Error In Simpson's 1/3 Rule

using equation,

the expression for the error is given by

since h = (b – a)/2, and a ≤ ξ ≤ b.

Since the method produces exact results, that is, R2(f, x) = 0, when f(x) is a polynomial of degree ≤ 3, the method is of order 3.

The bound for the error is given by

As in the case of the trapezium rule, if the length of the interval [a, b] is large, then b – a is also large and the error expression given in (3.41) becomes meaningless. In this case, we subdivide [a, b] into a number of subintervals of equal length and apply the Simpson’s 1/3 rule to evaluate each integral. The rule is then called the composite Simpson’s 1/3 rule.