User:EGM6341.s11.team1.Chiu/Mtg14

Mtg 14: Mon, 31 Jan 11

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HW 3.1:
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"f \left( \frac{7 \pi}{8} \right)- {f}^{T}_{n} \left( \frac{7 \pi}{8} \right) \right"

"from $ \underbrace{Theorem \ \color{blue} p.11-3:}_{ \color{red} LIET} e{}^{L}_{4} \left( t \right)= \frac{{q}_{4+1} \left( t \right)}{(4+1)!}\underbrace{{f}^{(4+1)} \left( \xi \right)}_{ \color{blue} {sin}^{(5)}\left( \xi \right)} \ $|undefined"

"e{}^{L}_{4} \left( t \right) \right"


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Proof of  LIET: (A. p.134)


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[[media: Nm1.s11.mtg10.djvu| (5) page10-2]], [[media: Nm1.s11.mtg8.djvu| (2) page8-3]]  $$ {f}^{L}_{n} \left( x \right)= \sum_{i=0}^{n} {l}_{i,n} \left( x \right)f \left(  {x}_{i} \right) $$
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$$  \displaystyle \color{blue} x= {x}_{j}, j=0,1, \ldots ,n \Rightarrow \color{black} {f}^{L}_{n} \left( {x}_{j} \right)= \sum_{i=0}^{n} \underbrace{{l}_{i,n} \left( {x}_{j}  \right)}_{ \color{blue} {\delta}_{ij} }f \left(  {x}_{i} \right)=f \left( {x}_{j} \right) $$
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$$  \displaystyle G \left( \underset{ \color{red} \underset{variant}{\uparrow}}x \right):=e{}^{L}_{n} \left( f; \underset{ \color{red} \underset{variant}{\uparrow}}x \right)- \frac{{q}_{n+1} \left( x \right)}{{q}_{n+1} \left( t \right)}e{}^{L}_{n} \left( f;\underset{ \color{red} \underset{fixed}{\uparrow}}t \right) $$
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$$  \displaystyle =e \left( x \right)- \underbrace{\frac{{q}_{n+1} \left( x \right)}{{q}_{n+1} \left( t \right)}e \left( t \right)}_{\color{blue} \frac{e \left( t \right)}{{q}_{n+1} \left( t \right)}{q}_{n+1} \left( x \right)} $$
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Scaling error $$ e(.) \ $$ relative to $$ {q}_{n+1}(.) \ $$:


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$$  \displaystyle \left.\begin{matrix} 1. \frac{e \left( x \right)}{{q}_{n+1} \left( x \right)} \\ 2. \frac{e \left( t \right)}{{q}_{n+1} \left( t \right)} \end{matrix}\right\}$$ Difference of scaled errors: $$d \left( x \right)= \color{blue}1.-2. $$
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"$ \Rightarrow {q}_{n+1} \left( x \right) \cdot d \left( x \right)=G \left( x \right) \ $" End Proof of LIET

Remember: $$ G \left( {x}_{i} \right)\underset{\underset{ \color{blue} HW 3.2}{\uparrow}}=0 \ $$ for $$ i=0,1, \ldots ,n \ $$

Derivative MVT & Rolle's Theorem

End Remember