, Cambridge, 1997. MR 1464941 (99d:65364)
- 2.
- K. Atkinson and W. Han, Theoretical numerical analysis: A functional analysis framework, Springer, New York, NY, 2005. MR 2153422 (2006a:65001)
- 3.
- A. Böttcher, Infinite matrices and projection methods, Lectures on Operator Theory and Its Applications, American Mathematical Society, Providence, RI, 1996, pp. 3-72. MR 1364444 (96m:47016)
- 4.
- A. Böttcher and B. Silbermann, Analysis of Toeplitz operators, Springer-Verlag, New York, NY, 1990. MR 1071374 (92e:47001)
- 5.
- J.B. Conway, A course in functional analysis, 2nd ed., Springer, New York, NY, 1990. MR 1070713 (91e:46001)
- 6.
- B. Deconinck and J. N. Kutz, Computing spectra of linear operators using the Floquet-Fourier-Hill method, J. Comput. Phys. 219 (2006), 296-321. MR 2273379 (2007g:37078)
- 7.
- L.M. Delves and T.L. Freeman, Analysis of global expansion methods: Weakly asymptotically diagonal systems, Academic Press, New York, NY, 1981. MR 624434 (84a:65002)
- 8.
- I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of linear operators: Vol. I, Birkhäuser-Verlag, Basel, 1990. MR 1130394 (93d:47002)
- 9.
- I. Gohberg, S. Goldberg, and N. Krupnick, Traces and determinants of linear operators, Birkhäuser-Verlag, Basel, 2000. MR 1744872 (2001b:47035)
- 10.
- G. W. Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon, Acta Math. 8 (1886), no. 1, 1-36. MR 1554690
- 11.
- P.D. Hislop and I.M. Sigal, Introduction to spectral theory: With applications to Schrödinger operators, Springer, New York, NY, 1996. MR 1361167 (98h:47003)
- 12.
- T. Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, New York, N.Y., 1976. MR 0407617 (53:11389)
- 13.
- M.A. Krasnosel'skiı, G.M. Vainikko, P.P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko, Approximate solution of operator equations, Wolters-Noordhoff, Groningen, 1972. MR 0385655 (52:6515)
- 14.
- P.D. Lax, Functional analysis, Wiley, New York, NY, 2002. MR 1892228 (2003a:47001)
- 15.
- J. Locker, Functional analysis and two-point differential operators, Longman Scientific and Technical, New York, NY, 1986. MR 865984 (88d:34001)
- 16.
- A.W. Naylor and G.R. Sell, Linear operator theory in engineering and science, Springer-Verlag, New York, NY, 1982. MR 672108 (83j:46001)
- 17.
- H. Poincaré, Sur les déterminants d'ordre infini, Bull. Soc. Math. France 14 (1886), 77-90. MR 1503979
- 18.
- M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, NY, 1972. MR 0493419 (58:12429a)
- 19.
- H. Sandberg, E. Mollerstedt, and B. Bernhardsson, Frequency-domain analysis of linear time-periodic systems, IEEE Trans. of Auto. Control 50 (2005), 1971-1983. MR 2186267 (2006h:93067)
- 20.
- B. Sandstede, Stability of travelling waves, Handbook of Dynamical Systems, Vol. 2 (B. Fiedler, ed.), Elsevier Science B.V., 2002, pp. 983-1055. MR 1901069 (2004e:37121)
- 21.
- M. Schechter, Principles of functional analysis, 2nd ed., American Mathematical Society, Providence, RI, 2001. MR 1861991 (2002j:46001)
- 22.
- H. von Koch, Sur les systèmes d'une infinité d'équations linéaires à une infinité d'inconnues, C. R. Cong. d. Math. Stockholm (1910), 43-61.
- 23.
- J. Zhou, Zeros and poles of linear continuous-time periodic systems: Definitions and properties, IEEE Trans. of Auto. Control 53 (2008), 1998-2011. MR 2459573
- 24.
- J. Zhou, T. Hagiwara, and M. Araki, Spectral characteristics and eigenvalues computation of the harmonic state operators in continuous-time periodic systems, Systems and Control Letters 53 (2004), 141-155. MR 2091839 (2005k:93047)
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Additional Information:
Christopher
W.
Curtis
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3420
Email:
curtchr@amath.washington.edu
Bernard
Deconinck
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, Washington 98195-3420
DOI:
10.1090/S0025-5718-09-02277-7
PII:
S 0025-5718(09)02277-7
Received by editor(s):
October 7, 2008
Received by editor(s) in revised form:
February 13, 2009
Posted:
July 6, 2009
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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