User:Egm6341.s11.team5.shin/HW4

= Problem 4.5 - Proof of e(1) in Simple Simpson Error = From [[media:nm1.s11.mtg20.djvu|Mtg 20-3]]

Given

 * {| style="width:100%" border="0" align="left"

e(1)=-\frac{1}{90}F^{4}(\zeta_4) $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-1) $$
 * }
 * }

Find
Prove $$\displaystyle e(1)=-\frac{(b-a)^4}{1440}f^{(4)}(\xi)$$ and find the relationship between $$\displaystyle \zeta_4 $$ and $$\displaystyle \xi $$.

Solution
Before solving this problem, the following should be noted.
 * {| style="width:100%" border="0" align="left"

\begin{align} F(t) &= f(x(t)) \\ x&=x_1+ht \\ \mathrm{d}x &=h \mathrm{d}t \\ \xi&=x_1+h \zeta_4 \mbox{, } (t=\zeta_4,x=\xi) \\ \end{align} $$ With the equations above, the proof can be made as below.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-2)$$
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\begin{align} F^{(4)}(\zeta_4)&=\frac{\mathrm{d}^4}{\mathrm{d}t^4}F(\zeta_4) \\ &=\frac{\mathrm{d}^4}{(\frac{\mathrm{d}x}{h})^4}f(\xi) \\ &=h^4\frac{\mathrm{d}^4}{\mathrm{d}x^4}f(\xi) \\ &=(\frac{b-a}{2})^4f^{(4)}(\xi) \mbox{ ,} h=\frac{b-a}{2} \end{align} $$ Substitute Eq.(5-3) into Eq.(5-1).
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-3)$$
 * }
 * }
 * {| style="width:100%" border="0" align="left"

\begin{align} e(1)&=-\frac{1}{90}\frac{(b-a)^4}{16}f^{(4)}(\xi) \\ &=-\frac{(b-a)^4}{1440}f^{(4)}(\xi) \end{align} $$.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (5-4)$$
 * }
 * }

Author and proof-reader
[Author] Shin

[Proof-reader] = Problem 4.10 - Evaluation of the error using both composite Trapezoidal and composite Simpson rule = From [[media:Nm1.s11.mtg22.djvu|Mtg 22-3]]

Given
Part a,b.
 * $$\displaystyle

I=\int_0^{\pi} \! e^x \sin{x} \, \mathrm{d}x. $$ Part c.
 * $$\displaystyle

I=\int_0^{1} \! x^{3.7} \, \mathrm{d}x. $$

Find
Part a.
 * Produce Table 5.1, page 255, Atkinson text book using the composite Trapezoidal rule.

Part b.
 * Produce Table 5.3, page 258, Atkinson text book using the composite Simpson rule.

Part c.
 * Produce Table 5.4, page 261, Atkinson text book using the composite Trapezoidal & composite Simpson rule.

Part a.

 * From the class notes (p.7-4), the composite Trapezoidal rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=h(\dfrac{1}{2}f(x_0)+f(x_1)+f(x_2)+...+f(x_{n-1})+\dfrac{1}{2}f(x_{n})) \mbox{, }h=\dfrac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-1)
 * }
 * The asymptotic error is defined as the following equation.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\overline{E_n^T}=-\frac{h^2}{12}[f^{(1)}(b)-f^{(1)}(b)] $$ $$ The exact integration was performed using Wolfram Alpha as below.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-2)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

I = \int_0^{\pi} \! e^x \sin{x} \, \mathrm{d}x = 12.0703 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-3)
 * }
 * }

 Table: 

Matlab code:
 * {| style="width:100%" border="0" align="left"

.
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

Part b.

 * From the class notes (p.7-4), the composite Simpson's rule is defined as below.
 * {| style="width:100%" border="0" align="left"

I_n=\dfrac{h}{3}(f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+...+2f(x_{n-2})+4f(x_{n-1})+f(x_n))\mbox{, }h=\frac{b-a}{n} $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-4)
 * }
 * The asymptotic error is defined as the following equation.
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

\overline{E_n^S}=-\frac{h^4}{180}[f^{(3)}(b)-f^{(3)}(b)] $$ $$  Table: 
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-5)
 * }
 * }

Matlab code:
 * {| style="width:100%" border="0" align="left"

.
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

Part c.
The exact integration was performed using Wolfram Alpha as below.
 * {| style="width:100%" border="0" align="left"

I = \int_0^{1} \! x^{3.7} \, \mathrm{d}x = 0.212766 $$ $$
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (10-6)
 * }
 * }

 Table: 

Matlab code:
 * {| style="width:100%" border="0" align="left"

.
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * style="width:50%; padding:10px; border:1px solid #888888" align="center" |
 * }
 * }
 * }

Author and proof-reader
[Author] Shin [Proof-reader]