Truncation and roundoff errors in three-point approximations of first and second derivatives

We use linear combinations of Taylor expansions to develop three-point finite difference expressions for the first and second derivative of a function at a given node. We derive analytical expressions for the truncation and roundoff errors associated with these finite difference formulae. Using these error expressions, we find optimal values for the stepsize and the distribution of the three points, relative to the given node. The latter are obtained assuming that the three points are equispaced. For the first derivative approximation, the distribution of the points relative to the given node is not symmetrical, while it is so for the second derivative approximation. We illustrate these results with a numerical example in which we compute upper bounds on the roundoff error.