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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Algorithms for piecewise polynomials and splines with free knots
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by G. Meinardus, G. Nürnberger, M. Sommer and H. Strauss PDF
Math. Comp. 53 (1989), 235-247 Request permission


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.
  • Carl de Boor, Good approximation by splines with variable knots, Spline functions and approximation theory (Proc. Sympos., Univ. Alberta, Edmonton, Alta., 1972) Internat. Ser. Numer. Math., Vol. 21, Birkhäuser, Basel, 1973, pp. 57–72. MR 0403169
  • Carl de Boor, Good approximation by splines with variable knots. II, Conference on the Numerical Solution of Differential Equations (Univ. Dundee, Dundee, 1973) Lecture Notes in Math., Vol. 363, Springer, Berlin, 1974, pp. 12–20. MR 0431606
  • Carl de Boor, A practical guide to splines, Applied Mathematical Sciences, vol. 27, Springer-Verlag, New York-Berlin, 1978. MR 507062
  • Hermann G. Burchard, Splines (with optimal knots) are better, Applicable Anal. 3 (1973/74), 309–319. MR 399708, DOI 10.1080/00036817408839073
  • D. S. Dodson, Optimal Order Approximation by Polynomial Spline Functions, Ph. D. Thesis, Purdue University, West Lafayette, IN, 1972.
  • Charles L. Lawson, Characteristic propertiesof the segmented rational minmax approximation problem, Numer. Math. 6 (1964), 293–301. MR 176278, DOI 10.1007/BF01386077
  • Günter Meinardus, Approximation of functions: Theory and numerical methods, Expanded translation of the German edition, Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. Translated by Larry L. Schumaker. MR 0217482
  • Günter Meinardus and Gerhard Merz, Praktische Mathematik. I, Bibliographisches Institut, Mannheim, 1979 (German). Für Ingenieure, Mathematiker und Physiker. MR 535443
  • Günther Nürnberger and Manfred Sommer, A Remez type algorithm for spline functions, Numer. Math. 41 (1983), no. 1, 117–146. MR 696554, DOI 10.1007/BF01396309
  • G. Nürnberger, M. Sommer, and H. Strauss, An algorithm for segment approximation, Numer. Math. 48 (1986), no. 4, 463–477. MR 834333, DOI 10.1007/BF01389652
  • 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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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Math. Comp. 53 (1989), 235-247
  • MSC: Primary 65D07; Secondary 41A15
  • DOI:
  • MathSciNet review: 969492