Editing Simpson's rule (section)

== Alternative extended Simpson's rule ==
This is another formulation of a composite Simpson's rule: instead of applying Simpson's rule to disjoint segments of the integral to be approximated, Simpson's rule is applied to overlapping segments, yielding{{sfn|Weisstein|loc=Equation 35}}

<math display="block">

\begin{aligned}
\int_a^b f(x)\, dx \approx \frac{1}{48} h \bigg[
    & 17f(x_0) + 59f(x_1) + 43f(x_2) + 49f(x_3) \\
    +& 48 \sum_{i=4}^{n-4} f(x_i) \\

    +& 49f(x_{n-3}) + 43f(x_{n-2}) + 59f(x_{n-1}) + 17f(x_n)

\bigg]
\end{aligned}

</math>
The formula above is obtained by combining the composite Simpson's 1/3 rule with the one consisting of using Simpson's 3/8 rule in the extreme subintervals and Simpson's 1/3 rule in the remaining subintervals. The result is then obtained by taking the mean of the two formulas.

=== Simpson's rules in the case of narrow peaks ===
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 as trapezoidal rule.{{sfn|Kalambet|Kozmin|Samokhin|2018}} Composite Simpson's 3/8 rule is even less accurate. Integration by Simpson's 1/3 rule can be represented as a weighted average with 2/3 of the value coming from integration by the trapezoidal rule with step ''h'' and 1/3 of the value coming from integration by the rectangle rule with step 2''h''. The accuracy is governed by the second (2''h'' step) term. Averaging of Simpson's 1/3 rule composite sums with properly shifted frames produces the following rules:
<math display="block">\int_a^b f(x)\, dx \approx \frac{1}{24} h\left[-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})\right],</math>
where two points outside of the integrated region are exploited, and 
<math display="block">\int_a^b f(x)\, dx \approx \frac{1}{24} h\left[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)\right],</math>
where only points within integration region are used. Application of the second rule to the region of 3 points generates 1/3 Simpson's rule, 4 points - 3/8 rule.

These rules are very much similar to the alternative extended Simpson's rule. The coefficients within the major part of the region being integrated are one with non-unit coefficients only at the edges. These two rules can be associated with [[Euler–MacLaurin formula]] with the first derivative term and named '''First order''' '''Euler–MacLaurin integration rules'''.{{sfn|Kalambet|Kozmin|Samokhin|2018}} The two rules presented above differ only in the way how the first derivative at the region end is calculated. The first derivative term in the Euler–MacLaurin integration rules accounts for integral of the [[second derivative]], which equals the difference of the first derivatives at the edges of the integration region. It is possible to generate higher order Euler–Maclaurin rules by adding a difference of 3rd, 5th, and so on derivatives with coefficients, as defined by [[Euler–MacLaurin formula]].