20.5 Numerical integration 575
The derivative of the integrand is
f′(x) 1 = 1 e
−x2(1 1 − 12 x
2),
so thatf
0′ 1 = 1 f′(0) 1 = 11 andf
n′ 1 = 1 f′(1.2) 1 = 1 −0.445424. Thenε 1 = 1 0.120452h
2. The results
are collected in Table 20.8.
Table 20.8 Values of
n 3612
T(h) 0.361939 0.376698 0.380330
ε 0.019272 0.004818 0.001205
T(h) 1 + 1 ε 0.381211 0.381516 0.381535
The exact value of the integral is (1 1 − 1 e
−1.44) 1 = 1 0.381536.
0 Exercise 19
Simpson’s rule. Quadratic interpolation
Simpson’s rule has been one of the most popular simple numerical quadrature methods
for over two centuries.
6It is obtained by dividing the intervala 1 ≤ 1 x 1 ≤ 1 binto an even
number of subintervals, 2 n, each with widthh 1 = 1 (b 1 − 1 a) 22 n, and approximating the
integrand by piecewise quadratic interpolation. Then
(20.35)
or
The error in Simpson’s rule behaves likeh
4when his small, so that halving h(or
doubling the number of subintervals n) reduces the error by a factor of sixteen.
Simpson’s rule therefore usually converges much more quickly that the (uncorrected)
trapezoidal rule.
Z
abfxdx
h
() ≈++( ) ( ) (
∑∑
3
end points 42 odd points eveen points)
∑
Z
abnnfxdx
h
() ≈ ++++++fffff f f+
−3
4242 4
01234 212
12Ixedx
x==
−−Z
01.2 26Thomas Simpson (1710–1761), professor of mathematics at the Royal Military Academy at Woolwich, was a
self-taught mathematician. He was brought up as a weaver and in 1735 he joined the Mathematical Society at
Spitalfields, a weaving community, of which it was the duty of every member ‘if he be asked any mathematical or
philosophical question by another member, to instruct him in the plainest and easiest manner he is able.’ The rule
named for him appeared in his Mathematical dissertations on a variety of physical and analytical subjectsof 1743,
but had appeared previously in work by Cavalieri (1639) and Gregory (1668).