The Chemistry Maths Book, Second Edition

576 Chapter 20Numerical methods

EXAMPLE 20.11Find estimates of the integral by means of

Simpson’s rule (20.35) for 2 n 1 = 1 2, 4, 6,and 12.

We use the values of the integrand in Table 20.7 and construct Table 20.9.

Table 20.9 Simpson’s rule for

jx

j

f(x

j

)2n 1 = 122 n 1 = 142 n 1 = 162 n 1 = 112

0 0.0 0.000000 × 1 × 1 × 1 × 1

1 0.1 0.099005 × 4

2 0.2 0.192158 × 4 × 2

3 0.3 0.274179 × 4 × 4

4 0.4 0.340858 × 2 × 2

5 0.5 0.389400 × 4

6 0.6 0.418606 × 4 × 2 × 4 × 2

7 0.7 0.428838 × 4

8 0.8 0.421834 × 2 × 2

9 0.9 0.400372 × 4 × 4

10 1.0 0.367879 × 4 × 2

11 1.1 0.328017 × 4

12 1.2 0.284313 × 1 × 1 × 1 × 1

Totals 1.958737 3.819729 5.724269 11.446227

0.391747 0.381973 0.381618 0.381541

The exact value is 0.381536, and Simpson’s rule is accurate enough for many

applications even without correction.

0 Exercises 20–23

The Euler–MacLaurin formula

When the integrandf(x)can be expanded as a Taylor series in the intervala 1 ≤ 1 x 1 ≤ 1 b,

the approximate expression (20.32) for the trapezoidal rule can be replaced by the

Euler–MacLaurin formula

(20.36)

−

!

′− ′ −

!

′′′

−

′′′

−

!

B

h

ff B

h

ff B

h

2 nn

2

04

4

06

6

24 6

() ( ) (()ff

n

vv

−−

0



Z

a

b

nn

fxdx h f f f() =+++++f f

−





















1

2

1

2

012 1



×=

h

I

3

Ixedx

x

==

−−

Z

0

1.2 2

Ixedx

x

=

−

Z

0

12. 2