1. Closed formulas
By dividing the closed interval into uniform subintervals with length of , the extended Simpson’s rule is given by (1)
where for , and is in . The number of points, , must be odd, meaning that there are even number of subintervals. This requirement is a consequence of the fact that the coefficients alternate in a specific pattern, namely there are pairs of coefficient terms plus an extra -coefficient term and two end-point terms. Therefore, there are terms in total, while is always odd.
To relax the constraint of odd number of points, which is important for situations when it is not convenient to freely choose the number of points such as in experiments or numerical simultaions, we can use
Note that the coefficients are no longer alternate except the first and last three terms. See Arif Zaman’s Note (2) for a derivation of the exact form of the error. Other non-alternate forms of fourth-order quadrature formulas are possible. An example from Wolfram Mathworld (3)
2. Open and semi-open formulas
Sometimes open and semi-open formulas are required. Below is a fourth-order open formula for
And here is a fourth-order semi-open formula for
Note that above formulas share the same discrete scheme as the closed formulas.