User:Egm6341.s10.Team4.nimaa&m/HW3

= Problem 2: Error for composite Simpson's rule =

Given
Use composite Simpson's equation (3) on slide [[media:Egm6341.s10.mtg7.pdf|7-2]], which is:


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$$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle I_n=\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]


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Find
To demonstrate error of composite Simpson's rule is equal to the following:


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$$ $$
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 * $$\displaystyle
 * $$\displaystyle 	\left | {E_n}^2\right |\leqslant\dfrac{(b-a)^5}{2880n^4}M_4=\dfrac{(b-a)h^4}{2880}M_4



M_4:=max\left |f^{(4)}(\xi)\right | $$



\xi\in[a,b] $$
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Solution
The error definition can be done as:



{E_n}^2=I-I_n=\int\limits_{a}^{b} f(x)dx-I_n $$

From the composite form of Simpson's rule, we have:



I_n=\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)] $$



\Rightarrow {E_n}^2=\int\limits_{a}^{b} f(x)dx-\dfrac{h}{3}[f(x_0)+4f(x_1)+2f(x_2)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n)]= $$



=\sum_{i=1}^{n/2} [\int\limits_{x_{2i-2}}^{x_{2i}} f(x)dx-\dfrac{h}{3}[f(x_{2i-2})+4f(x_{2i-1})+f(x_{2i})]] $$

where,

h=\frac{b-a}{n} = x_i-x_{i-1} $$

According to Lagrange interpolation error for simple Simpson's rule (refer to [14-1]) for $$\displaystyle [x_{i-1},x_{i+1}] $$ ;



E_2=\dfrac{x_{i+1}-x_{i-1}}{2} f^{(4)}(\xi) $$



\xi\in [x_{i-1},x_{i+1}] $$



\Rightarrow \left |{E_n}^2 \right | \le \sum_{i=1}^{n/2} max \left | \dfrac {(x_{i+1}-x_{i-1})^5}{90\times 2^5} f^{(4)}(\xi)\right | =\dfrac{h^5}{90}\times \sum_{i=1}^{n/2} max \left | f^{(4)}(\xi)\right | $$



h=\dfrac {x_{i+1}-x_{i-1}}{2}=\dfrac {b-a}{n} $$



M_4:=max\left | f^{(4)}(\xi) \right | $$



\xi\in [x_{i-1},x_{i+1}] $$



\overline{M_4}:=\sum_{i=1}^{n/2} max\left |f^{(4)}(\xi) \right | $$



\overline{M_4}\le max\left | \dfrac {n}{2} .M_4 \right | $$



\Rightarrow\left |{E_n}^2\right |\le\dfrac{h^5}{90}\times \dfrac {n}{2}.M_4=\dfrac{h^5}{90}\times \dfrac {\dfrac{b-a}{h}}{2}.M_4 $$


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$$\displaystyle \Rightarrow\left |{E_n}^2\right | \le \dfrac{(b-a)h^4}{180} M_4 $$
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'''This answer is different from the asked question, However, this answer is confirmed in the Atkinson's text book "An Intro, Numerical Analysis" 2nd edition (page 257~258)
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Author
Solved and typed by - Egm6341.s10.Team4.nimaa&amp;m 15:31, 17 February 2010 (UTC) Reviewed by - Egm6341.s10.Team4.andy 16:23, 17 February 2010 (UTC) .