Errors of Interpolating Polynomials

Theorem

Suppose are distinct numbers in the interval and . Then, for each in , a number (generally unknown) between , and hence in , exists with:

where is the interpolating polynomial. is the error term. The above equation can only be used for known and differentiable functions, which is usually not the case.

An alternative formula to find error is by applying a finite divided difference to approximate the th derivative. Thus, we have:

Since this contains the unknown , it cannot be solved for the error. However, using an additional data point , the error can be estimated as follows.