9.4.1 Finite-difference interpolation formulas  9. Numerical Methods  9.3 Finite-difference derivatives

9.4 Interpolation and extrapolation

Often experimental results are available for selected conditions, but values are needed for intermediate conditions. For example, the fall speeds of raindrops are measured for specific diameters, but calculation of a rainrate from a measured drop size distribution requires fall speeds for intermediate diameters. The estimation of such intermediate values is called interpolation. Another common use for interpolation is when the functional dependence is so complicated that explicit evaluation is costly in terms of computer time. In such cases, it may be more efficient to evaluate the function at selected points spanning the region of interest and then use interpolation, which can be very efficient, to determine intermediate values.

Extrapolation is the extension of such data beyond the range of the measurements. It is much more difficult, and can result in serious errors if not used and interpreted properly. For example, a high-order polynomial may provide a very good fit to a data set over its range of validity, but if higher powers than needed are included, the polynomial may diverge rapidly from smooth behavior outside the range of the data.
 

   
 9.4.1 Finite-difference interpolation formulas  9. Numerical Methods  9.3 Finite-difference derivatives