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Mathematics of Computation

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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
MathSciNet review: 572849
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Abstract: It is shown that simple eigenvalues of an mth order ordinary differential equation are approximated within $ \mathcal{O}(\vert\Delta {\vert^{2k}})$ by collocation at Gauss points with piecewise polynomial functions of degree $ < m + k$ on a mesh $ \Delta $. 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.

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Keywords: Eigenvalues, compact linear map, ordinary differential equation, collocation, piecewise polynomial, superconvergence
Article copyright: © Copyright 1980 American Mathematical Society