# Romberg Method For The Simpson's 1/3 Rule

This is the same method as used in Trapezium rule.

**Romberg method for the Simpson's 1/3 rule :** We can apply the same procedure as in trapezium rule to obtain the Romberg’s extrapolation procedure for the Simpson’s 1/3 rule.

Let I denote the exact value of the integral and IS denote the value obtained by the composite Simpson’s 1/3 rule.

The error, I – I_{S,}, in the composite Simpson’s 1/3 rule in computing the integral is given by

or

...............1.1

As in the trapezium rule, to illustrate the extrapolation procedure, first consider two error terms.

......................1.2

Let I be evaluated using two step lengths h and qh, 0 < q < 1. Let these values be denoted by I_{S}(h) and I_{S}(qh). The error equations become

.....................1.3

...................1.4

From (1.3), we obtain

......................1.5

From (1.4), we obtain

....................1.6

Multiply (1.5) by q^{4} to obtain

......................1.7

Eliminating c_{1}q^{4}h^{4} from (3.66) and (3.67), we obtain

Note that the error term on the right hand side is now of order O(h6). Solving for I, we obtain

Neglecting the O(h^{6}) error term, we obtain the new approximation to the value of the integral as

..................1.8

Again, we note that this value is obtained by suitably using the values of the integral obtained with step lengths h and qh, 0 < q < 1. This computed result is of order, O(h^{6}), which is higher than the order of the Simpson’s 1/3 rule, which is of O(h^{4}).

For q = 1/2, that is, computations are done with step lengths h and h/2, the formula (1.8) simplifies to