User:Egm6341.s10.team3.heejun/Meeting 46 transcript

 '''MTG 46: Thu. 10 May 10'''


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Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:57, 11 August 2010 (UTC)


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[[media: Egm6341.s10.mtg46.djvu | Page 46-1]] 

Clencurt code by Trefethen




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 * $$ \displaystyle \left \{ \theta_{k} := k \frac{\pi}{n},\,\, k=0,1,2,...,n \right \} $$
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 * $$ \displaystyle \left \{ x_{k} := cos \theta_{k},\,\, k=0,1,2,...,n \right \} $$
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 * $$ \displaystyle \underbrace{mod(n,2)}_{\color{blue}{module}} = n\,\, mod\,\, 2 \color{blue}{\in \left \{0,1 \right \}} $$
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 * $$ \displaystyle \underbrace{mod(n,m)}_{\color{blue}{remainder\,\, of\,\, \frac{n}{m}}} = m-\underbrace{q}_{\color{blue}{quotient}}m $$
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 * $$ \displaystyle q = argmax(j,m)\,\, such\,\, that\,\, n-jm > 0 $$
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[[media: Egm6341.s10.mtg46.djvu | Page 46-2]] 


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 * $$ \displaystyle n= \underbrace{q}_{\color{blue}{quotient\,\, of\,\, \frac{n}{m}}}m + \underbrace{mod(n,m)}_{\color{blue}{remainder\,\, of\,\, \frac{n}{m}}} $$
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$$ \displaystyle n = 2q + \underbrace{0}_{\color{blue}{mod(n,2)}} $$
 * even number
 * even number
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$$ \displaystyle n = 2q + \underbrace{1}_{\color{blue}{mod(n,2)}} $$
 * odd number
 * odd number
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Mtg 42: Clenshow-Curtis Quadrature


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 * $$ \displaystyle I = \int^{b}_{a} g(t) dt = \int^{+1}_{-1} \underbrace{f(x)}_{\color{blue}{g(t(x))J}}dx $$
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 * $$ \displaystyle x = (t - \frac{a+b}{2}) / \frac{b-a}{2} = [2t - (a+b)] / (b-a) $$
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 * $$ \displaystyle t(x) = [(b-a)x + (a+b)] / 2 $$
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 * $$ \displaystyle dt = \underbrace{\frac{(b-a)}{2}dx}_{\color{blue}{J}} $$
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 * $$ \displaystyle I = \int^{+1}_{-1} f(x) dx = \sum^{n}_{i=0} \underbrace{w_{i}}_{\color{blue}{weight}} f(x_{i}) $$
 *   (1)
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 * Transform $$ \displaystyle \underbrace{x=cos \theta}_{\color{blue}{Chebyshev\,\, polynomial}} $$
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 * $$ \displaystyle dx = -sin \theta d \theta $$
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 * $$ \displaystyle I = \int^{\pi}_{0} \underbrace{f(\underbrace{cos \theta}_{\color{blue}{even}}) \underbrace{sin \theta}_{\color{blue}{odd}} d \theta}_{\color{red}{y(\theta) \color{blue}{odd,\,\, periodic\,\, on\,\, [0,2 \pi]\,\, or\,\, [- \pi, + \pi] }}} $$
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Cosine series of $$ /displaystyle f(cos \theta) $$


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 * $$ \displaystyle f(cos \theta) = \frac{a_{0}}{2} + \sum^{\infty}_{k=1} a_{k} cos(k \theta)$$
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 * $$ \displaystyle \Rightarrow I = a_{0} + \sum^{\infty}_{k=1} \frac{2a_{2k}}{1-(2k)^{2}} $$
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 * $$ \displaystyle a_{k} = \frac{2}{\pi} \underbrace{\int^{\pi}_{0} \underbrace{f(cos \theta) cos k \theta}_{\color{blue}{Z(\theta)}} d \theta}_{\color{blue}{\alpha\,\, can\,\, be\,\, integrated}} $$
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efficiently with Discrete Cosine Transform(DCT, Trapezoidal Rule)


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 * $$ \displaystyle Z(\theta)\,\, \underline{periodic}\,\, on\,\, [0, 2 \pi]\,\, or\,\, [-\pi, +\pi] $$
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 * $$ \displaystyle \bar{\alpha} := \int^{+\pi}_{-\pi} \underbrace{Z(\theta)}_{\color{blue}{even\,\, \Rightarrow\,\, Z(-\theta)=Z(\theta)}} d \theta = 2 \alpha $$
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 * $$ \displaystyle \theta_{k} = k \frac{\pi}{n},\,\, k=-1, -(n-1),...,-1,0,1,2,...,(n-1),n $$
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Trapezoidal Rule for $$ \displaystyle \bar{\alpha} $$


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 * $$ \displaystyle \bar{\alpha} = \underbrace{h}_{\color{blue}{\frac{\pi}{n}}} [\frac{1}{2} Z(\theta_{-n}) + Z(\theta_{-(n-1)}) + ... + Z(\theta_{-1}) + Z(\theta_{0}) + Z(\theta_{1}) +...+ Z(\theta_{n-1}) + \frac{1}{2} Z(\theta_{n})] $$
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 * $$ \displaystyle \bar \alpha = h \color{blue}{\underset{k=-n}{\overset{n}{{\sum}^{\prime\prime}}}} \color{black}{Z(\theta_k)} $$
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[[media: Egm6341.s10.mtg46.djvu | Page 46-5]] 


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& Z(\theta_{0}) = \frac{1}{2} [Z(\theta_{0}) + Z(\theta_{0})] \\\end{align} $$
 * $$ \displaystyle Z(\cdot)\,\, even\,\, \Rightarrow\,\, use \begin{align} & Z(\theta_{k}) = Z(\theta_{-k}) \\
 * $$ \displaystyle Z(\cdot)\,\, even\,\, \Rightarrow\,\, use \begin{align} & Z(\theta_{k}) = Z(\theta_{-k}) \\
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 * $$ \displaystyle \bar{\alpha} = 2h \underbrace{[\frac{1}{2} Z(\theta_{0}) + Z(\theta_{1}) + ... + Z(\theta_{n-1}) + \frac{1}{2} Z(\theta_{n})]}_{\color{blue}{ \underset{k=0}{\overset{n}{{\sum}^{\prime\prime}}} Z(\theta_k) }} $$
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 * $$ \displaystyle \Rightarrow\,\, \alpha = h \underset{k=0}{\overset{n}{{\sum}^{\prime\prime}}} Z(\theta_k) $$
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 * $$ \displaystyle a_{k} = \underbrace{\frac{2}{\pi} h}_{\color{blue}{\frac{2}{n}}} \underset{i=0}{\overset{n}{{\sum}^{\prime\prime}}} Z(\theta_{i}) $$
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