Numerical Integration - Numerical Methods in Engineering with MATLAB - page 210

6.3 Romberg Integration

Romberg integration combines the composite trapezoidal rule with Richardson ex-
trapolation (see Art. 5.3).Let us first introduce the notation

I i
where, as before, I i represents the approximate valueof a f ( x ) dx computedbythe
recursivetrapezoidal rule using 2 i − 1 panels.Recall that the errorinthis approximation
is E

R i , 1 =

= c 1 h 2 + c 2 h 4 +··· , where

b a

2 i − 1

−

h =

is the width of a panel.
Romberg integration starts with the computation of R 1 , 1 = I 1 (one panel) and

I 2 (two panels) from the trapezoidal rule. The leading error term c 1 h 2 is then
eliminatedbyRichardson extrapolation. Using p

R 2 , 1 =

2 (the exponent in the error term)
in Eq. (5.9) and denoting the result by R 2 , 2 , weobtain

=

2 2 R 2 , 1 − R 1 , 1 4

3 R 2 , 1 −

1
3 R 1 , 1

R 2 , 2 = = (a)

2 2 − 1

It isconvenient to store the results in an array of the form
R 1 , 1
R 2 , 1 R 2 , 2

I 3 (four panels) and repeat Richardson extra-
polationwith R 2 , 1 and R 3 , 1 , storing the result as R 3 , 2 :

The next stepistocalculate R 3 , 1 =

4
3 R 3 , 1

1
3 R 2 , 1

R 3 , 2 = − (b)

The elements of array R calculated so far are
⎡
⎣

⎤
⎦

R 1 , 1
R 2 , 1 R 2 , 2
R 3 , 1 R 3 , 2

Both elements of the second column have an error of the form c 2 h 4 , which can also
beeliminatedwith Richardson extrapolation. Using p = 4 in Eq. (5.9), we get

2 4 R 3 , 2 − R 2 , 2 16

15 R 3 , 2 −

1
15 R 2 , 2

R 3 , 3 = = (c)

2 4 − 1

O

( h 6 ). The array has now expanded to
⎡
⎣

This result has an error of

⎤
⎦

R 1 , 1
R 2 , 1 R 2 , 2
R 3 , 1 R 3 , 2 R 3 , 3