User:Egm6341.s10.team3.heejun/Meeting 16 transcript

 MTG 16: Thu. 28 Jan 10

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Note: Open Intervals


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& end\,\, points\,\, a,b\,\, \in\,\, [a,b] \\\end{align}\ $$
 * $$ \begin{align} \displaystyle [a,b] &:\,\, closed\,\, interval \\
 * $$ \begin{align} \displaystyle [a,b] &:\,\, closed\,\, interval \\
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& \underbrace{end\,\, points\,\, a,b}_{\color{blue}{i.e.,\,\, excluded\,\, from\,\, the\,\, open\,\, interval}}\,\, \notin\,\, ]a,b[ \\\end{align}\ $$
 * $$ \begin{align} \displaystyle ]a,b[ \,\, \equiv (a,b)&:\,\, open\,\, interval \\
 * $$ \begin{align} \displaystyle ]a,b[ \,\, \equiv (a,b)&:\,\, open\,\, interval \\
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HW: Runge Phenomenon S+M p.208
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$$ \displaystyle I = \int^{5}_{-5} \frac{1}{1+x^2} dx $$

(i) Find $$ \displaystyle I_n $$ using Newton-Cotes for n=1, 2,. . ., 15

(ii) Plot $$ \displaystyle f, f_n $$ $$ \displaystyle \mbox{,  } $$ for n=1, 2, 3, 8, 12

(iii) Plot $$ \displaystyle I_n $$ Vs. $$ \displaystyle n $$ and observe that $$ \displaystyle I_n $$ does not converge as $$ \displaystyle n \rightarrow \infty $$

(iv) Observe weight $$ \displaystyle W_{i,n}:=\int^{b}_{a} l_{i,n}(x),\ dx $$ are not all positive for n $$ \displaystyle \ge $$ 8. Then plot $$ \displaystyle l_{i,n}(x) $$ for i=1, 2, ... ,8 & n=8


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Motivation for Comp. rules :

Runge Pheno. $$ \displaystyle \Rightarrow \,\, keep\,\, n\,\, small\,\, (i.e.,\,\, n=1,2),\,\, and\,\, subdivide\,\, [a,b]\,\, into\,\, smaller\,\, subinterv. $$
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$$ \displaystyle n=2\,\, Simple\,\, Simpson $$
 * $$ \displaystyle n=1\,\, Trap.\,\, Simple $$
 * $$ \displaystyle n=1\,\, Trap.\,\, Simple $$
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Error for Comp. Trap :


 * $$ \displaystyle h $$ is the step size on $$ \displaystyle [a,b] $$


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&= \int_{a}^{b} f(x)dx - h[\frac{1}{2}f(x_{0})+ f(x_{1}) + ... + f(x_{n-1}) + \frac{1}{2}f(x_{n})] \\ &= \sum_{i=1}^{n} [\int_{x_{i-1}}^{x_{i}}f(x)dx - \frac{h}{2}[f(x_{i-1})+f(x_{i})]] \\\end{align}\ $$
 * $$ \displaystyle \begin{align} \underbrace{E^{1}_{n}}_{\color{blue}{Trap.}}\, &:=\, I-I_{n} \\
 * $$ \displaystyle \begin{align} \underbrace{E^{1}_{n}}_{\color{blue}{Trap.}}\, &:=\, I-I_{n} \\
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 * $$ \displaystyle \left| E^{1}_{n} \right| \leq \frac{h^{3}}{12} \underbrace{ \sum_{i=1}^{n} (max \left| f^{(2)}(\zeta) \right|\,\,\,\, \zeta \in [x_{i-1},\,\, x_{i}] ) }_{\color{blue}{\bar{M}_{2}}} $$
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 * $$ \displaystyle \Rightarrow M_{2} = max \left| f^{(2)}(\zeta) \right|\,\,\,\, \zeta \in [a,b]\,\,\,\, $$
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 * $$ \displaystyle \bar{M}_{2} \leq n \cdot M_{2}$$
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 * $$ \displaystyle \Rightarrow \left| E^{1}_{n} \right| \leq \frac{(b-a)^{3}}{12n^{3}} n M_{2} \,\,\,\, = \frac{(b-a)^{3}}{12n^{2}} M_{2}  \,\,\,\, = \underbrace{ \frac{(b-a)h^{2}}{12} M_{2}}_{\color{blue}{A.\,\, p.253}} $$
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 * $$ \displaystyle M_{2}\,\, Vs.\,\, \bar{M_{2}} $$

$$ \displaystyle M_{2} $$ is the max second order deriv. within whole interval $$ \displaystyle [a,b] $$

$$ \displaystyle \bar{M_{2}} $$ is the max second order deriv. within each subinterval