User:EGM6341.s11.team1.Chiu/Mtg12

Mtg 12: Wed, 26 Jan 11


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Appl. of LIET p.11-2

$$Let \ {f}^{L}_{n}(t) \ be \ Lagrange \ interpolation \ of \ order \ n \ to \ f(t). \ \Rightarrow \ {f}^{L}_{n}(.) \ \in \ \color{blue}  \underset{set \ of \ poly \ of \ degree \  \leqslant \ n}{ \underbrace{ {\color{black}P_{n} }}}$$

$${\color{blue} n=1} \ {f}^{L}_{1}(.) \ interpolation \ exactly$$

$${\color{blue} P_{1}(x) \rightarrow}straight \ lines \ (linear \ functions) \ i.e., \ e_{1}(P_{1},t) \ \equiv \ 0 \ \forall \ t$$


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$${\color{blue}n=2} \ {f}^{L}_{2}(.) \ interpolation \ exactly \ parabolas \ (polynomial \ of \ {\color{blue}\underset{degree}{ \underbrace{\color{black}order}}}) \ 2$$

$${\color{blue}n=3} \ cubic \ polynomial$$

$$\color{blue} \vdots$$

$${\color{blue}n} \ {f}^{L}_{n}(.) \ interpolation \ exatcly \ \color{blue}\overset{ \in P_{n}}{ \overbrace}$$

$$i.e., \ e_(p_;t) \ =0 \ \forall t$$

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LIET: (2) p.11-3

$$IF \ f \ = \ p \in \ P_{n} \ \Rightarrow \ f^{(n+1)} \ \equiv \ 0$$

$${\color{red} (2)} \ {\color{blue} \ p.11-3} \ \Rightarrow \ e_(p,t)=0 \ \forall t$$

$$i.e., \ {f}^{L}_{n}(.) \ interpolation \ {\color{red}exactly} \ p\in P_{n}$$

$$\color{blue} Method \ to \ comp. \ \left\{w_{i,n}; \ i=0,..,n \right\} \ given \ \left\{x_{i}; \ i=0,..,n \right\} \ \underline {not} \ necessarily \ equidistant.$$

$${\color{blue}\underline {j=0}} \ consider \ f=p_{\color{blue}j}=p_{0}\in P_{0} \ {\color{red}constant}$$

$$e_{\color{blue}n} \left( \underset{{\color{blue} \underbrace{{\color{black}p_{\color{red}0}}}};t} \right) \ = {\color{blue}\underset{1}{ \underbrace}}- {f}^{L}_{{\color{blue}n}(t)}={\color{blue}1}-{f}^{L}_{n}(t)=0$$

$$\Rightarrow {\color{blue} \underset{ \sum_{i=0}^{n}l_{i,n}(t){\color{red} \underset{1}{ \underbrace}}}{ \underbrace}}=1 \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \left( \underset{{f}^{L}_{n}(t)}{ \underbrace}\right)}dt \ = \ {\color{blue} \underset{b-a}{ \underbrace}}$$

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$$\Rightarrow \ {\color{blue} \underset{w_{i,n}}{ \underbrace{ \left( {\color{red} \int_{a}^{b}}l_{i,n}(t)dt\right)}}.{\color{red}1}}=b-a$$

$$\Rightarrow$$

$${\color{blue} \underline {j=1}} \ let \ f=p_=p_{1}\in P_{1}$$

$$choose \ f(x)=p_{1}(x)= \ {\color{red}x} \in P_{1}$$

$$e_{\color{blue} \left( \underset{ \underbrace}{\color{black};t }\right)} \ = \ {\color{blue} \underset{ \underbrace}}-{f}^{L}_{\color{blue}n}(t) ={\color{blue}t}-f_{n}(t)=0$$

$$\Rightarrow \underset{ {\color{blue}\underbrace{\color{black}{f}^{L}_{n}(t)}}}={\color{red}t} \ \Rightarrow \ {\color{red} \int_{a}^{b}}{\color{blue} \underset{{f}^{L}_{n}(t)}{ \underbrace{ \left( {\color{black} \sum_{i=0}^{n}l_{i,n}(t).}{\color{red}x_{i}}\right)}dt}}={\color{blue} \underset{(b^{2}-a^{2})/2}{ \underbrace}}$$

$$\Rightarrow$$