Encyclopedia of Operations Research and Management Science

2001 Edition
| Editors: Saul I. Gass, Carl M. Harris

Splines

  • Sharon A. Johnson
Reference work entry
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...
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References

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    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
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    de Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag, New York.Google Scholar
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    Dierckx, P. (1993). Curve and Surface Fitting with Splines. Oxford University Press, New York.Google Scholar
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    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
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    Schumaker, L.L. (1981). Spline Functions: Basic Theory. John Wiley, New York.Google Scholar
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Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • Sharon A. Johnson
    • 1
  1. 1.Worcester Polytechnic InstituteWorcesterUSA