Iterated Interpolation

The main problem with using the Lagrange Polynomial for interpolation is that one has only an idea of the error once the Lagrange Polynomial has been calculated, and increasing the accuracy requires calculating the Lagrange Polynomial for more data points all over again.

Here, we look at a different strategy: start with few data points, calculate the Lagrange Polynomial, then add data points and calculate the new Lagrange Polynomial, making use of a formula that links the Lagrange Polynomial of degree with those of degree through a smaller sample of data points:

Theorem 11

Proof

Neville's method