Page 160 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 7 INTEGRATION 137

Figure 7-11. Some results returned by the Integrates custom function.
(folder 'Chapter 07 Examples', workbook 'Integration', sheet Simpson Integration Fn')

Because some functions may require a large number of iterations, there may
be a noticeable delay in calculation.

Gaussian Quadrature
The preceding methods for numerical integration employ evenly spaced
values of x at which the function is evaluated. Other formulas have been
developed whereby the function is evaluated at specially selected values of x.
These Gaussian quadrature formulas are significantly more efficient, in terms of
the accuracy of the evaluation.
Gaussian quadrature formulas involve the evaluation of the function at a set
of x, values (nodes), with the use of a corresponding set of weights w,, in the
following formula
1 N
1.
A = IF(x)dx = c WiF(Xi) (7-9)
-1 i=l
The nodes and weights can be derived from certain kinds of polynomials.
The Legendre polynomials will be used here to determine the values of xi and wi.
The Legendre polynomials are a set of polynomials of degree N. Increasing N
provides an increase in accuracy of evaluation but requires a concomitant
increase in computation time. Values of Legendre polynomials for N up to 100
have been published.
The integration need not be limited solely to the interval -1 to 1. By
employing a change of variable
2~ - (a + b)
Z= (7-10)
(b - a)
the integral expression is