We have seen how to use interpolation to approximate values
between points *x*_{1}, ..., *x*_{n},
and in many cases, the error of the approximations of the
points near the center of the *x* values is quite accurate.
However, if we are trying to approximate a value at a value
outside the range of *x* values, the error increases
significantly when using interpolation. However, if model
information is available, for example, that the data is linear,
quadratic, or exponential, we may use least-squares to find
a best-fitting curve. It can be shown that the error associated
with extrapolating a least-squares fitting curve is significantly
less than the error associated with extrapolating an interpolating
polynomial.

Thus, for example, if you are given the points

and wish to approximate the value at 2.0, then we can do
nothing which may give us a good approximation. If, however,
we are told that this data is linear, then we may find the
least-squares fitting line (y(x) = -0.60830 x + 0.89531), then
we may approximate the value at *x* = 2 by evaluating
this function: y(2) = -0.60830⋅2 + 0.89531 = -0.32130.

The data, the interpolating polynomial (blue), and the least-squares line (red) are shown in Figure 1. The appropriateness of the extrapolating estimator should be apparent.

Figure 1. Extrapolation of exponentially decaying points in Example 4.

If you take nothing else from this topic, remember: **you
cannot use an interpolating polynomial to extrapolate a value**.
To successfully extrapolate data, you must have correct model information,
and if possible, use the data to find a best-fitting curve of
the appropriate form (e.g., linear, exponential) and evaluate the
best-fitting curve on that point.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.