User:EGM6341.s11.team1.Chiu/Mtg26

Mtg 26: Wed, 2 Mar 11

[[media: Nm1.s11.mtg26.djvu| page26-1]]


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HW5.2: [[media: Nm1.s11.mtg25.djvu| page25-3]] continued

2) find equilibrated point as


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$$ \displaystyle \lim_{k \to \infty } \mathbf x_{k+1}=\lim_{k \to \infty } \mathbf F^{k+1} \mathbf x_{0}=: \hat{\mathbf x} $$
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Plot $$\left.\begin{matrix} \hat{\mathbf x} \ as \ { \color{red} big \ red \ dot} \\ \mathbf x_{ \color{red} 0} \ as \ { \color{red} big} \ { \color{blue} blue} \ { \color{red} dot} \end{matrix}\right\} \ $$ in same plot for $$ \left \{ \mathbf x_{j}, j=1,2, \cdots \right \} \ $$ small dots
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3) Gaussian random noise :


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Let $$ \mathbf G =\begin{Bmatrix} 1 \\ 1 \end{Bmatrix}_{ \color{red} 2\times 1} \ { \color{blue} \alpha}$$, thus $$ \mathbf w_{k+1}=\left \{ w_{k+1} \right \}_{ \color{red} 1 \times 1} \ $$. Use matlab randn to generate $$ \left \{ w_{j}, j=0,1,2, \cdots \right \} \ $$.
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Plot $$ \left \{ \mathbf x_{j}, j= \underset{ \color{blue} \underset{big \ blue \ dot}{\uparrow}} 0, \underbrace{1,2, \cdots}_{small \ dots} \right \} \ $$ for $$ { \color{blue} \alpha}= 0.5,1,2 \ $$.
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4) Cauchy random noise : Same as 3) but with Cauchy. Hint: Find a matlab command to generate $$ \left \{ \theta_{j}, j=0,1, \cdots \right \} \ $$ in single-slit diffraction.


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[[media: Nm1.s11.mtg26.djvu| page26-2]]


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HW5.3: Cauchy heavy tails

Quartile points: $$ Q_{1}, Q_{2}, Q_{3} \ $$


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$$ \mathbb{P}(x<Q_{ \color{red}1})={ \color{red}0.25}= \int^{Q_{ \color{red}1}}_{-\infty} \underbrace{f(x)}_{ \color{blue} pdf} dx= \underset{ \color{blue} \underset{cdf}{\uparrow}} F(Q_{ \color{red}1}) \ $$
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$$ \mathbb{P}(x<Q_{ \color{red}3})={ \color{red}0.75}= \int^{Q_{ \color{red}3}}_{-\infty} \underbrace{f(x)}_{ \color{blue} pdf} dx= \underset{ \color{blue} \underset{cdf}{\uparrow}} F(Q_{ \color{red}3}) \ $$
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Cauchy probability density function = $$ C(x_{0}, \gamma):= \frac{ \gamma}{ \pi [ \gamma^{2}+(x-x_{0})^{ \color{red} 2}]} \ $$
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Gauss (normal) probability density function = $$ N( \mu, \sigma)= \frac{1}{\sigma \sqrt{2\pi}}exp \left[ \frac{{ \color{red}-}(x-\mu)^{2}}{2\sigma^{2}} \right] \ $$
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$$ \left.\begin{matrix} {\color{blue}1)} Find \ \left \{ Q_{1}, Q_{3} \right \} \ for \ { \color{blue} C(x_{0}, \gamma)} \\ {\color{blue}2)} Find \ \left \{ Q_{1}, Q_{3} \right \} \ for \ { \color{blue} N( \mu, \sigma)} \end{matrix}\right\} \ $$ Hint: See WA.
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Newton-Raphson-Simpson: $$ F(Q)=0 \ $$


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$$ Q^{(k+1)} \underset{ \underset{{ \color{red} (1)} \ { \color{blue} p.26-3}}{\uparrow}}= Q^{(k)}- \underbrace{\frac{F(Q^{(k)})}{F{ \color{red}'}(Q^{(k)})}}_{ \color{blue} f(Q^{(k)})} { \color{blue}\to} { \color{red} \hat{Q}} \ $$  as $$ \color{blue} k \to \infty \ $$
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[[media: Nm1.s11.mtg26.djvu| page26-3]]


 * nm1.s11.Mtg26.pg3.fig1.svg$$ \frac{F(Q^{(k)})}{Q^{(k)}-Q^{(k+1)}} \underset{ \color{red} \underset{(1)}{\uparrow}}= \frac{F {\color{red}'} (Q^{(k)})}{1}\ $$

3) Let $$ x_{0}= \mu= 0 \ $$ and $$ \gamma^{ \color{blue} C}=1 \ $$. Find $$ \sigma^{ \color{red} 1} \ $$ at $$ \gamma^{ \color{blue} G}={ \color{red}1} \ $$,


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$$ \gamma^{ \color{blue} C}= \ $$ half width of $$ C(x_{0}, \gamma^{ \color{blue} C}) \ $$
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$$ \gamma^{ \color{blue} G}= \ $$ half width of $$ N( \mu, \sigma) \ $$
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Plot $$ C(0,1) \ $$ and $$ N(0, \sigma^{ \color{red} 1}) \ $$
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4) Find $$ \left \{ Q_{1}^{ \color{blue}C}, Q_{3}^{ \color{blue}C} \right \} \ $$ for $$ C(x_{0}, \gamma) \ $$ and $$ C(0,1) \ $$


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$$ \ \left \{ Q_{1}^{ \color{blue}G}, Q_{3}^{ \color{blue}G} \right \} \ $$ for $$ N( \mu, \sigma) \ $$ and $$ N(0,\sigma^{ \color{red}1}) \ $$
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Plot $$ \left.\begin{matrix} \ \left \{ Q_{1}^{ \color{blue}C}, Q_{3}^{ \color{blue}C} \right \} \ for \ C(0,1) \\ \ \ \left \{ Q_{1}^{ \color{blue}G}, Q_{3}^{ \color{blue}G} \right \} \ for \ N(0,\sigma^{ \color{red}1}) \end{matrix}\right\} \ $$ on same plot. Comment on results.
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[[media: Nm1.s11.mtg26.djvu| page26-4]]


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HW*5.1:[[media: Nm1.s11.mtg23.djvu| page23-4]] continued

3) Periodic functions ( IMPORTANT: Chebyshev Clenshaw-Curtis)


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$$ I= \int^{2\pi}_{0}exp \left [ cos x \right ] dx \ $$ s10
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$$ I= \int^{2\pi}_{0}exp \left [ \color{blue} sin x \right ] dx \ $$ s11
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$$ f(x)=exp \left [ \cos x \right ] \ $$ s10
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(from s10 HW3 Team3)

much better than $$ \color{blue} \theta(h^{ \color{red}2}) \ $$ composite trapezoidal, $$ \color{blue} \theta(h^{ \color{red}4}) \ $$ composite simpson

Composite Trapezoidal Rule beats Composite Simpson's Rule for periodic functions.


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