Algorithms for piecewise polynomials and splines with free knots
Authors:
G. Meinardus, G. Nürnberger, M. Sommer and H. Strauss
Journal:
Math. Comp. 53 (1989), 235247
MSC:
Primary 65D07; Secondary 41A15
MathSciNet review:
969492
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Abstract: We describe an algorithm for computing points which solve certain nonlinear systems , . In contrast to Newtontype 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.
 [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
(53 #6982)
 [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
(55 #4603)
 [3]
Carl
de Boor, A practical guide to splines, Applied Mathematical
Sciences, vol. 27, SpringerVerlag, New YorkBerlin, 1978. MR 507062
(80a:65027)
 [4]
Hermann
G. Burchard, Splines (with optimal knots) are better,
Applicable Anal. 3 (1973/74), 309–319. MR 0399708
(53 #3551)
 [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
(31 #553)
 [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,
SpringerVerlag New York, Inc., New York, 1967. MR 0217482
(36 #571)
 [8]
Günter
Meinardus and Gerhard
Merz, Praktische Mathematik. I, Bibliographisches Institut,
Mannheim, 1979 (German). Für Ingenieure, Mathematiker und Physiker. MR 535443
(80g:65002)
 [9]
Günther
Nürnberger and Manfred
Sommer, A Remez type algorithm for spline functions, Numer.
Math. 41 (1983), no. 1, 117–146. MR 696554
(85d:65013), http://dx.doi.org/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
(87g:65023), http://dx.doi.org/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 (40 #3126)
 [1]
 C. de Boor, "Good approximation by splines with variable knots," in Spline Functions and Approximation Theory (A. Meir and A. Sharma, eds.), BirkhäuserVerlag, Basel, 1973, pp. 5772. MR 0403169 (53:6982)
 [2]
 C. de Boor, "Good approximation by splines with variable knots II," in Numerical Solution of Differential Equations (G. A. Watson, ed.), Lecture Notes in Math., vol. 363, SpringerVerlag, Berlin and New York, 1974, pp. 1220. MR 0431606 (55:4603)
 [3]
 C. de Boor, A Practical Guide to Splines, SpringerVerlag, Berlin and New York, 1978. MR 507062 (80a:65027)
 [4]
 H. G. Burchard, "Splines (with optimal knots) are better," Applicable Anal., v. 3, 1974, pp. 309319. MR 0399708 (53:3551)
 [5]
 D. S. Dodson, Optimal Order Approximation by Polynomial Spline Functions, Ph. D. Thesis, Purdue University, West Lafayette, IN, 1972.
 [6]
 C. L. Lawson, "Characteristic properties of the segmented rational minimax approximation problem," Numer. Math., v. 6, 1964, pp. 293301. MR 0176278 (31:553)
 [7]
 G. Meinardus, Approximation of Functions: Theory and Numerical Methods, SpringerVerlag, Berlin and New York, 1967. MR 0217482 (36:571)
 [8]
 G. Meinardus & G. Merz, Praktische Mathematik I, B. I.Verlag, Mannheim, 1979. MR 535443 (80g:65002)
 [9]
 G. Nürnberger & M. Sommer, "A Remez type algorithm for spline functions," Numer. Math., v. 41, 1983, pp. 117146. MR 696554 (85d:65013)
 [10]
 G. Nürnberger, M. Sommer & H. Strauss, "An algorithm for segment approximation," Numer. Math., v. 48, 1986, pp. 463477. MR 834333 (87g:65023)
 [11]
 T. J. Rivlin, An Introduction to the Approximation of Functions, Blaisdell, Waltham, Mass., 1969. MR 0249885 (40:3126)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909694927
PII:
S 00255718(1989)09694927
Article copyright:
© Copyright 1989
American Mathematical Society
