User:Egm6341.s10.team3.heejun/Meeting 14 transcript

 MTG 14: Tue. 26 Jan 10

[[media: Egm6341.s10.mtg14.djvu | Page 14-1]] 

Theorem: Simple Simpson's Rule


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&= - \frac {h^{5}}{90}f^{4}(\xi),\,\,\,\, h:= \frac {b-a}{2} \\\end{align}\ $$
 * $$ \begin{align} \displaystyle E_{2} &= -\frac{(b-a)^{5}}{2880}f^{4}(\xi),\,\,\,\, \xi \in [a,b] \\
 * $$ \begin{align} \displaystyle E_{2} &= -\frac{(b-a)^{5}}{2880}f^{4}(\xi),\,\,\,\, \xi \in [a,b] \\
 * }
 * }

Reference: 1) Atkinson p.257, proof based on divided differences 2) Suli & Mayers = S&M, p.205 * E-book of Suli & Mayers, An introduction to numerical analysis (2003), can be found on UF libraries catalog

Use Lagrange  (L.) interpolating error

Remark: If $$ \displaystyle f \in \mathbb{P}_{3}\,\,\,\, \Rightarrow f^{(4)} = 0\,\,\,\, \Rightarrow E_{2}=0 $$

Simple Simpson exact for polynomial degree $$ \displaystyle \leq 3 $$

Proof: (Similar technique of proof for L. interpolating error) Shift origin of $$ \displaystyle x $$ axis to point $$ \displaystyle x_{1} $$: Transformation of valiable (fig. [[media: Egm6341.s10.mtg13.djvu | Page 13-3]])


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$$ \displaystyle h:= \frac {(b-a)}{2} $$
 * $$ \displaystyle x(t) = x_{1} + ht, \,\,\,\, t \in [-1,1] $$
 * $$ \displaystyle x(t) = x_{1} + ht, \,\,\,\, t \in [-1,1] $$
 * }.
 * }.

[[media: Egm6341.s10.mtg14.djvu | Page 14-2]] 


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$$ \displaystyle t=+1,\,\,\,\, x(+1)=x_{1} + h = x_{2} $$ $$ \displaystyle t=-1,\,\,\,\, x(-1)=x_{0} $$
 * $$ \displaystyle t=0,\,\,\,\, x(0)=x_{1} $$
 * $$ \displaystyle t=0,\,\,\,\, x(0)=x_{1} $$
 *  (1)
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 * $$ \displaystyle E_{2} = I - I_{2} = \underbrace{\frac{b-a}{2}}_ e(1) \underbrace{=}_ h \cdot e(1) $$
 *  (2)
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 * $$ \displaystyle \underbrace{e(t)}_ := \int_{-t}^{+t} \underbrace{f(x(t))}_ dt - \frac {t}{3} [f(-t) + 4F(0) + F(t)] = error\,\, of\,\, int.\,\, on \,\, [-t, +t] $$
 *  (3)
 * }.
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 * $$ \displaystyle \underbrace{e(1)}_ = error\,\, of\,\, int.\,\, on \,\, [-1, +1]$$
 * }.
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Def:


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 * $$ \displaystyle G(t):= e(t) - t^{5} e(1) $$
 *  (4)
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 * }.

Goal:


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 * $$ \displaystyle G^{(3)}(\xi_{3}) = 0 $$
 * }.
 * }.
 * }.


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 * $$ \displaystyle = - \frac {2(\xi_{3})^{2}}{3} [F^{4}(\xi_{4}) + 90 e(1)] $$
 *  (5)
 * }.
 * }.
 * }.


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 * $$ \displaystyle \Rightarrow e(1) = - \frac {1}{90} F^{4} (\xi_{4}) = - \frac {(b-a)^4}{1440}f^{4}(\xi) $$
 *  (6)
 * }.
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[[media: Egm6341.s10.mtg14.djvu | Page 14-3]] 


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 * $$ \displaystyle E(2) = h \cdot e(1) = - \frac {(b-a)^{5}}{2880}f^{4}(\xi) $$
 *  (1)
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Remark: Similarity with proof for L. interpolating error


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 * $$ \displaystyle G(t) := e(t) - t^{5} \cdot e(1) $$
 *  (4)   p.14-2
 * }.
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 * $$ \displaystyle G(x) := E(x) - \frac {q_{n+1}(x)}{q_{n+1}(t)} E(t) $$
 *  (1)   [[media: Egm6341.s10.mtg10.djvu | p.10-3]]
 * }.
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Error:


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 * $$ \displaystyle Simpson:\,\,\,\, (b-a)^{5} \leftrightarrow t^5 $$
 *  (1) <p style="text-align:right;">  p.14-3
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 * $$ \displaystyle L.\,\, interpolating\,\, error:\,\,\,\, \underbrace{q_{n+1}(x)}_ \leftrightarrow \underbrace{q_{n+1}(x)}_ $$
 * <p style="text-align:right;"> (3) <p style="text-align:right;">  [[media: Egm6341.s10.mtg10.djvu | p.10-1]]
 * }.
 * }.
 * }.

Montesquieu: Darwin NOVA: "What Darwin never knew."