In topic composite trapezium rule I have shown you what is the so-called trapezium rule and the so-called composite trapezium rule. Recall that the idea of trapezium rule is the following approximation
The right hand side of the above approximation is nothing but the integration of one-order Lagrange interpolation polynomial of at the nodes and . If we add one more node, we then obtain a new approximation. This leads to the so-called Simpson rule.
In numerical analysis, Simpson’s rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation
One derivation replaces the integrand by the quadratic polynomial which takes the same values as at the end points and and the midpoint . One can use Lagrange polynomial interpolation to find an expression for this polynomial,
An easy calculation shows that
Next we will show you the error of Simpson rule. Firstly, we assume that is of fourth continuously differentiable. Similarly to the trapezium rule, by considering the following function
We firstly see that , then with the aid of Hermite interpolation, there exists a such that
then we deduce that
where . Regarding to composite Simpson rule, if the interval of integration is in some sense “small”, then Simpson’s rule will provide an adequate approximation to the exact integral. By small, what we really mean is that the function being integrated is relatively smooth over the interval . For such a function, a smooth quadratic interpolant like the one used in Simpson’s rule will give good results.
However, it is often the case that the function we are trying to integrate is not smooth over the interval. Typically, this means that either the function is highly oscillatory, or it lacks derivatives at certain points. In these cases, Simpson’s rule may give very poor results. One common way of handling this problem is by breaking up the interval into a number of small subintervals. Simpson’s rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. This sort of approach is termed the composite Simpson’s rule.
Suppose that the interval is split up in subintervals, with an even number. Then, the composite Simpson’s rule is given by
where for with ; in particular, and . The above formula can also be written as
The error committed by the composite Simpson’s rule is bounded (in absolute value) by
where is the “step length”.