×

Algorithms for piecewise polynomials and splines with free knots. (English) Zbl 0668.41009

We describe an algorithm for computing points \(a=x_ 0<x_ 1<...<x_ k<x_{k+1}=b\) which solve certain nonlinear systems \(d(x_{i-1},x_ i)=d(x_ i,x_{i+1})\), \(i=1,...,k\). In contrast to Newton-type methods, the algorithm converges when starting with arbitrary points. The method is applied to compute best piecewise polynomial approximations with free knots. The advantage is that in the starting phase only simple expressions have to be evaluated instead of computing best polynomial approximations. We finally discuss the result to the computation of good spline approximations with free knots.

MSC:

41A15 Spline approximation
41A10 Approximation by polynomials
65D07 Numerical computation using splines
41A50 Best approximation, Chebyshev systems
65D15 Algorithms for approximation of functions
65H10 Numerical computation of solutions to systems of equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Birkhäuser, Basel, 1973, pp. 57 – 72. Internat. Ser. Numer. Math., Vol. 21.
[2] Carl de Boor, Good approximation by splines with variable knots. II, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Springer, Berlin, 1974, pp. 12 – 20. Lecture Notes in Math., Vol. 363. · Zbl 0255.41007
[3] Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. · Zbl 0406.41003
[4] Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 309 – 319. · Zbl 0313.41003 · doi:10.1080/00036817408839073
[5] D. S. Dodson, Optimal Order Approximation by Polynomial Spline Functions, Ph. D. Thesis, Purdue University, West Lafayette, IN, 1972.
[6] Charles L. Lawson, Characteristic propertiesof the segmented rational minmax approximation problem, Numer. Math. 6 (1964), 293 – 301. · Zbl 0121.34103 · doi:10.1007/BF01386077
[7] Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. · Zbl 0152.15202
[8] Günter Meinardus and Gerhard Merz, Praktische Mathematik. I, Bibliographisches Institut, Mannheim, 1979 (German). Für Ingenieure, Mathematiker und Physiker. · Zbl 0402.65002
[9] Günther Nürnberger and Manfred Sommer, A Remez type algorithm for spline functions, Numer. Math. 41 (1983), no. 1, 117 – 146. · Zbl 0489.65011 · doi:10.1007/BF01396309
[10] G. Nürnberger, M. Sommer, and H. Strauss, An algorithm for segment approximation, Numer. Math. 48 (1986), no. 4, 463 – 477. · Zbl 0569.65012 · doi:10.1007/BF01389652
[11] Theodore J. Rivlin, An introduction to the approximation of functions, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1969.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.