User:EGM6341.S11.team5.cavalcanti/Mtg19

Mtg 19: Mon, 14 Feb 11

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Pf of error of Simple Simpson's rule: p. 18-4 cont'd

Tighter bound

Definition:

Goal: $$\exists \left({\xi }_{3}\right)\in \mathrm{0,1}$$ such that $${G}^{3}\left({\xi }_{3}\right)\underbrace{=}_{\color{red}{(2)}}0=\frac{-2{(\xi )}^{2}}{3}\left\lbrack {F}^{\left(4\right)}\left({\xi }_{4}\right)+90e\left(1\right)\right\rbrack $$

see (5) p. 18-4

$$\rightarrow e\left(1\right)=\frac{-1}{90}\underbrace{{F}^{\left(4\right)}\left({\xi }_{4}\right)}_{\color{blue}{\frac{d^4}{dt^4}F}}\underbrace{=}_{\color{red}{(3)}}\frac{-{\left(b-a\right)}^{4}}{1440}{f}^{\left(4\right)}(\xi )$$

(4) p.18-3: $${E}_{2}=\mathit{he}\left(1\right)=\frac{-{\left(b-a\right)}^{5}}{2880}{f}^{\left(4\right)}(\xi )$$

→ SSET p.18-2

(end goal)

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Note: Comp. G(.) for SSET (1) p.19-1 to G(.) for LIET (1)p.14-3

(1)p.19-1: $$G\left(\color{red}{x}\right)\mathrm{\colon }=e\left(\color{red}{x}\right)-{\color{red}{x}}^{5}e\left(1\right)$$

SSET p.18-2 $${E}_{2}=\frac{-{\left(b-a\right)}^{5}}{2880}{f}^{\left(4\right)}(\xi )$$

(1) p.14-3 $$G\left(x\right)=e\left(x\right)-\frac{{q}_{n+1}\left(x\right)}{{q}_{n+1}\left(t\right)}e\left(t\right)$$

$${e}_{n}^{\color{red}{L}}\left(f;t\right)\mathrm{\colon }\underbrace{=}_{\color{red}{(2)} \ \color{blue}{p.11-3}}f\left(t\right)-{f}_{n}^{L}\left(x\right)=\frac{{q}_{n+1}\left(t\right)}{\left(n+1\right)!}{f}^{\left(n+1\right)}(\xi )$$

Error: SSET p.18-2

(b – a)5 in error ↔ t5 in G(.)

LIET p.11-3

qn+1(x) in error ↔ qn+1(x) in G(.) "End Goal"

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Proof of SSET: (Similar to proof of LIET p. 16-2)

Apply Rolle's theorem.

(see (1) p.19-1)

(see pictorial interpretation p.20-1; also (5) p.18-3)