Divided differences

Summary

Divided differences are quotients built recursively from the values of a function at a list of points. First comes the slope between two points, then the difference of two slopes divided by the spread of three points, and so on. The divided difference $f[x_0, \dots, x_n]$ is the leading coefficient of the polynomial of degree at most $n$ that interpolates $f$ at the points. They are the coefficients of the Newton form of the interpolation polynomial. For smooth functions they behave like scaled derivatives: by the mean value theorem for divided differences, $f[x_0, \dots, x_n] = f^{(n)}(\xi)/n!$ for some $\xi$.