Numerical Calculus

Abstract

The derivative of a function f(x) is defined in theory by

$${\rm{f'(}}x{\rm{)}}\mathop {\lim }\limits_{h \to 0} {{{\rm{f}}(x + h) - {\rm{f}}(x)} \over h}$$

Therefore we can obtain a numerical approximation using

$${{{\rm{f}}(x + h) - {\rm{f}}(x)} \over h}$$

for small values of h The accuracy will clearly depend on the size of h. We can improve the approximation by letting h become small, but it is obvious we cannot let it become zero since we are dealing with numerical values. If h became zero we would be dividing by zero. However, before this becomes a problem the difference between f(x) and f(x+h) would become indistinguishable on the computer and this would lead to gross inaccuracies. Therefore, by reducing h in the classical theoretical manner, we can only achieve an accuracy within the limits of that allowed by the computer we are using.