Chapter 7 - Numerical integration and differentiation

Abstract

The fundamental theorem of calculus shows that integration is the inverse process to differentiation. The trapezoidal rule is usually applied in a composite form. To estimate the integral of f over [a, b], the [a, b] is divided into N subintervals of equal length h= (b–a)/N. The composite trapezoidal rule consists of the composite rectangular rule plus the correction term h(fN– f0)/2. Rounding errors does not affect the accuracy of the quadrature rule. This is generally true of numerical integration, unlike numerical differentiation. There is a generalization of the composite trapezoidal rule called the Gregory' s formula. Like the trapezoidal rule, Simpson's rule is usually applied in composite form. The composite form of the Simpson's rule requires an even number of subintervals. For a rule of comparable accuracy which permits an odd number of subintervals, the Gregory formula can be used.