In this chapter we are concerned with a common problem from calculus: Find the value of the definite integral
We are interested in finding numerical methods that yield an accurate approximation to the exact value of I(f). Typically, the approximation will be of the form
where i0 = 0 or i0 = 1 is almost always the case. The weights wi and nodes or abscissas xi define the method. Schemes for approximating integrals are often called quadrature rules, and different schemes use different rules for defining the weights and abscissas. In Chapter 2 we saw perhaps the simplest example of a reasonable quadrature scheme, the trapezoid rule. Here we will look at other methods, some of which are substantially more accurate than the trapezoid rule.
In addition to constructing various quadrature schemes and analyzing their accuracy, we will also look at ways to estimate and improve the accuracy of existing quadrature methods, as well as the idea of adaptive quadrature, by which we try to estimate the value of the integral to within a user-specified tolerance.
Note that we seem to be suggesting that the integral can be viewed (approximately, at least) as a sum of function values. This is not surprising, considering the theoretical foundations of the integral, which ...