Computing Integrals - Elements of Scientific Computing - page 13

Tabl e 1. 1 The table shows
the number of intervals, n,the
length of the intervals, h,the
error, E h , and finally E h =h 2 .
The latter term seems to
converge toward a constant

E h =h 2

n h E h

1 1:0000 0:5000 0:5000

2 0:5000 0:1274 0:5096

4 0:2500 0:0320 0:5121

8 0:1250 0:0080 0:5127

16 0:0625 0:0020 0:5129

32 0:0313 0:0005 0:5129

64 0:0156 0:0001 0:5129

E h 0:5129h 2 ;

(1.20)

which means that we can get as accurate an approximation as we want just by choos-
ing h sufficiently small which is done by increasing the number of intervals n.Say,
we want

10 5 :

E h 6 (1.21)

Then it follows from (1.20)thath must satisfy

0:5129h 2 6 10 5 ;

or

h 6 0:0044:

Using (1.19), we have

n D 1= h > 226:47:

Since n has to be an integer, we conclude that any n > 227 willyieldanerrorless

than 10 5 .
We have seen that the trapezoidal method gives good results for the sine-function
and for the function given by (1.18). Let us now challenge the method by applying
it to a few other functions. In Fig. 1.6 we have plotted the functions x 4 , x 20 ,and
p x, and in Table 1.2 we have given numerical results obtained by applying the
trapezoidal method to these functions. We observe from Table 1.2 that for x 4 and for
x 20 , the error is about 0:33h 2 and 1:67h 2 , respectively. So for these functions, the
trapezoidal method gives an error of basically the same form as we observed above.
We note that the error is about five times larger for x 20 compared to that of x 4 .If
you consider the plot above and recall the derivation of the trapezoidal method, it
is not hard to realize that the lower curve is more difficult, since the second-order
derivative is much larger. Compare this with Fig. 1.2 where it is apparent that large
second-o rd er derivatives may cause large errors. Now, this observation also applies

to the p x function. In this case the second derivative is infinite as x approaches
zero, and we see from Table 1.2 that E h =h 2 does not converge toward any constant.
These issues will be given further consideration in Project 1.7.1 below.