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Newton-Cotes formulas

Newton-Cotes formulas (so named after Isaac Newton and Roger Cotes) are a group of formulas used for numerical integration (quadrature), a subject of numerical analysis. They deal with classifying a number of possible solutions to this problem under the assumption that you have gained samples of a function $f(x)$ on $n$ equidistant points $x_0, x_1, x_2, ..., x_n$ such that $x_0 < x_1 < x_2 ... x_n$ and $x_1-x_0 = x_2-x_1 = ... x_{n}-x_{n-1}= h$. Under this assumption Newton-Cotes formula holds that:

$\int_a^b f(x) dx \approx \sum_0^n w_i f(x_i)$

The $w_i$:s are called weights and are derived from computing the corresponding Lagrange polynomials thusly:

$\int_a^b f(x) dx \approx \int_a^b P(x) = \int_a^b \sum_{i=0}^n f(x_i) L_i(x) dx =$
$\sum_{i=0}^n \int_{x_{i-1}}^{x_i} f(x_i) L_i(x) dx = \sum_{i=0}^n f(x_i) \underbrace{\int_{x_{i-1}}^{x_i} L_i(x) dx}_{w_i}$

Here the $w_i$:s are fix constants depending only on the chosen $x_i$:s. Computing this integral for different numbers of points $n$ will yield some famous quadrature formulas:

 $n$ Common name Short form Exact for 0 Midpoint rule $M(x)$ $f \in \pi_1$ 1 Trapezoid rule $T(x)$ $f \in \pi_1$ 2 Simpson's rule $S(x)$ $f \in \pi_3$ 3 ? ? $f \in \pi_3$ 4 Bode's rule[?] ? $f \in \pi_5$ ... ... ... ...

These are the actual Newton-Cotes formulas. The $\pi$:s on the left signify on which monomial[?] bases the solution is exact, e.g. a solution that is exact for $\pi_2$ will be exact on the monomials in the set $\{1, x, x^2\}$. Note that the formulas' exactness increase in odd steps, so using the trapezoid rule for integrating is actually no better than using the midpoint rule, and just a waste of time. In the same manner, Simpson's rule will be exact for monomials in $\{1, x, x^2, x^3\}$ which will suffice for most functions found in practice.

If you also let the distance $h$ between each sample point vary, you obtain another group of quadrature formulas, the best of which is know as Gaussian quadrature.

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