Interpolation

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Part of the book series: Graduate Texts in Physics ((GTP))

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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.

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Notes

1. 1.

$$\omega(x)=\prod_{i=0}^{n}(x-x_{i})$$ as in (2.39).

2. 2.

It can be shown that any rational interpolant can be written in this form.

3. 3.

The opposite case can be treated by considering the reciprocal function values 1/f(x i ).

4. 4.

Bilinear means linear interpolation in two dimensions. Accordingly linear interpolation in three dimensions is called trilinear.

5. 5.

A typical task of image processing.

References

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