User:EGM6341.s11.team1.Chiu/Mtg8

Mtg 8: Wed, 19 Jan 11

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Legendre poly. Pn(x):


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$$ {P}_{n}(x)= \sum_{i=0}^{{\color{red}[{\color{black}n/2}]} } {(-1)}^{i} \frac{(2n-2i)! {x}^{n-2i}}{ {2}^{n}i!(n-i)!(n-2i)!}$$

(1)
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$$ {\color{red}[ {\color{black} n/2}]} = integer \ part \ of \ n/2$$

(2)
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e.g., m = 5, n/2 = 2.5 , [2.5] = 2

$$ {\color{red}(3)} \ {P}_{0}=1 \color{blue} \in {P}_{0} $$

$$ {\color{red}(4)} \ {P}_{1}=x \color{blue} \in {P}_{1} $$

$$ {\color{red}(5)} \ {P}_{2}= \frac{1}{2}(3{x}_{2}-1)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(6)} \ {P}_{3}= \frac{1}{2}(5{x}_{3}-3x)  \color{blue} \in {P}_{1} $$

$$ {\color{red}(7)} \ {P}_{4}= \frac{35}{8}{x}^{4}-\frac{15}{4}{x}^{2}+\frac{3}{8}  \color{blue} \in {P}_{1} $$

$${\color{blue}{P}_{n}} \ = \ set \ of \ poly. \ of \ degree \ \leqslant \ n$$

HW 2.6:Verify(3)-(7) using(1)-(2)

$${\color{red}(5)} \Rightarrow \ {P}^{2}=0 \Rightarrow {x}_{1,2} = {\color{red}-} \frac{1}{\sqrt{3}} \ or {\color{red}+} \frac{1}{\sqrt{3}}$$

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Weights wi, i = 1,...,n ((1)p.7-5)

$$ \underline { \color{blue} {NOTE:}} \ {E}_{n}(f)=0 \ \forall f \in {P}_{2n-1} \ since \ f^{\color{red}(2n)}(x)=0 \. \ Only \ need \ to \ use \ n \ int. \ pts\ \left\{x_{i}, \ i=1,...n\right\} \ to \ int. \ exactly \ any \ poly \ in \ P_{2n-1} \ i.e., \ of \ degree \ \leqq \ 2n-1$$

$${\color{blue} {Newton-cotes \ method:}} \ I(f)$$

$$History \ Newton \ cotes \ \color{blue}  \rightarrow \ Lecture \ plan \ suli+Meyers(2003)$$

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$${\color{blue}1)} \ Approx. \ f(.) \ using \ {\color{blue} { \underline { {\color{black} Lagrange}}}} \ interp. \  funcs \  \Rightarrow \  {f}^{ {\color{red} L}}_{n}(x) $$

$$ Int. \ exactly \ {f}^{ {\color{red} L}}_{n}(x) \ \Rightarrow \ \color{blue}  \underbrace{ \color{black} I( {f}^_{n})}  $$

$$I_{n}(f) \ := I( {f}^_{n})=\int {f}^_{n}(x)dx \ \color{red} (1) $$

$${f}^_{n}(x)=P_{n}(x)= \sum_{i=0}^{\infty} {\color{blue} \underset{Lagrange \ interp. \ func}{ {\color{black} \underbrace{l_{i,n}(x)}}}} \ f(x_{i}) \ \color{red} (2)$$

NOTE: Demonstrated Wolfram Alpha(WA) e.g., (debt usa)/(gdp usa) integrate x from 0 to 1 link WA comp. results in HW reportsAvoid plagiarism