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Algorithms for piecewise polynomials and splines with free knots

Authors: G. Meinardus, G. Nürnberger, M. Sommer and H. Strauss
Journal: Math. Comp. 53 (1989), 235-247
MSC: Primary 65D07; Secondary 41A15
MathSciNet review: 969492
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Abstract: We describe an algorithm for computing points $ a = {x_0} < {x_1} < \cdots < {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, \ldots ,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 relation to the computation of good spline approximations with free knots.

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  • [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. MR 0403169
  • [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. MR 0431606
  • [3] Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
  • [4] Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 309–319. MR 0399708
  • [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. MR 0176278
  • [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. MR 0217482
  • [8] Günter Meinardus and Gerhard Merz, Praktische Mathematik. I, Bibliographisches Institut, Mannheim, 1979 (German). Für Ingenieure, Mathematiker und Physiker. MR 535443
  • [9] Günther Nürnberger and Manfred Sommer, A Remez type algorithm for spline functions, Numer. Math. 41 (1983), no. 1, 117–146. MR 696554, 10.1007/BF01396309
  • [10] G. Nürnberger, M. Sommer, and H. Strauss, An algorithm for segment approximation, Numer. Math. 48 (1986), no. 4, 463–477. MR 834333, 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. MR 0249885

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Article copyright: © Copyright 1989 American Mathematical Society