Gauss Elimination by Segments and Multivariate Polynomial Interpolation
The construction of a polynomial interpolant to data given at finite pointsets in ℝ d (or, most generally, to data specified by finitely many linear functionals) is considered, with special emphasis on the linear system to be solved. Gauss elimination by segments (i.e., by groups of columns rather than by columns) is proposed as a reasonable means for obtaining a description of all solutions and for seeking out solutions with ‘good’ properties. A particular scheme, due to Amos Ron and the author, for choosing a particular polynomial interpolating space in dependence on the given data points, is seen to be singled out by requirements of degree-reduction, dilation-invariance, and a certain orthogonality requirement. The close connection, between this particular construction of a polynomial interpolant and the construction of an H-basis for the ideal of all polynomials which vanish at the given data points, is also discussed.
KeywordsFormal Power Series Polynomial Interpolation Gauss Elimination Polynomial Ideal Diagonal Block
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- Birkhoff G. The algebra of multivariate interpolation. In Constructive approaches to mathematical models, pages 345–363. Academic Press, New York, 1979. C. V. Coffman and G. J. Fix, eds.Google Scholar
- de Boor C. Polynomial interpolation in several variables. In Proceedings of the Conference honoring Samuel D. Conte, pages xxx–xxx. Plenum Press, New York, 199x. R. DeMillo and J. R. Rice.Google Scholar
- Lorentz R. A. Multivariate Birkhoff Interpolation. Lecture Notes in Mathematics, v. 1516, Springer-Verlag, Heidelberg, 1992.Google Scholar