Numerical Calculus

Abstract

This chapter has two primary topics: numerical differentiation and numerical integration. In some respects the methods of numerical differentiation are similar to those of numerical integration, in that they are typically based on using (in this case, differentiating) an interpolation polynomial. One major and important difference between numerical approaches to integration and differentiation is that integration is numerically a highly satisfactory operation with results of high accuracy being obtainable in economical ways. This is because integration tends to smooth out the errors of the polynomial approximations to the integrand. Unfortunately the reliability and stability we find in numerical integration is certainly not reflected for differentiation which tends to exaggerate the error in the original polynomial approximation. In the case of numerical integration we begin with the simple, and often already familiar approaches like the trapezoid rule, including its relation to the fundamental concept of a Reimann sum. The trapezoid rule and Simpson’s rule are explored in more detail which then leads to a discussion of so-called composite integration rules where the interval of integration is broken into many smaller pieces of the same size. The behavior of the errors in such rules is compared to that for the simple versions and this allows us to develop methods of achieving guaranteed precision at the desired level. These composite rules in turn provide a basis for the introduction to practical approaches to numerical integration which is our final major topic. Python implementation and the practical application of numerical calculus techniques are discussed throughout.