📂Numerical Analysis

Newton-Cotes Integration Formulas

Definition ¹

Assume that $f : [a,b] \to \mathbb{R}$ is integrable over $[a,b]$ and $[a,b]$ is divided into nodes with a constant interval of $\displaystyle h:= {{b-a} \over {n}}$, like in $a = x_{0} < \cdots < x_{n} = b$. The numerical integration operator $I_{n}^{p}$ defined as follows is called the Newton-Cotes formula. $$ I_{n}^{p} (f) := \sum_{i=0}^{n} w_{i} f ( x_{i} ) $$

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• For $i=0,1,\cdots , n$, $x_{i} := a + i h$, and $l_{i}$ represents the polynomial $\displaystyle l_{i} (x) := \prod_{i \ne j} \left( {{ x - x_{j} } \over { x_{i} - x_{j} }} \right)$ used in the Lagrange formula.
• The weight $w_{i}$ is defined as $\displaystyle w_{i} := \int_{a}^{b} l_{i} (x) dx$.

Error

If $f \in C^{n+2} [a,b]$, then $$ C_{n} := \begin{cases} \displaystyle {{1} \over {(n+2)! }} \int_{0}^{n} \mu^2 ( \mu - 1 ) \cdots ( \mu - n ) d \mu & , n \text{ is even} \\ \displaystyle {{1} \over {(n+1)! }} \int_{0}^{n} \mu ( \mu - 1 ) \cdots ( \mu - n ) d \mu & , n \text{ is odd} \end{cases} $$ and for some $ \xi \in [a,b]$, $$ E_{n}^{p} (f) = \begin{cases} C_{n} h^{n+3} f^{(n+2)} ( \xi ) & , n \text{ is even} \\ C_{n} h^{n+2} f^{(n+1)} ( \xi ) & , n \text{ is odd} \end{cases} $$

Specialization

If the Trapezoidal Rule uses the $1$th polynomial interpolation and the Simpson’s Rule uses $2$th polynomial interpolation, it is natural to consider generalizing for the $p$th degree. The Newton-Cotes integration formula includes all rules that can be made by increasing the degree of the polynomial used for approximation.

Trapezoidal Rule

• (1) $p=1$: $$I^{1} (f) := h [ f(a) + f(b) ]$$

Simpson’s Rule

• (2) $p=2$: $$I^{2} (f) := {{h} \over {3}} \left[ f(a) + 4 f \left( {{a + b} \over {2}} \right) + f(b) \right]$$

$3-8$ Rule

• (3) $p=3$: $$I^{3} (f) := {{3h} \over {8}} \left[ f(a) + 3 f ( a + h ) + 3 f ( b - h ) + f(b) \right]$$

Boole’s Rule

• (4) $p=4$: $$I^{4} (f) := {{2h} \over {45}} \left[ 7 f(a) + 32 f ( a + h ) + 12 f \left( {{a + b} \over {2}} \right) + 32 f(b - h) + 7 f(b) \right]$$