The Chemistry Maths Book, Second Edition

20.5 Numerical integration 573

Figure 20.8. This demonstrates that curve fitting with splines produces a continuous

curve that closely follows the straight lines of piecewise linear interpolation. This

does notguarantee however that the computed curve is a true representation of the

function of which the nodes are a sample. Thus in the present case, the points in

Table 20.4 have been computed from the functionf(x) 1 = 111 − 1 (x 1 − 1 1)

2

(x 1 − 1 3) 1 sin 1 πx.

This has many local maxima and minima. The ‘unrealistic’ 8th degree polynomial fit

shown in Figure 20.7 is in fact, quite fortuitously, a fairly good representation of the

function, whereas the spline fit is only as good as the poor sample of points allows.*

20.5 Numerical integration

Numerical integration (quadrature) is the numerical evaluation of the integral

(20.31)

when the integrandf(x)is either given by a table of numbers or when the integral

cannot be evaluated by the methods described in Chapters 5 and 6 in terms of a finite

number of standard functions. Geometrically, the problem is to estimate the area

under the curve. The standard method is to replace the functionf(x)by a different

function that can be integrated, and one way is to use the interpolation formulas

discussed in the previous section. This gives the class of numerical integration methods

called Newton–Cotes quadratures, of which the trapezoidal rule and Simpson’s rule

are the simplest examples.

5

The trapezoidal rule. Linear interpolation

The simplest estimate of an integral (20.31), as an area, is obtained by dividing

the intervala 1 ≤ 1 x 1 ≤ 1 binto nequal subintervals each of widthh 1 = 1 (b 1 − 1 a) 2 n, and

approximating the integrand by piecewise linear interpolation; that is, by joining the

corresponding adjacent points on the graph by straight lines as shown in Figure 20.9.

Ifxdx

a

b

=Z ()

*Garbage in, Garbage out(Wilf Hey, d. 2007, computer programmer and writer).

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The Newton–Cotes formulas first appeared in Newton’s letter to Leibniz of 24 October 1676 and in the

Principia, although particular examples were known long before. They were included in the collection of work

by Cotes, the Harmonia mensurarum, published posthumously in 1722. Roger Cotes (1682–1716), Cambridge

mathematician, spent much of 1709–1713 preparing the second edition of the Principia. His work includes one of

the earliest discussions of the calculus of logarithmic and trigonometric functions, with tables of integrals.

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Figure 20.9