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

DOI:
https://doi.org/10.1090/S0025-5718-1989-0969492-7

MathSciNet review:
969492

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Abstract: We describe an algorithm for computing points which solve certain nonlinear systems , . 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.

**[1]**C. de Boor, "Good approximation by splines with variable knots," in*Spline Functions and Approximation Theory*(A. Meir and A. Sharma, eds.), Birkhäuser-Verlag, Basel, 1973, pp. 57-72. 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, Springer-Verlag, Berlin and New York, 1974, pp. 12-20. MR**0431606 (55:4603)****[3]**C. de Boor,*A Practical Guide to Splines*, Springer-Verlag, 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. 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]**C. L. Lawson, "Characteristic properties of the segmented rational minimax approximation problem,"*Numer. Math.*, v. 6, 1964, pp. 293-301. MR**0176278 (31:553)****[7]**G. Meinardus,*Approximation of Functions: Theory and Numerical Methods*, Springer-Verlag, 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. 117-146. MR**696554 (85d:65013)****[10]**G. Nürnberger, M. Sommer & H. Strauss, "An algorithm for segment approximation,"*Numer. Math.*, v. 48, 1986, pp. 463-477. 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:
https://doi.org/10.1090/S0025-5718-1989-0969492-7

Article copyright:
© Copyright 1989
American Mathematical Society