Numerical Integration and Differentiation - Numerical Structural Analysis - page 100

y′ = f(x)

f i f i+1

x

∆x

y

y i+1

y i+1 - y i

y i

x

x i x i+1

Figure 3.2. Romberg integration.

The Taylor series for f i+ 1 expanded about x i is as follows:

′′

2

f fh fh Higherorder termsi

′

f =+ +

i

+

i + 1 i i

2!

′′

2

fh

fh f ′ =−− f

i

− Higherorder termsi

(3.2)

i i + 1 i

2!

Substituting Equation 3.2 into Equation 3.1, the following can be derived:

′′

2

fh Higherorder termsh

f −− − f

i

i + 1 i

2 !

y −= + y fh

i + 1 i i

2 !

′′

3

fh Higherorder termsi

+

i

+

3!

( )

′′

f + f

3

fh Higherorder termsi

i i +1

(3.3)

y −= y exact area = h −

i

+

i + 1 i

2 12