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# Interpolation

• Philipp O. J. Scherer
Chapter
Part of the Graduate Texts in Physics book series (GTP)

## Abstract

Interpolation is necessary, if additional data points are needed, for instance to draw a continuous curve. It can also be helpful to represent a complicated function by a simpler one or to develop more sophisticated numerical methods for the calculation of numerical derivatives and integrals. In the following we concentrate on the most important interpolating functions which are polynomials, splines and rational functions. The interpolating polynomial can be explicitly constructed with the Lagrange method. Newton’s method is numerically efficient if the polynomial has to be evaluated at many interpolating points and Neville’s method has advantages if the polynomial is not needed explicitly and has to be evaluated only at one interpolation point.

For interpolation over a larger range, Spline functions can be superior which are piecewise defined polynomials. Especially cubic splines are often used to draw smooth curves. Curves with poles can be represented by rational interpolating functions whereas a special class of rational interpolants without poles provides a rather new alternative to spline interpolation.

## References

1. 12.
C.T.H. Baker, Numer. Math. 15, 315 (1970)
2. 13.
G.A. Baker Jr., P. Graves-Morris, Padé Approximants (Cambridge University Press, New York, 1996)
3. 23.
J.P. Berrut, Comput. Math. Appl. 14, 1 (1988)
4. 24.
J.-P. Berrut, L.N. Trefethen, Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)
5. 25.
J.P. Berrut, R. Baltensperger, H.D. Mittelmann, Int. Ser. Numer. Math. 151, 27 (2005)
6. 90.
M.S. Floater, K. Hormann, Numer. Math. 107, 315 (2007)
7. 137.
H. Jeffreys, B.S. Jeffreys, Lagrange’s interpolation formula, in Methods of Mathematical Physics, 3rd edn. (Cambridge University Press, Cambridge, 1988), p. 260, § 9.011 Google Scholar
8. 138.
H. Jeffreys, B.S. Jeffreys, Divided differences, in Methods of Mathematical Physics, 3rd edn. (Cambridge University Press, Cambridge, 1988), pp. 260–264, § 9.012 Google Scholar
9. 180.
E.H. Neville, J. Indian Math. Soc. 20, 87 (1933) Google Scholar
10. 186.
G. Nürnberger, Approximation by Spline Functions (Springer, Berlin, 1989). ISBN 3-540-51618-2
11. 227.
C. Schneider, W. Werner, Math. Comput. 47, 285 (1986)
12. 228.
I.J. Schoenberg, Q. Appl. Math. 4, 45–99 and 112–141 (1946)
13. 244.
J. Stör, R. Bulirsch, Introduction to Numerical Analysis, 3rd revised edn. (Springer, New York, 2010). ISBN 978-1441930064 Google Scholar
14. 271.
W. Werner, Math. Comput. 43, 205 (1984)

## Copyright information

© Springer International Publishing Switzerland 2013

## Authors and Affiliations

• Philipp O. J. Scherer
• 1
1. 1.Physikdepartment T38Technische Universität MünchenGarchingGermany