User:EGM6341.S11.team5.cavalcanti/Mtg7

'''Mtg. 7: Mon, 17 Jan 11 [[media: Nm1.s11.mtg7.djvu | Page 7-1]] '''

(2) p. 6-5:

Error:

$$\begin{array}{c}{E}_{n}\left(f\right)=I\left(f\right)-{I}_{n}\left(f\right)\\ ={\int }_{0}^{1}\left\lbrack f\left(x\right)-{f}_{n}\left(x\right)\right\rbrack \mathit{dx}\\ ={\int }_{0}^{1}\underbrace{\underbrace{\frac{{x}^{n}}{\left(n+1\right)!}}_{\color{blue}{W(x)}} \underbrace{\exp \left\lbrack \xi \left(x\right)\right\rbrack }_{\color{blue}{g(x)}}\mathit{dx}}_{\color{blue}{IMVT} \ \color{red}{(1)} \ p.6-3}\end{array}$$

$${E}_{n}=\underbrace{g(\alpha )}_{\color{blue}{exp[\xi (\alpha)]}}\underbrace{{\int }_{0}^{1}W\left(x\right)\mathit{dx}}_{\color{blue}{\frac{1}{(n+1)!(n+1)}}},\alpha \in \left\lbrack \mathrm{0,1}\right\rbrack $$

[[media: Nm1.s11.mtg7.djvu | Page 7-2]]  $$\begin{array}{c}\mathit{\min } \ g(\alpha )=\mathrm{1,}\forall \alpha \in \left\lbrack \mathrm{0,1}\right\rbrack \\ \mathit{\max } \ g(\alpha )=e,\forall \alpha \in \left\lbrack \mathrm{0,1}\right\rbrack \end{array}$$

$$\frac{1}{\left(n+1\right)!\left(n+1\right)}\le {E}_{n}\le \frac{e}{\left(n+1\right)!\left(n+1\right)}$$

$${I}_{6}=1.3178\mathrm{...}$$ A. p.250 $$\color{blue}{n = 6 \to} $$$$2.83 \times 10^{-5} \leqslant E_{6}=I-I_{6} \leqslant 7.70 \times 10^{-5} $$

[[media: Nm1.s11.mtg7.djvu | Page 7-3]] Trapezoidal Rule:

Simple Rule: [[media: Nm1.s11.mtg7.djvu | Page 7-4]]

Composite rule:

$${f}_{i}=f\left({x}_{i}\right),i=\mathrm{0,1,.}\mathrm{..},n$$

Simpson's Rule: Use 2nd order polu (parabola) to approx. f(.).

Simple rule:

2 intervals: $$\left\lbrack {x}_{0},{x}_{1}\right\rbrack ,\left\lbrack {x}_{1},{x}_{2}\right\rbrack $$

Composite Rule:

$$n=\mathrm{2k},k=\mathrm{0,1,2,.}\mathrm{..}h=\left(b-a\right)/n$$

[[media: Nm1.s11.mtg7.djvu | Page 7-5]]  Gauss-Legendre quadrature: See FEA1 F10 for more info.

$$\left\lbrace {x}_{i},i=\mathrm{1,}\mathrm{...},n\right\rbrace $$ = roots of Pn(x), Legendre poly. of order n.

$$\left\lbrace {W}_{i},i=\mathrm{1,}\mathrm{...},n\right\rbrace $$ = weights associated w. {x i}

http://en.wikipedia.org/wiki/Gaussian_quadrature