User:Egm6341.s10.team3.heejun/Meeting 10 transcript

 MTG 10: Tue. 19 Jan 10

[[media: Egm6341.s10.mtg10.djvu | Page 10-1]] 

Theorem of interpolation error (Atkinson. p.134)

This theorem is very important to know how big the error is committed after interpolating function (Lagrange Interpolation in previous lectures)

$$ \displaystyle f: \underbrace{\mathbb{R}}_{\color{blue}{domain}} \rightarrow \underbrace{\mathbb{R}}_{\color{blue}{range}}  $$  (set of real numbers)


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 * $$ \displaystyle f $$  differentiable with $$ \displaystyle (n+1) $$   continuous derivative on
 * $$ \displaystyle f $$  differentiable with $$ \displaystyle (n+1) $$   continuous derivative on

 $$ \displaystyle I_{t} := \mathbb{H} (t, x_{0}, x_{1}, ..., x_{n}) := $$ smallest interval containing points $$ \displaystyle (t, x_{0}, x_{1}, ... , x_{n})\,\,\,\, where\,\, x_{0} < x_{1} < ... < x_{n}\,\, and\,\, t\,\, is\,\, anywhere $$
 *  (1)
 * }.
 * }.


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 * Case 1: $$ \displaystyle \color{blue}{ t < x_{0}} $$
 * Case 1: $$ \displaystyle \color{blue}{ t < x_{0}} $$

$$ \displaystyle I_{t}=[t,x_{n}] $$
 * }
 * }


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 * Case 2: $$ \displaystyle \color{blue}{t \in [x_{0}, x_{n}]} $$
 * Case 2: $$ \displaystyle \color{blue}{t \in [x_{0}, x_{n}]} $$

$$ \displaystyle I_{t}=[x_{0},x_{n}] $$
 * }
 * }


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$$ \displaystyle I_{t}=[x_{0},t] $$   
 * Case 3: $$ \displaystyle \color{blue}{x_{n} < t} $$
 * Case 3: $$ \displaystyle \color{blue}{x_{n} < t} $$
 * }.
 * }.

Then, (Then means "imply", denoted by $$\displaystyle \Rightarrow $$ )


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 * $$ \displaystyle f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$ 
 * $$ \displaystyle f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$ 


 *  (2)
 * }
 * }


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 * $$ \displaystyle f(t) - f_{n}(t)= \underbrace{\frac{q_{n+1}(t)}{(n+1)!}f^{(n+1)}(\xi)}_{\color{blue}{error\,\, term}} $$
 * $$ \displaystyle f(t) - f_{n}(t)= \underbrace{\frac{q_{n+1}(t)}{(n+1)!}f^{(n+1)}(\xi)}_{\color{blue}{error\,\, term}} $$


 *  (3)
 * }
 * }


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 * $$ \displaystyle \xi \in I_{t}, $$ 
 * }
 * }
 * }


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&= \prod_{j=0}^{n}(x-x_{j}) \in \color{blue}{\mathbb{P}_{n+1}} \\\end{align}\ $$ 
 * $$ \begin{align} \displaystyle q_{n+1}&:=(x-x_{0})(x-x_{1})...(x-x_{n}) \\
 * $$ \begin{align} \displaystyle q_{n+1}&:=(x-x_{0})(x-x_{1})...(x-x_{n}) \\
 * <p style="text-align:right;"> (4)
 * }.
 * }.

<p style="text-align:left;">[[media: Egm6341.s10.mtg10.djvu | Page 10-2]] 

Proof: (Similar proof technique used in other error analyses)  <br\>

Note:

1) Montesquieu

Similarity and difference between  (3)   p.10-1 (i.e, approx error) and Taylor series remainder   (1)   [[media: Egm6341.s10.mtg2.djvu | P.2-3]]  <br\>

2) Consider $$ \displaystyle t=x_{j},\,\,\,\, j=0,1,...,n $$

RHS  (3)   p.10-1 = 0 since $$ \displaystyle q_{n+1}(x_{j})=0\,\,\,\, j=0,1,...,n $$

LHS  (3)   p.10-1 = 0 since


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 * $$ \displaystyle f(x_{j})- \underbrace{f_{n}(x_{j})}_{\color{blue}{=\,\, f(x_{j})}} = 0 $$ <br\>
 * }
 * }
 * }


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 * $$ \displaystyle f_{n}(x_{j})= \sum_{i=0}^{n} \underbrace{l_{i,n}(x_{j})}_{\color{blue}{=\,\, \delta_{ij}}} f(x_{j}) = f(x_{j}) $$ <br\>
 * }
 * }
 * }


 * Refer to fig. [[media: Egm6341.s10.mtg7.djvu | P.7-3]] for $$ \displaystyle f_{n}(x_{j})\,\, =\,\, f(x_{j}) $$ and  (1) & (4)  [[media: Egm6341.s10.mtg8.djvu | P.8-1 & 3]] for $$ \displaystyle l_{i,n}(x_{j})\,\, =\,\, \delta_{ij} $$  <br\>


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 * $$ \displaystyle E(x) := f(x)-f_{n}(x)\,\,\,\, \color{blue}{interpolating\,\, error} $$ <br\>
 * }.
 * }.
 * }.

<p style="text-align:left;">[[media: Egm6341.s10.mtg10.djvu | Page 10-3]] 


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 * $$ \displaystyle G(x) := E(x)- \frac{q_{n+1}(x)}{q_{n+1}(t)}E(t)\,\,\,\, x \in I_{t} $$ <br\>
 * }
 * }
 * }


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 * $$ \displaystyle G(\cdot)\,\, has\,\, (n+1)\,\, cont.\,\, deriv.\,\, $$ <br\>
 * }
 * }
 * }