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 f(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 f(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 f(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

Consider the definite integral

(6.1)

We divide the range of integration (a, b) into n equal intervals...