TABLE OF CONTENTS In this chapter we deal with techniques for approximating numerically the two fundamental operations of calculus, differentiation, and integration. Both of these problems may be approached in the same way. Although both numerical differentiation and numerical integration formulas will be discussed, it should be noted that numerical differentiation is inherently much less accurate than numerical integration, and its application is generally avoided whenever possible. Nevertheless, it has been used successfully in certain applications.
Engineers are frequently confronted with the problem of differentiating functions that are defined in tabular or graphical form rather than as explicit functions. The interpretation of experimentally obtained data is a good example of this. A similar situation involves the integration of functions, which have explicit forms that are difficult or impossible to integrate in terms of elementary functions. Graphical techniques, employing the construction of tangents to curves and the estimation of areas under curves, are commonly used in solving such problems, when great accuracy is not a prerequisite for results. However, there are occasions when a higher degree of accuracy is desired, and, for these, various numerical methods are available.
Firstly, we discuss the numerical process for approximating the derivative of the function f(x) at the given point. A function f(x), known either explicitly or as a set of data points, is replaced by a simpler function. A polynomial p(x) is the obvious choice of approximating function, since the operation of differentiation is then easily performed. The...
