6.1 Introduction

Numerical integration, also known as quadrature, is intrinsically a much more accurate procedure than numerical differentiation. Quadrature approximates the definite integral

by the sum

where the nodal abscissas x _i and weights A _i depend on the particular rule used for the quadrature. All rules of quadrature are derived from polynomial interpolation of the integrand. Therefore, they work best if ( x) can be approximated by a polynomial.

Methods of numerical integration can be divided into two groups: Newton-Cotes formulas and Gaussian quadrature. Newton-Cotes formulas are characterized by equally spaced abscissas, and include well-known methods such as the trapezoidal rule and Simpson s rule. They are most useful if ( x) has already been computed at equal intervals, or can be computed at low cost. Since Newton-Cotes formulas are based on local interpolation, they require only a piecewise fit to a polynomial.

In Gaussian quadrature the locations of the abscissas are chosen to yield the best possible accuracy. Because Gaussian quadrature requires fewer evaluations of the integrand for a given level of precision, it is popular in cases where ( x) is expensive to evaluate. Another advantage of Gaussian quadrature is its ability to handle integrable singularities, enabling us to evaluate expressions such as

provided that g(x) is a well-behaved function.

6.2 Newton-Cotes Formulas

Figure 6.1: Polynomial approximation of ( x).

Consider the definite...