Neville's method

We can now apply Theorem 11 on subsets of consecutive data points. Let us call , with . Then .

For example: for , i.e. a linear polynomial, we take the data points in pairs: , , ... , . These linear polynomials can then be used to calculate the quadratic polynomials over consecutive triplets , , ...

This method has also been illustrated in Worksheet 3.

As was shown in Worksheet 3, one can apply Neville's Method to immediately find the interpolated value at a given -value. In this case, the scheme applies on numerical values rather then functions.

If higher accuracy is required, one can add a data point at the end of the set and just calculate the succesive polynomials over intervals which include that extra data point. Overall, this works out to be more efficient then recalculating the highest order Lagrange Polynomial from scratch.