User:EGM6341.S11.team5.cavalcanti/Mtg17

Mtg 17: Wed, 9 Feb 11

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Error in Newton-Cotes formula:

Thm:

Let

(1) – (4):

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Appl: Error Trap. Rule (Newton-Cotes w/ $${\color{blue}{f}}_{\color{red}{1}}^{\color{blue}{L}}\color{blue}\left(x\right)$$ → 1st order, linear

n = 1 → $$q_{2} \left( x \right)=\left( x - \underbrace{a}_{\color{blue}x_{0}}\right)\left(x-\underbrace{b}_{\color{blue}x_{1}}\right)$$

(5) p17-1: $$\begin{array}{c}|{E}_{n}| \leqslant \frac{M_{2}}{2!}\int _{a}^{b}|{q}_{2}\left(x\right)|\mathit{dx}\\ =\frac{M_{2}}{2!}\underset{\leqslant 0}{{\int }_{a}^{b}\left(x-a\right)}|\underset{\leqslant 0}{\left(x-b\right)}|\mathit{dx}\end{array}$$

(3) p.7-4: $$h\mathrm{\colon }=\left(b-a\right)/1=b-a$$ See Atkinson p.253

Application: Error for Simple Simpson (ie, Newton-Cotes with $${\color{blue}{f}}_{\color{red}{2}}^{\color{blue}{L}}\color{blue}{\left(x\right)}$$

n = 2 → $${q}_{3}\left(x\right)=\left(x-\underbrace{{x}_{0}}_{\color{blue}{a}}\right)\left(x-\underbrace{x_{1}}_{\color{blue}{\frac{\left(a+b\right)}{2}}}\right)\left(x-\underbrace{{x}_{2}}_{\color{blue}{b}}\right)$$

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$$|{E}_{\color{red}{2}}^{\color{blue}{S}}| \leqslant \frac{{M}_{3}}{3!}{\int }_{a}^{b}|\left(x-a\right)\left(x-\frac{a+b}{2}\right)\left(x-b\right)|\mathit{dx}$$

Reference: Suli & Meyers 2003, p.205

Simpson's rule can int. exactly poly. of deg $$ \leqslant 3!$$ Cubic (not just quadratic)

Note: Misprint in S&M 2003 p.205: 192 in (1), not 196.