Newton's Divided Difference Interpolation Formula

Let

$$\pi_n(x)\equiv \prod_{i=1}^n (x-x_n),$$ (1)

then

$$f(x)=f_0+\sum_{k=1}^n x_{k-1}(x)[x_0, x_1, \ldots, x_k]+R_n,$$ (2)

where $[x_1, \dots]$ is a Divided Difference, and the remainder is

$$R_n(x)=\pi_n(x)[x_0, \ldots, x_n, x]=\pi_n(x){f^{(n+1)}(\xi)\over (n+1)}$$ (3)

for $x_0<\xi<x_n$.

See also Divided Difference, Finite Difference

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 880, 1972.

Hildebrand, F. B. Introduction to Numerical Analysis. New York: McGraw-Hill, pp.43-44 and 62-63, 1956.