# An Introduction to Non-Linear Splines

• Helmut Werner
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 49)

## Abstract

We introduce functions termed non-linear splines because they belong to a class of k-times differentiable funtions (k N) and their restrictions to certain subintervals (defined by knots) are non-linearly dependent on the parameters in contrast to the splines usually considered. It is shown that use of these non-linear splines in interpolation, approximation and numerical quadrature or ordinary differential equations is widely parallel to the linear case as far as numerical work and stability is concerned, while it allows us to take into account special properties of the functions which the spline is used to replace.

## References

1. [1]
Arndt, H., Interpolation mit regulären Spline-Funktionen, Dissertation, Münster, 1974.Google Scholar
2. [2]
Baumeister, J., Extremaleigenschaften nichtlinearer Splines, Dissertation, München, 1974.Google Scholar
3. [3]
Braess, D. and Werner, H., Tschebyscheff-Approximation miteiner Klasse rationaler Splinefunktionen II, J. Approx-imation Theory 10, 379–399, (1974).
4. [4] Lambert, J.D. and Shaw, B., On the numerical solution of y′ = f(x,y)
by a class of formulae based on rational approximation, Math. Comp. 19, 456–462, (1965).
5. [5] Lambert, J.D. and Shaw, B., A method for the numerical solution of y y′ = f(x,y)
based on a self-adjusting non- polynomial interpolant, Math. Comp. 20 11–20, (1966).
6. [6]
Loscalzo, F.R. and Talbot, T.D., Spline function approximation for solutions of ordinary differential equations, SIAM J. Num. Anal. 4, 433–445, (1967).
7. [7]
Micula, G., Bemerkungen zur numerischen Lösung von Anfangs-wert-problemen mit Hilfe nichtlinearer Splinefunktionen, In Lecture Notes in Mathematics 501, 200-209, Spline Functions, Karlsruhe 1975, Springer Verlag, Berlin-Heidelberg-New York, (1976).Google Scholar
8. [8]
Runge, R., Lösung von Anfangswertproblemen mit Hilfenichtlinearer Klassen von Spline-Funktionen, (Dissertation Münster, 1972 ).Google Scholar
9. [9]
Schaback, R., Spezielle rationale Splinefunktionen, J. Approximation Theory 7,281-292, (1973). (Dissertation Münster, 1969 ).Google Scholar
10. [10]
Schaback, R., Interpolation mit nichtlinearen Klassen von Splinefunktionen, J. Approximation Theory 8, 173–188, (1973).
11. [11]
Schömberg, H., Tschebyscheff-Approximation durch rationale Splinefunktionen mit freien Knoten, Dissertation, Münster, 1973.Google Scholar
12. [12]
Werner, H., Tschebyscheff-Approximation mit einer Klasserationaler Splinefunktionen, J. Approximation Theory 10, 74–92, (1974).
13. [13]
Werner, H., Tschebyscheff-Approximation nichtlinearer Splinefunktionen, in K. Böhmer-G. Meinardus-W. Schempp Spline-Funktionen, BI-Verlag Mannheim-Wien-Zürich, 303–313, (1974).Google Scholar
14. [14]
Werner, H., Interpolation and Integration of Initial Value Problems of Ordinary Differential Equations by Regular Splines, SIAM J. Num. Anal. 12, 255–271, (1975).
15. [15]
Werner, H., Numerische Behandlung gewöhnlicher Differential gleichungen mit Hilfe von Splinefunktionen, ISNM 32, 167–175, (1976).Google Scholar
16. [16]
Werner, H., Approximation by Regular Splines with Free Knots, Austin, Symposium on Approximation Theory 1976, S. 567 - 573.Google Scholar
17. [17]
Werner, H. and Loeb, H., Tschebyscheff-Approximation by Regular Splines with Free Knots, in Approximation Theory, Bonn 1976, Lecture Notes in Math. 556, 439 - 452, Springer Verlag, Berlin-Heidelberg-New York (1976).Google Scholar
18. [18]
Werner, H., Praktische Mathematik I, 2. Auflage, Springer-Verlag, Berlin-Heidelberg-New York 1976.Google Scholar
19. [19]
Werner, H. and Schaback, R., Praktische Mathematik II, 2. Auflage, Springer-Verlag, Berlin-Heidelberg-New York 1978.Google Scholar
20. [20]
Werner, H. and Wuytack, L., Nonlinear Quadrature Rules in the Presence of a Singularity, Universiteit Antwerpen, to appear in CAMWA 1978.Google Scholar
21. [21]
Werner, H. and Zwick, D., Algorithms for Numerical Integration with Regular Splines, Rechenzentrum der Universität Münster Schriftenreihe Nr. 27, (1977).Google Scholar
22. [22]
Wuytack, L., Numerical Integration by Using Nonlinear Techniques, J. of Comp, and Appl. Math. 267–272, (1975).Google Scholar