A Complex Variables Techniques for Approximating the Derivatives of Real-Valued Functions.

The numerical solution of an n-th order differential equation relies on an accurate approximation of the derivative. A standard method of approximating the derivative of a real-valued function f at a point Xo is to use the central difference formula f \u27 (xo) [f (x, + h) - f (x, - h)] / 2h . An error analysis of this formula shows that the truncation error is 0(h2) while the round off error is 0(h-1). A major dilemma in using this formula is the fact that using a small h increases the round off error. In this thesis, a method of approximating the derivative of real-valued functions via complex variables is presented. This method avoids the high round off error inherent in the standard method. Numerical examples are presented to illustrate the power of this method of approximation