The Chemistry Maths Book, Second Edition

574 Chapter 20Numerical methods

The area of the trapezium betweenx

j

andx

j+ 1

is(f

j

1 + 1 f

j+ 1

)h 22 , and the total area is

(20.32)

wheref

j

1 = 1 f(a 1 + 1 jh).

For an integrand that is continuous and differentiable in the intervala 1 ≤ 1 x 1 ≤ 1 b, it

can be shown that the error in this formula is

(20.33)

wheref′′(c)is the second derivative of the function at some pointx 1 = 1 cin the interval.

Alternatively, when his small enough, the error can be approximated as (see the

Euler–MacLaurin formula (20.39) below)

(20.34)

wheref

0

′andf

n

′are the derivatives off(x)at the end points. It follows that halving

h(or doubling the number of subintervals n) reduces the error by a factor of four.

EXAMPLE 20.10Find estimates of the integral by means of the

trapezoidal rule (20.32) and the error formula (20.34) forn 1 = 1 3, 6and 12.

The required values of the integrandf(x) 1 = 1 xe

−x

2

are listed in the following table.

Table 20.7 Values off(x) 1 = 1 xe

−

x

2

x 0.0 0.1 0.2 0.3 0.4 0.5 0.6

f(x) 0.000000 0.099005 0.192158 0.274179 0.340858 0.389400 0.418606

x 0.7 0.8 0.9 1.0 1.1 1.2

f(x) 0.428838 0.421834 0.400372 0.367879 0.328017 0.284313

Then, for example, forn 1 = 13 we have

h 1 = 1 0.4, f

0

1 = 1 0, f

1

1 = 1 0.340858, f

2

1 = 1 0.421834, f

3

1 = 1 0.284313

and

T(h) 1 = 1 0.4[0 1 + 1 0.340858 1 + 1 0.421834 1 +× 1 0.284313] 1 = 1 0.361939

1

2

Ixedx

x

=

−

Z

0

12. 2

ε≈− ′− ′













1

12

2

0

hf f

n

Z

a

b

fxdx Th

ba

() −==−() hf c()

−

ε ′′

12

2

Z

a

b

nn

fxdx Th h f f f() ≈= +++++() f f

−











1

2

1

2

012 1











