Neville’s algorithm

Neville s algorithm constructs the same unique interpolating polynomial and improves the straightforward Lagrange implementation by the addition of an error estimate. 

Neville s algorithm recursively calculates the preceding columns from left to right as... 

The evaluation of Pj.x) at a point other than at the defining points can be made with Neville s algorithm [Press et al. (1986)]. Let be the value at x of the unique function passing through the point (xi, y 1) or Pi = yi. Let P 2 be the value at x of the unique polynomial passing through the points Xi and x2. Likewise, Plfi, is the unique polynomial... 

---