Composite Simpson's 1/3 Rule

Composite Simpson's 1/3 rule :  We note that the Simpson’s rule derived earlier uses three nodal points. Hence, we subdivide the given interval [a, b] into even number of subintervals of equal length h. That is, we obtain an odd number of nodal points. We take the even number of intervals as 2N.

The step length is given by,

h = (b – a)/(2N)

The nodal points are given by,

The given interval is now written as

Note that there are N integrals. The limits of each integral contain three nodal points. Using the Simpson’s 1/3 rule to evaluate each integral, we get the composite Simpson’s 1/3 rule as.

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

The composite Simpson’s 1/3 rule is also of order 3.

The error expression,

becomes

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

where,

The bound on the error is given by

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

or,

This expression is a true representation of the error in the Simpson’s 1/3 rule. We observe that as N increases, the error decreases.