Romberg Method For The Simpson's 1/3 Rule

.....................1.9

In practical applications, we normally use the sequence of step lengths h, h/2, h/2², h/2³, ...

Suppose, the integral is computed using the step lengths h, h/2, h/2². Using the results obtained with the step lengths h/2, h/2², we get

......................2.0

Both the results IT⁽¹⁾ h, IT⁽¹⁾h(1/2) are of order, O(h⁶). Now, we can eliminate the O(h⁶) terms of these two results to obtain a result of next higher order, O(h⁸). The multiplicative factor is now (1/2)⁶ = 1/64. The formula becomes

......................2.1

Therefore, we obtain the Romberg extrapolation procedure for the composite Simpson’s 1/3 rule as

.......................2.2

where,

The computed result is of order O(h^{2m 4}).

The extrapolations using three step lengths h, h/2, h/2², are given in Table

Figure : Romberg method for Simpson’s 1/3 rule

Note that the most accurate values are the values at the end of each column.