University of Florida/Egm6321/f09.team1.gzc/Mtg22

Mtg 30: Mon, 14 Mar 11


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Comp. Trap. rule error: $$ h={\color{red}\frac{n}} {\color{blue}\leftarrow}{\color{red}(3)}{\color{blue}p.7-4}$$


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$${E}^_{n} \underset{:=} \ I-{I}^{T}_{n}= \int_{a}^{b}f(x)dx-h[{\color{red}\frac{1}{2}}f(x_)+f(x_{1})+ \cdots+f(x_{n-1}){\color{red}\frac{1}{2}}f(x_)] \ {\color{red}(1)}$$

$$= \sum_{i=1}^{n}{\color{blue} \left\{{\color{black} \int_{x_{i-1}}^{x_{i}}f(x)dx-\frac{h}{2}[f(x_{i-1})+f(x_{i})]} \right\}}$$

(1)p.17-2:

$$ \left|{E}^{T}_{n} \right| \leqslant \frac{h^{3}}{12} \underset{ {\color{blue}\underset{ \overset{=: \overline{M_{2}}}}}}{\sum_{i=1}^{n}(max{\color{red} \left|{\color{black}f^(\xi)} \right|})}{\color{red} \ (2)}$$

(4)p.17-1:

$$ \overline{M_{2}} \leqslant M_{2} \ {\color{red}(4)}$$

$${\color{red}(5)}$$
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HW*4.8: Comp. Simpson error

$$ \left|{E}^_ \right| \leqslant \frac{(b-a)^{5}}{2880n^{4}}{\color{blue} \underset{{\color{red}(4)}p.17-1}{ \underbrace}}=\frac{(b-a)h^}{2880}M_{4} \ \color{red}(1)$$

HW*4.8: See HW*2.4 p.7-3

<P>1)Use error estimate for Taylor Series, Compare Trap., Compare Simpson, to findn</P> <P>Q(10<SUP>-6</SUP>), and compare to number results.</P> <P>2)Numerically find the power ofhin error : Plot</P> <P>logerror vslogh, and meas. slop with least square</P> <P>(lin. regression).</P> <P> </P>


 * nm1.s11.Mtg22.pg2.fig1.svg$$E=ah^+Q(h^{k+1}), \ $$   $$ \ logE=loga+{\color{red}k}logh$$

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<P><U>HW</U><SUP>*</SUP><U>4.9:</U> pf. of SSET, G(.) in (1) p.19-1</P><P>A)Redo the pf for 2 cases</P><P></P><P></P>

$${\color{blue}1)} \ G(t):= \ e(t)-t^{\color{red}4}e(1) \ {\color{red}(1)}$$

$${\color{blue}2)} \ G(t):= \ e(t)-t^{\color{red}6}e(1) \ {\color{red}(2)}$$

Print out where pf breaks down.

<P>B)For G(t) as in(1)p.19-1(w/t<SUP>5</SUP>), find</P> <P>G<SUP>(3)</SUP>(0)and follow same steps in pf to</P> <P>see what happen.</P> <P> </P> <P>HW<SUP>*</SUP>4.10:1)Don'tUse matlabtrapzfor<U>compare Trap.</U>in</P> <P>your code to produce Table 5.1 n A.p.255(p.22-4)</P> <P> </P> $${\color{red}(3)}I= \int_{0}^{\pi}e^{x}{\color{blue}sinx}dx$$(use WAto find I)