User:Egm6341.s10.team3.sa/Mtg17

Mtg 17: Tue, 02 Feb 10

 Typeset of transparencies, not lecture transcript. Subramanian Annamalai 13:52, 12 August 2010 (UTC)

[[media: Egm6341.s10.mtg17.djvu | Page 17-1]] 

Note: Runge phenomenon on [[media: Egm6341.s10.mtg16.djvu | Page 16-1]]

Oscillations of $$\displaystyle f_n $$ near end points as $$\displaystyle n \rightarrow 0 $$; motivation for Chebyshev plolynomial expansion and Clenshaw-Curtis quadrature (Discrete cosine transform, JPEG).


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

HW:
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

 Part 1  Replicate the proof of tighter error bound for Simple Simpson for two cases:
 * {| style="width:100%" border="0" align="left"

a) G(t) = e(t) - t^{4}e(1) $$
 * $$ \displaystyle \color{blue}
 * $$ \displaystyle \color{blue}
 * }
 * {| style="width:100%" border="0" align="left"

b) G(t) = e(t) - t^{6}e(1) $$ Point out where proof breaks down |}  Part 2  $$ \displaystyle \color{blue} G(t) = e(t) - t^{6}e(1) $$
 * $$ \displaystyle \color{blue}
 * $$ \displaystyle \color{blue}

Find $$ \displaystyle \color{blue} G^{(3)}(0) $$ and follow same steps in proof and see what happens.


 * style = |
 * }
 * }

[[media: Egm6341.s10.mtg17.djvu | Page 17-2]] 


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

HW:
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

$$ \displaystyle \color{blue} Show\, the\, error\, for\, Composite\, Simpson's\, Rule,\, $$ $$ \displaystyle \color{blue} \left| {E}_{n}^{2} \right| \leq \frac{(b-a)^{5}}{2880 n^{4}} M_{4} = \frac{(b-a)h^{4}}{2880} M_{4} $$ $$ \displaystyle \color{blue} where\,\, M_{4} := max \left| f^{4}(\xi) \right| \,\, and \,\, \xi \in [a,b]$$


 * style = |
 * }
 * }


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

HW:
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |

(i) Use Error estimate for Taylor series, composite trapezoidal and composite Simpson's rule to find n such that $$ \displaystyle \color{blue} E_n = I - I_n = O(10^{-6}) $$ and compare to numerical results.

(ii) Numerically find the power of h in the error: Plot the numerical error vs h on a log-log plot and measure the slope using the method of least squares.


 * style = |
 * }
 * }

[[media: Egm6341.s10.mtg17.djvu | Page 17-2]] 


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

HW:
 * style="width:100%; padding:10px; border:2px solid #8888aa" |
 * style="width:100%; padding:10px; border:2px solid #8888aa" |


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

$$\displaystyle \color{blue} Evaluate\, \int_{a}^{b} f(x)\,dx \,using\, Composite\,  Trapezoidal\, rule\, with\,\,  n\;\; =2,4,8,16,32,64\, and\,128.$$ $$\displaystyle \color{blue} Also\, evaluate\,\,  asymptotic\,  errors\,.$$
 * $$\displaystyle \color{blue} (i)\; f(x) = e^{x}cos(x) \,\,, x\in [a,b] = [0,\pi]$$
 * $$\displaystyle \color{blue} (i)\; f(x) = e^{x}cos(x) \,\,, x\in [a,b] = [0,\pi]$$
 * }
 * }


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

$$\displaystyle \color{blue} Evaluate\, \int_{a}^{b} f(x)\,dx \,using\, Composite\,  Simpson\, rule\, with\,\,  n\;\; =2,4,8,16,32,64\, and\,128.$$ $$\displaystyle \color{blue} Also\, evaluate\,\,  asymptotic\,  errors\,.$$
 * $$\displaystyle \color{blue} (ii)\; f(x) = e^{x}cos(x) \,\,, x\in [a,b] = [0,\pi]$$
 * $$\displaystyle \color{blue} (ii)\; f(x) = e^{x}cos(x) \,\,, x\in [a,b] = [0,\pi]$$
 * }
 * }


 * style = |
 * }
 * }