User:Egm6341.s10.team3.heejun/Meeting 18 transcript

 '''MTG 18: Tue. 2 Feb 10'''


 * {| style="width:100%" border="0" align="left"


 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

Note:  Typeset of Transparencies, not lecture transcript  Heejun Chung 20:59, 11 August 2010 (UTC)


 * style = |
 * }
 * }

[[media: Egm6341.s10.mtg18.djvu | Page 18-1]] 


 * {| style="width:100%" border="0" align="left"

HW: Continue from [[media: Egm6341.s10.mtg17.djvu | Page 17-3]]
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

(3) $$\displaystyle  f(x) = x^3 \sqrt{x} \,\,, x\in [a,b] = [0,1]$$ Find the Peano Kernel error estimates for the Composite Trapezoidal and Composite Simpson rules with $$ \displaystyle n=2,4,8,16,32,64\, and\,128.$$ Then Comment about results.
 * Reference: Table 5.4   A. p. 261

(4) $$\displaystyle f(x) = \frac{1}{1+(x-\pi)^2} \,\,, x\in [a,b] = [0,5]$$ Find the Peano Kernel error estimates for the Composite Trapezoidal and Composite Simpson rules with $$ \displaystyle n=2,4,8,16,32,64\, and\,128.$$ Then Comment about results.
 * Reference: Table 5.5   A. p. 262

(5) $$\displaystyle f(x) = \sqrt{x} \,\,, x\in [a,b] = [0,1]$$ Find the Peano Kernel error estimates for the Composite Trapezoidal and Composite Simpson rules with $$ \displaystyle n=2,4,8,16,32,64\, and\,128.$$ Then Comment about results.
 * Reference: Table 5.6   A. p. 262

(6)   Periodic functions   $$\displaystyle  f(x) = e^{cos(x)} \,\,, x\in [a,b] = [0,2\pi]$$ Find the Peano Kernel error estimates for the Composite Trapezoidal and Composite Simpson rules with $$ \displaystyle n=2,4,8,16,32,64\, and\,128.$$ Then Comment about results.
 * Reference: Table 5.7   A. p. 263
 * style = |
 * }.
 * }.

Motivation: Higher-order error analysis of Trapezoidal rule (Simple + Composite), Euler-Maclaurin series.

[[media: Egm6341.s10.mtg18.djvu | Page 18-2]] 

* Corrected Trapezoidal rules (not intuitive, need mathematics)

* Chebyshev polynomial expension, Runge phenomenon  (use transfomation of variable to a periodic function to fix Runge phenomonon)

* Clenshaw-Curtis quadrature discrete cosine transform

* Richardson extrapolation Romberg integration table

Thm: Higher-order error for Trapezoidal rule

Error $$ \displaystyle E_{n}^{1} := I-\underbrace{I_{n}}_{\color{blue}{T(n) \equiv T_{n}}} $$ is a (Euler-Maclaurin) series of   even orders (even powers)  of $$ \displaystyle h := \frac{b-a}{n},\,\,  i.e.,\,\, h^{2i}\,\,\,\, (i=1,2,...) $$

[[media: Egm6341.s10.mtg18.djvu | Page 18-3]] 
 * $$ \displaystyle \color{blue}{T} $$ means Trapezoidal rule (Composite)


 * {| style="width:100%" border="0" align="left"

&= \sum_{i=1}^{\infty} a_{i}h^{2i} \\\end{align}\ $$
 * $$ \displaystyle \begin{align} E^{1}_{n} &= a_{1}h^{2 \cdot 1} + a_{2}h^{2 \cdot 2} + ... \\
 * $$ \displaystyle \begin{align} E^{1}_{n} &= a_{1}h^{2 \cdot 1} + a_{2}h^{2 \cdot 2} + ... \\
 *   (1)
 * }.
 * }.

Richardson extrapolation


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle E^{1}_{n} := I-T(n) = a_{1}h^{2} + \underbrace{a_{2}h^{4}  + ... }_{\color{blue}_{O(h^{4})}} $$
 *   (2)
 * }.
 * }.
 * }.

In this case, O means every higher order terms such as $$ \displaystyle \color{blue}{O}: a_{2}h^{4} + ... $$
 * Def. of O : known as big O notation (also known as Big Oh notation, Landau notation, Bachmann–Landau notation, and asymptotic notation in mathematics or computer science.


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle E^{1}_{2n} = I-T(2n) = \underbrace{a_{1}{(\frac{h}{2})^{2}}}_{\color{blue}_{\frac{a_{1}}{4}h^{2}}} + \color{blue}{O(h^{4})} $$
 *   (3)
 * }.
 * }.
 * }.



$$ \displaystyle T(2n) $$ composite based on $$ \displaystyle T(n), $$ and $$ \displaystyle f(x_{i}),\,\,\,\, i=1,3,5,... $$

If   (3)  is multiplied by number 4,

$$ \displaystyle 4 \times \color{red}{(3)} - \color{red}{(2)} \color{black}{:} 3I - 4T(2n) + T(n) = O(h^{4})$$

[[media: Egm6341.s10.mtg18.djvu | Page 18-4]] 


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle \Rightarrow I = \underbrace{\frac{4T(2n) - T(n)}{3}}_{\color{blue}_{T_{1}(n)}} + O(h^{4}) $$
 *   (1)
 * }.
 * }.
 * }.


 * Question: Cancel 4th order term $$ \displaystyle a_{2}h^{4}? $$


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle I-T_{1}(n) = a_{2}h^{4} + O(h^{6}) $$
 *   (2)
 * }
 * }
 * }


 * {| style="width:100%" border="0" align="left"


 * $$ \displaystyle I-T_{1}(2n) = a_{2} (\frac{h}{2})^{4} + O(h^{6}) $$
 *   (3)
 * }.
 * }.
 * }.


 * $$ \displaystyle 2^{4} \times \color{red}{(3)} - \color{red}{(2)} $$