Simpsom's 3/8 Rule

Simpsom's 3/8 Rule : Simpson's 3/8 rule is a method for approximating a definite integral by evaluating the integrand at finitely many points.Simpson’s 3/8 method uses a third degree polynomial (i.e. a cubic) to estimate the curve you are trying to find the integral.

For interpolating by a cubic polynomial, we require four nodal points. Hence, we subdivide the given interval [a, b] into 3 equal parts so that we obtain four nodal points. Let h = (b – a)/3. The nodal points are given by

Using the Newton’s forward difference formula, the cubic polynomial approximation to f(x), interpolating at the points

is given by,

Substituting in

and integrating, we obtain the Simpson’s 3/8 rule as

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

 

The error expression is given by

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

Since the method produces exact results, that is, R3(f, x) = 0, when f(x) is a polynomial of degree ≤ 3, the method is of order 3.

As in the case of the Simpson’s 1/3 rule, if the length of the interval [a, b] is large, then b – a is also large and the error expression given in (3.47) becomes meaningless. In this case, we subdivide [a, b] into a number of subintervals of equal length such that the number of subintervals is divisible by 3. That is, the number of intervals must be 6 or 9 or 12 etc., so that we get 7 or 10 or 13 nodal points etc. Then, we apply the Simpson’s 3/8 rule to evaluate each integral. The rule is then called the composite Simpson’s 3/8 rule. For example, if we divide [a, b] into 6 parts, then we get the seven nodal points as

The Simpson’s 3/8 rule becomes

The error in this composite Simpson’s 3/8 rule becomes

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