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On the condition number of certain Rayleigh-Ritz-Galerkin matrices

Published online by Cambridge University Press:  17 April 2009

Bernard J. Omodei
Bernard J. Omodei
Affiliation:
Department of Mathematics, University of Manchester, Manchester, England.
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Abstract

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Martin H. Schultz [Bull. Amer. Math. Soc. 76 (1970), 840–844] has investigated the spectral condition number of the Rayleigh-Ritz-Galerkin matrices that arise when normalized B-spline coordinate functions are used to approximate the solution of a class of linear, self-adjoint, elliptic boundary value problems in one dimension. This paper shows how results analogous to those of Schultz [op. cit.] can be established under weaker assumptions. We also extend the results to boundary value problems in higher dimensions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 3 , December 1976 , pp. 329 - 338
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]de Boor, Carl, “On uniform approximation by splines”, J. Approximation Theory 1 (1968), 219–235.CrossRefGoogle Scholar
[2]de Boor, Carl, “The quasi-interpolant as a tool in elementary polynomial spline theory”, Approximation theory, 269–276 (Proc. Internat. Sympos., Univ. Texas, Austin, Texas, 1973. Academic Press, New York, London, 1973).Google Scholar
[3]de Boor, C. and Fix, G.J., “Spline approximation by quasiinterpolants”, J. Approximation Theory 8 (1973), 19–45.CrossRefGoogle Scholar
[4]Curry, H.B. and Schoenberg, I.J., “On Pólya frequency functions IV: the fundamental spline functions and their limits”, J. Analyse Math. 17 (1966), 71–107.CrossRefGoogle Scholar
[5]Михлин, С.Г., Численная реалиэация вариационных методов (Izdat. “Nauka”, Moscow, 1966). Mikhlin, S.G., The numerical performance of variational methods (translated by R.S., Anderssen. Wolters-Hoordhoff, Groningen, 1971).Google Scholar
[6]Omodei, Bernard J., “Stability of the Rayleigh-Ritz-Galerkin procedure for elliptic boundary value problems” (PhD thesis, Australian National University, Canberra, 1976). See also: Abstract, Bull. Austral. Math. Soc. 14 (1976), 1471–472.Google Scholar
[7]Schultz, Martin H., “The condition number of a class of Rayleigh-Ritz-Galerkin matrices”, Bull. Amer. Math. Soo. 14 (1970), 840–844.CrossRefGoogle Scholar