Collocation approximation to eigenvalues of an ordinary differential equation: the principle of the thing

Authors:
Carl de Boor and Blair Swartz

Journal:
Math. Comp. **35** (1980), 679-694

MSC:
Primary 65L15

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572849-1

MathSciNet review:
572849

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that simple eigenvalues of an *m*th order ordinary differential equation are approximated within by collocation at Gauss points with piecewise polynomial functions of degree on a mesh . The same rate is achieved by certain averages in case the eigenvalue is not simple. The argument relies on an extension and simplification of Osborn's recent results concerning the approximation of eigenvalues of compact linear maps.

**[1]**Kendall Atkinson,*Convergence rates for approximate eigenvalues of compact integral operators*, SIAM J. Numer. Anal.**12**(1975), 213–222. MR**0438746**, https://doi.org/10.1137/0712020**[2]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328**, https://doi.org/10.1137/0710052**[3]**C. DE BOOR & B. SWARTZ, "Collocation approximation to eigenvalues of an ordinary differential equation: Numerical illustrations." (Submitted to*Math. Comp*.)**[4]**C. DE BOOR &. B. SWARTZ, "Local piecewise polynomial projection methods for an ODE which give high-order convergence at knots." (Submitted to*Math. Comp*.)**[5]**J. H. Bramble and J. E. Osborn,*Rate of convergence estimates for nonselfadjoint eigenvalue approximations*, Math. Comp.**27**(1973), 525–549. MR**0366029**, https://doi.org/10.1090/S0025-5718-1973-0366029-9**[6]**John H. Cerutti and Seymour V. Parter,*Collocation methods for parabolic partial differential equations in one space dimension*, Numer. Math.**26**(1976), no. 3, 227–254. MR**0433922**, https://doi.org/10.1007/BF01395944**[7]**Jim Douglas Jr. and Todd Dupont,*Collocation methods for parabolic equations in a single space variable*, Lecture Notes in Mathematics, Vol. 385, Springer-Verlag, Berlin-New York, 1974. Based on 𝐶¹-piecewise-polynomial spaces. MR**0483559****[8]**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****[9]**Heinz-Otto Kreiss,*Difference approximations for boundary and eigenvalue problems for ordinary differential equations*, Math. Comp.**26**(1972), 605–624. MR**0373296**, https://doi.org/10.1090/S0025-5718-1972-0373296-3**[10]**JAMES F. LATHROP, "Using B-splines to solve the one-dimensional Schrödinger equation,"*J. Comput. Phys*. (Submitted.)**[11]**John E. Osborn,*Spectral approximation for compact operators*, Math. Comput.**29**(1975), 712–725. MR**0383117**, https://doi.org/10.1090/S0025-5718-1975-0383117-3**[12]**G. M. Vainikko,*Rapidity of convergence of approximation methods in eigenvalue problems*, Ž. Vyčisl. Mat. i Mat. Fiz.**7**(1967), 977–987 (Russian). MR**0221746****[13]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422****[14]**R. Winther,*A collocation method for eigenvalue problems*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 96–105. MR**0388791****[15]**R. WINTHER,*En Kollokasjonsmetode for Egenverdiproblemer*, Thesis for the Cand. Real Degree, University of Oslo, Norway, 1973.**[16]**K. A. Wittenbrink,*High order projection methods of moment- and collocation-type for nonlinear boundary value problems*, Computing (Arch. Elektron. Rechnen)**11**(1973), no. 3, 255–274 (English, with German summary). MR**0400724**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L15

Retrieve articles in all journals with MSC: 65L15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1980-0572849-1

Keywords:
Eigenvalues,
compact linear map,
ordinary differential equation,
collocation,
piecewise polynomial,
superconvergence

Article copyright:
© Copyright 1980
American Mathematical Society