Numerical Differentiation of Analytic Functions

Abstract

It is well known that the classical difference formulas for evaluating high derivatives of a real function f(ζ) are very ill-conditioned. However, if the function f(ζ) is analytic and can be evaluated for complex values of ζ, the problem can be shown to be perfectly well-conditioned. An algorithm that performs this evaluation for an arbitrary analytic function f(~) is described. A short FORTRAN program for generating up to 50 leading derivatives is to be found in the algorithm section of this issue. To use this program, no knowledge is required either of the method or of the analytical nature (e.g., position of nearest singularity, its type) of the function.

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