Derivative formulas and errors for non-uniformly spaced points

Lagrange interpolation estimates a function at a reference position χ from known values of the function at distinct non-uniformly spaced points x1, . . . , xn. Here, the corresponding n-point finite-difference formulas are derived to estimate derivatives up to order n − 1 at χ. A recurrence relation permits the errors to be determined to any accuracy as a Taylor series. The error coefficient multiplying the n + j’th derivative is a polynomial of order j +1 in the elementary symmetric functions for the displacements x1−χ, . . . , xn−χ. Appendices state finite difference formulas to estimate the derivatives and the first four error terms for n = 1, . . . , 5.