User:Egm6341.s10.team3.heejun/HW4

=  6.1 b) Comparison 2nd column of Romberg Table to Composite Simpson's rule =

Problem Statement
$$ \displaystyle Compare\, the\, second\, column\, of\, Romberg\, Table\, Romberg\, to\, Simpson's\, rule\, and \, derive any relationship,\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-1]]

Solution
$$ \displaystyle a)\, Formula\, for\, Composite\, Trap.\, rule\, is\, given\, by\, $$ $$ \int_{a}^{b}f(x)dx=\frac{b-a}{2n}*[f(x_{0})+2f(x_{1})+2f(x_{2})+.........+2f(x_{n-1})+f(x_{n})] $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg7.pdf|p.7-1]]

$$ \displaystyle b)\, Formaula\, for.\, Romberg\, table\, is\, given\, by\, $$ $$ T_{k}(n)=\frac{2^{2k}T_{k-1}(n)-T_{k-1}(\frac{n}{2})}{2^{2k}-1} $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg19.pdf|p.19-2]]

$$ \displaystyle Since\, the\, 2nd\, column\, of\, Romberg\, table\, corresponds\, to\, k=1\, $$ $$ T_{1}(n)=\frac{4T_{0}(n)-T_{0}(\frac{n}{2})}{3} $$

$$ \displaystyle T_{0}(n)\, and\, T_{0}(\frac{n}{2})\, are\, composite\, trap.\, rule\, $$

$$ \displaystyle i)\, T_{0}(\frac{n}{2})= \frac{b-a}{\frac{n}{2}}[\frac{1}{2}f_{0}+f_{2}+...+f_{n-2}+\frac{1}{2}f_{n}]\, =\, \frac{b-a}{n}[f_{0}+2f_{2}+...+2f_{n-2}+2f_{n-1}+f_{n}] $$ $$ \displaystyle Where\, the\, number\, of\, pannels\, =\, \frac{n}{2}\,, \,\, h=\frac{b-a}{\frac{n}{2}}\, $$

$$ \displaystyle ii)\, T_{0}(n)= \frac{b-a}{n}[\frac{1}{2}f_{0}+f_{1}+f_{2}+...+f_{n-2}+f_{n-1}+\frac{1}{2}f_{n}] $$ $$ \displaystyle Where\, the\, number\, of\, pannels\, =\, n\,, \,\, h=\frac{b-a}{n}\, $$

$$ \displaystyle iii)\, T_{1}(n)= \frac{1}{3}\underbrace{[4T_{0}(n)-T_{0}(\frac{n}{2})]}_{call\, p} $$

$$ \displaystyle Where\, p\, =\, 4T_{0}(n)-T_{0}(\frac{n}{2})\, =\, \frac{b-a}{n}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}] $$

$$ \begin{align} \displaystyle Therefore,\, T_{1}(n)&=\frac{1}{3}p \\ &=\frac{1}{3}(\frac{b-a}{n}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}]) \\ &=\underbrace{\frac{h}{3}[f_{0}+4f_{1}+2f_{2}+...+2f_{n-2}+4f_{n-1}+f_{n}])}_{Exactly same with the formula of Comp. Simp. rule} \\\end{align}\ $$

=  6.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-1 & 25-2]]

Solution
As above table shown, Romberg Table has the shortest computational time for  $$\displaystyle I = \int_{0}^{1}\frac{e^{x}-1}{x}dx$$

However, note that Comp. Simpson's rule and Clencurt have almost same computational time with the Romberg Table even though the Romberg Table has slightly shorter computational time.

Therefore, ''' The Romberg Table, Comp. Simpson's rule and Clencurt are efficient ways to get a numerical integration value for the given function.'''

=  7.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Solution
As above table shown,''' The Comp. Simpson's rule has the shortest computational time for ''' $$\displaystyle I = \int_{-5}^{5}\frac{1}{1+x^2}dx$$ However, note that Romberg Table has almost same computational time with Simpson's rule even though Simpson's rule has slightly shorter computational time than Romberg Table. Therefore,  both Simpson's rule and Romberg Table are efficient ways to get a numerical integration value for the given function.

=  8.1 e) Time Comparison and Results =

Problem Statement
$$ \displaystyle Compare\, the\, time\, for\, above\, each\, method\, with\, comment\, on\, the\, results\, $$

Ref: Lecture Notes [[media:Egm6341.s10.mtg25.djvu|p.25-2]]

Solution
As above table shown,''' Comp. Simpson's Rule has the shortest computational time for ''' $$\displaystyle I = \int_{0}^{2\pi}\frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}d\theta$$

However, we should note that the computational time does not really different from each other for three method. Especially, Comp. Trap Rule has a shorter computational time than Romberg Table (in general, computational time of Romberg Table is much shorter than that of Comp. Trap. Rule.). Since the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ is a periodic function, the numerical integration of this function is converged very fast by Comp. Trap. rule.

Therefore, ''' Comp. Trap. rule is an efficient way to numerically integrate the given function $$\displaystyle f(\theta) = \frac{1-sin^{2}(\frac{\pi}{12})}{1-sin(\frac{\pi}{12})cos\theta}$$ because this function is a periodic function.'''