# Splines

Reference work entry

First Online: 25 October 2005

**DOI:**https://doi.org/10.1007/1-4020-0611-X_982

Splines are an important class of mathematical functions used for approximation. A spline is a piecewise polynomial function that is commonly described as being “as smooth as it can be without reducing to a polynomial” (
de Boor, 1978, p. 125). For example, the cubic spline shown as the solid line in
Figure 1 is composed of individual cubic polynomials, each defined between two adjacent data points, such that the function values and first and second derivatives of adjoining polynomial pieces are the same. In general, a function defined on an interval [

*a*,*b*] is defined as a*polynomial spline of degree k*, having knots*t*_{1}, ...,*t*_{r}, if the following three conditions hold: (i)*a*<*t*_{1}< ⃛ <*t*_{r}<*b*, so the knots*t*_{1}, ...,*t*_{r}partition the interval [*a*,*b*] into*r*+ 1 smaller subintervals, (ii) on each subinterval [*t*_{i},*t*_{i+1}], the spline is given by a polynomial function of at most degree*k*, and (iii) the spline and its derivatives up to order*k*− 1 are all continuous on [*a*,*b*]. The definition of...This is a preview of subscription content, log in to check access.

## References

- [1]Chen, V.C., Ruppert, D., and Shoemaker, C.A. (1999). “Applying Experimental Design and Regression Splines to High-Dimensional Continuous-State Dynamic Programming,” Opns. Res. 47, 38–53.Google Scholar
- [2]de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag, New York.Google Scholar
- [3]Dierckx, P. (1993). Curve and Surface Fitting with Splines. Oxford University Press, New York.Google Scholar
- [4]Johnson, S.A., Stedinger, J.R., Shoemaker, C.A., Li, Y., Tejada-Guibert, J.A. (1993). “Numerical Solution of Continuous-State Dynamic Programs Using Linear and Spline Interpolation,” Opns. Res. 41, 484–500.Google Scholar
- [5]Schumaker, L.L. (1981). Spline Functions: Basic Theory. John Wiley, New York.Google Scholar
- [6]Schweitzer, P.J. and Seidmann, A. (1985). “Generalized Polynomial Approximations in Markovian Decision Processes,” Jl. Math. Anal. & Appl. 110, 568–582.Google Scholar
- [7]Seber, G.A.F. and Wild, C.J. (1989). Nonlinear Regression. John Wiley, New York.
*Sports*775 Google Scholar - [8]Smith, P.L. (1979). “Splines as a Useful and Convenient Statistical Tool,” The American Statistician, 33, 57–62.Google Scholar

## Copyright information

© Kluwer Academic Publishers 2001

## How to cite

- Cite this entry as:
- Johnson S.A. (2001) Splines. In: Gass S.I., Harris C.M. (eds) Encyclopedia of Operations Research and Management Science. Springer, New York, NY

## About this entry

- First Online 25 October 2005
- DOI https://doi.org/10.1007/1-4020-0611-X
- Publisher Name Springer, New York, NY
- Print ISBN 978-0-7923-7827-3
- Online ISBN 978-1-4020-0611-1
- eBook Packages Springer Book Archive