INTERPOLATION

Extrapolation is the process of interpolation when the unknown point falls outside of the interval of known points, x₁, …, x_n = [A, B]. In interpolation, the most accurate values are computed when they lie midway between two known points. Thus by extension extrapolation can include much larger errors if care is not taken.

For the interpolation methods discussed previously, extrapolation can be determined by asked for a point outside of the known points. The programmer must include the possibility of extrapolation by not limiting the requested points to lie within x₁, …, x_n = [A, B] for this to be possible. No guarantees are given in terms of the accuracy of the extrapolation.

If curve fitting is not used, then extrapolation is typically based on Taylor Series.

The simplest is the zeroth order extrapolation

while higher order extrapolation is determined by truncating the Taylor series. For a second-order extrapolation:

where the last term is the remainder. This extrapolation requires that first and second derivatives be computed (∂T)/(∂y) and (∂²T)/(∂y²) in a similar manner as discussed in the error discussion.