User:EGM6341.S11.team5.cavalcanti/Mtg11

Mtg11: (EDGE lect 8)

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Note on HW*2.12 p.10-4:

Do NOT use matlab commands “polyfit”, “polyeval”, “quad” to do this pb.; only to verify results at the end.

Evaluate $$\color{blue}{\left\lbrace {W}_{i,n};i=\mathrm{1,.}\mathrm{..},n\right\rbrace} $$in (2) p.9-2 exactly (use method in Mtg 12)

Form tables of $$\left\lbrace \left({x}_{i},{w}_{i}\right),i=\mathrm{1,.}\mathrm{..},n\right\rbrace $$for $$\left\lbrack a,b\right\rbrack =\left\lbrack -\mathrm{1,}+1\right\rbrack $$ n=1,2,4,8,16. Why? Uniform discretization.

Obseve on the construction of these tables.

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Why tables?

$$f(x)=f(x(y))=\overline{f}(y)$$ (end note on HW*2.12)
 * 1) Reuse tables for different integrals (with different integrand f(x) and different domain of integration [a,b])
 * 2) For integration with arbitrary domain [a,b] (e.g., [a,b]=[2,15]), do a tranf. Of variable x=x(y) such that

Lagrange interpolation Error: Theorem (LIET)

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

f(.) continuous with (n+1) continuous derivative on $${I}_{t}=F\left(t,{x}_{0},{x}_{1},\mathrm{...},{x}_{n}\right)$$

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 * = smallest interval containing points $$\left(t,{x}_{0},{x}_{1},\mathrm{...},{x}_{n}\right)\mathit{st}{x}_{0}< {x}_{1}< \mathrm{...}< {x}_{n}$$ (t anywhere in $$ \color{blue}\mathbb{R}$$)

Case 1: $$t < x_{0}$$

$$I_{t} = [t, x_{n}]$$



Case 2: $$\color{blue}x_{0} \leqslant t \leqslant x_{n}$$

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



Case 3: $$\color{blue}x_{n} \leqslant t $$

$$I_{t} = [x_{0}, t]$$ Then,