Simpson’s 1/24 Rule for Numerical Integration

Simpson’s 1/24 Rules:

In the task of estimation of full area of narrow peak-like functions, Simpson’s rules are much less efficient than trapezoidal rule. Namely, composite Simpson’s 1/3 rule requires 1.8 times more points to achieve the same accuracy^[7] as trapezoidal rule. Composite Simpson’s 3/8 rule is even less accurate. Integral by Simpson’s 1/3 rule can be represented as a sum of 2/3 of integral by trapezoidal rule with step h and 1/3 of integral by rectangle rule with step 2h. No wonder that error of the sum corresponds lo less accurate term. Averaging of Simpson’s 1/3 rule composite sums with properly shifted frames produces following rules:

$\int _{a}^{b}f(x)\,dx\approx {\tfrac {h}{24}}{\bigg [}-f(x_{-1})+12f(x_{0})+25f(x_{1})+24\sum _{i=2}^{n-2}f(x_{i})+25f(x_{n-1})+12f(x_{n})-f(x_{n+1}){\bigg ]}$

where two points outside of integrated region are exploited and

$\int _{a}^{b}f(x)\,dx\approx {\tfrac {h}{24}}{\bigg [}9f(x_{0})+28f(x_{1})+23f(x_{2})+24\sum _{i=3}^{n-3}f(x_{i})+23f(x_{n-2})+28f(x_{n-1})+9f(x_{n}){\bigg ]}$

Those rules are very much similar to Press’s alternative extended Simpson’s rule. Coefficients within the major part of the region being integrated equal one, differences are only at the edges. These three rules can be associated with Euler-MacLaurin formula with the first derivative term and named Euler-MacLaurin integration rules. They differ only in how the first derivative at the region end is calculated.