User:EGM6341.S11.team5.cavalcanti/Mtg29

 Mtg 29:  Fri, 4 Mar 11 [[media:Nm1.s11.mtg29.djvu | Page 29-1]] [[media:Nm1.s11.mtg29.djvu | Page 29-2]] Higher order trapezoidal rule error (HOTRE): Recall p. 26-4extraordinary performance of trapezoidal versus Simpson for periodic function.

Theorem: (HOTRE) Error {| style="width:100%" border="0"
 * style = "width:95%" | $$E_{n}^{T}=I - \underbrace{I_{n}^{T}}_{\color{blue}{T(n) \equiv T_{n}}}$$ ||
 * (1)|}
 * (1)|}

is an Euler-Maclauring series of ever orders (even powers) of {| style="width:100%" border="0"
 * style = "width:95%" | $$ h = \frac{b-a}{n}$$ ||
 * (2)|}
 * (2)|}

{| style="width:100%" border="0"
 * style = "width:95%" |i.e., $$ E_{n}^{T}= \sum_{i=1}^{\infty} a_{i}h^{\color{red}{2i}} $$ ||
 * (3)|}
 * (3)|}

{| style="width:100%" border="0" \right]$$ ||
 * style = "width:95%" | $$ a_{i}=d_{i} \left[ f^{(2i-1)}(b) - f^{(2i-1)}(a)
 * style = "width:95%" | $$ a_{i}=d_{i} \left[ f^{(2i-1)}(b) - f^{(2i-1)}(a)
 * (4)|}

{| style="width:100%" border="0"
 * style = "width:95%" | $$ d_{i} = \underbrace{-B_{2i}}_{\color{blue}{Bernoulli \ number}}/(2i)!$$ ||
 * (5)|}
 * (5)|}

[[media:Nm1.s11.mtg29.djvu | Page 29-3]] {| style="width:100%" border="0"
 * style = "width:95%" | $$d_{1}=\frac{-1}{12}, \ d_{2}=\frac{1}{720}, \ d_{3} = \frac{-1}{30240}, \ ... $$||
 * (1)|}
 * (1)|}

{| style="width:100%" border="0"
 * style = "width:95%" | Application: periodic functions.$$ \Rightarrow \underbrace{f^{(2i-1)}(b)}_{\color{blue}{odd \ derivative}} =\underbrace{f^{(2i-1)}(a)}_{\color{blue}{odd \ derivative}}$$ ||
 * (2)|}
 * (2)|}

$$\Rightarrow E_{n}^{T} = 0 \ \forall n$$ Application Richardson extrapolation "energy cascade' in turbulance numerical weather prediction in 1920s! {| style="width:100%" border="0"
 * style = "width:95%" | $$E_{n}^{T}=I-T_{\color{red}{0}}(n) = a_{1}^{\color{red}{0}}h^{2}+a_{2}^{\color{red}{0}}h^{4}+ ...$$ ||
 * (3)|}
 * (3)|}

{| style="width:100%" border="0"
 * style = "width:95%" | $$E_{2n}^{T}=I-T_{\color{red}{0}}(2n) = \underbrace{a_{1}^{\color{red}{0}}\left( \frac{h}{2} \right)^{2}}_{\color{blue}{\frac{a^{0}_{1}}{4}h^2}}+ \underbrace{\emph{O}(g^{4})}_{\color{blue}{a_{2}^{0}\left( \frac{h}{2} \right)^{4}}}$$ ||
 * (4)|}
 * (4)|}

[[media:Nm1.s11.mtg29.djvu | Page 29-4]] Reuse function values previously evaluated: T(2n). Computation based on T(n) and $$f(x_{i}), i=1, 3, 5...$$ Cancel 2nd order term: {| style="width:100%" border="0"
 * style = "width:95%" | $$\color{blue}{4} \color{red}{(4)-(3)} \color{black}{: 3I-4T(2n)+T(n) = \mathbb{O}(h^4)}$$ ||
 * (1)|}
 * (1)|}

$$ \Rightarrow$$
 * {| style="width:100%" border="0"

$$  I = \underbrace{\frac{4T(2n)-T(n)}{3}}_{\color{blue}{T_{1}(n)}} + \mathbb{O}(h^{4}) $$ Cancel 4th order term: (1) & (4)p. 29-3
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 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 *      (2) |}
 * {| style="width:100%" border="0"

$$  I - T_{\color{red}1}(n)= \underbrace{a_{\color{black}2}^{\color{red}0}\left [ 4\left ( \frac{1}{2} \right )^{4}-1 \right ]\frac{1}{3}}_{\color{blue}{a_{2}^{1} \ \color{black}{ rel. \ to \ T_{1}}}}h^{4}+\mathbb{O}(h^{6}) $$
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 *      (3) |}
 * {| style="width:100%" border="0"

$$ I - T_{\color{red}1}(2n)= a_{2}^{1}\left ( \frac{h}{2} \right )^{4}+\mathbb{O}(h^{6}) $$ $$2^{\color{red}{2.2}}\color{red}{(4)} \color{black}{-} \color{red}{(3)} = \color{black}{2^{4}}\color{red}{(4)}\color{black}{-}\color{red}{(3)} \color{black}\Rightarrow$$ [[media:Nm1.s11.mtg29.djvu | Page 29-5]]
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 *      (4) |}
 * {| style="width:100%" border="0"

$$  \displaystyle (2^{4}-1)I - 2^{4}T_{1}(2n) + T_{1}(n) = \underbrace{\left( \frac{2^{4}}{2^{4}}-1 \right)}_{\color{red}{\rightarrow 0 }}a_{\color{red}{2}}^{1}h^{\color{red}{4}}+\left( \frac{2^{4}}{2^{6}}-1 \right)a_{3}^{1}h^{6}+ \mathbb{O}(h^{8}) $$
 * style="width:95%" |
 * style="width:95%" |
 *      (1) |}
 * {| style="width:100%" border="0"

$$   \Rightarrow I= \underbrace{\frac{2^{2\cdot \color{red}{2}}T_{1}(2n)-T_{1}(n)}{2^{2\cdot \color{red}{2}}-1}}_{\color{blue}{T}_{\color{red}{2}}\color{blue}{(n)}} + \underbrace{a_{\color{black}{3}}^{\color{red}{2}}h^{\color{red}{6}}+ \mathbb{O}(h^8)}_{\color{blue}{\mathbb{O}(h^6)}} $$ In general:
 * style="width:95%" |
 * style="width:95%" |
 *      (2) |}
 * {| style="width:100%" border="0"

$$  \begin{matrix} I=T_{\color{red}{k}}(n)+ \underbrace{a_{\color{red}{k+1}}^{\color{red}{k}}h^{2(\color{red}{k+1}\color{black})}+\mathbb{O}(h^{(2k+2)})}_{\color{blue}{\mathbb{O}(h^{2(k+1)})}} \\ T_{\color{red}{k}}(n)= \frac{2^{2\cdot \color{red}{2}}T_{\color{red}{k-1}}(2n)- T_{\color{red}{k-1}}(n)}{2^{2 \cdot \color{red}{2}}-1} \end{matrix}
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |

$$ Note: Richardson extrapolation also applied to numerical integration of ODEs to estimate error.  "End Note"  [[media:Nm1.s11.mtg29.djvu | Page 29-6]] Romberg Integration table:
 *      (3) |}



[[Media:Nm1.s11.mtg29.djvu | Page 29-7]] Hw 5.4: comments/ hints Continuous Linear Control Model:
 * {| style="width:100%" border="0"

$$  \displaystyle \dot{\underline{x}}=\underline{A} \ \underline{x}+\underline{B} \ \underline{u} $$ Discrete Linear Control Model: Discretization:
 * style="width:95%" |
 * style="width:95%" |
 *      (1) |}


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

$$  \displaystyle \dot{\underline{x}}\cong \frac{\underline{x}_{k+1}-\underline{x}_k}{h}
 * style="width:95%" |
 * style="width:95%" |

$$
 *      (2) |}

Forward Euler Method:


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

$$  \displaystyle \frac{1}{h}\left [ \underline{x}_{k+1} -\underline{x}_{k}\right ]=\underline{A} \ \underline{x}_{k}+\underline{B} \ \underline{u}_{k} $$
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 * style="width:95%" |
 *      (3) |}


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

$$  \displaystyle \underline{x}_{k+1}=\underbrace{\left [ \underline{I}+h\underline{A} \right ]\underline{x}_{k}}_{\color{blue}{\underline{F}}}+\underbrace{h\underline{B} \ \underline{u}_{k}}_{\color{blue}{\underline{G}}} $$ Note: $$h \equiv \color{red}{\Delta}$$ in HW 5.2 p.25-2
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 *      (4) |}