TABLE OF CONTENTS The term numerical integration covers several different tasks, including numerical evaluation of integrals and numerical solution of ordinary differential equations. So we use the somewhat old-fashioned term quadrature for the simplest of these, the numerical evaluation of a definite integral. Modern quadrature algorithms automatically vary an adaptive step size.
Let f(x) be a real-valued function of a real variable, defined on a finite interval a ? x ? b. We seek to compute the value of the integral,
The word "quadrature" reminds us of an elementary technique for finding this area plot the function on graph paper and count the number of little squares that lie underneath the curve.
In Figure 6.1, there are 148 little squares underneath the curve. If the area of one little square is 3/512, then a rough estimate of the integral is 148 3/512 = 0.8672.
Adaptive quadrature involves careful selection of the points where f(x) is sampled. We want to evaluate the function at as few points as possible while approximating the integral to within some specified accuracy. A fundamental additive property of a definite integral is the basis for adaptive quadrature. If c is any point between a and b, then
The idea is that if we can approximate each of the two integrals on the right to within a specified tolerance, then the sum gives us the desired result. If not, we can recursively apply the...
