University of Florida/Egm6321/f09.team1.gzc/Mtg10

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page10-1 on  Template:Font. Plot the roots on [-1,+1] using matlab "plot" command (plot dots "." with coordinator (xi,yi), i = 1,...,5 : use "markersinge" 15) xi:Template:Font yi: Template:Font Repeat the above for PTemplate:FontTemplate:Font observe the location of the roots near end points -1 and +1   page10-2 Template:Font $[a,b]x_0=a,x_1=\frac{a+b}{2},x_2=b(1)$ $\underset{\_}{Method1:}{f}_2(x)=P_2(x)\stackrel{(2)}{=}{c}_2{x}^2+c_{1}x+c_0$ c0,  c1,  c2   unknowns p2=(xi) = f(xi)    i= 0, 1, 2               Template:Font Template:Font}   $(4)=\sum _{i=0}^{n=2}\underset{l_i(x)}{\underbrace{l_{i,2}(x)}}f(x_i)$ Equiv. of meth1 and meth 2:            Template:Font   $p_2(x_j)=\sum _{i=0}^2\underset{{\delta }_{ij}}{\underbrace{l_i(x_j)}}f(x_i)=f{x}_j(5)$ page10-3 $l_{0,2}=l_{0,2}=\prod _{j=0j\ne i}^{n=2}\frac{x-x_j}{x_0-x_j}=\frac{(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}\in P_2$ It can be verified that l0(x0)=1 , l0(x1) = l0(x2) = 0 li(xj) = δij                  i,j = 0. 1. 2 File:Nm1.s11.Mtg10.pg3.fig1.svg $\underset{l_1(x)}{\underbrace{l_{1(i=1),2(n=2)}(x)}}=\prod _{j=0,j\ne 1}^{2(n=2)}\frac{x-x_j}{x_{i(i=1)}-x_j}=\frac{(x-x_0)(x-x_2)}{\underset{>0}{\underbrace{(x_1-x_0)}}\underset{<0}{\underbrace{(x_1-x_2)}}}\in P_2$ $l_1(x_1)=1,l_1(x_0)=l_1(x_2)=0$ page10-4 File:Nm1.s11.Mtg10.pg4.fig2.svg $l_1^{{\text{'}}^{\prime }}(x_1)<0$ expression for {ci} in terms (xi, f(xi)) i=0,1,2. derive simple Simpson's rule Consider n=1Template:Font, 2Template:Font, 4, 8, 16   page10-5 Constrast fn(x) as in (2) p.8-3. Plot f , fn , n= 1, 2, 4, 8, 16 Compare $I_n=\underset{a}{\overset{b}{\int }}{f}_n(x)dx$ n=1, 2, 4, 8 and compare to I (use WA with more digits) For n=5 plot l0, l1, l2 How would l3, l4, l5  look like? $\underset{SimpleTrap.}{\underbrace{(1)p.7-3}}\Rightarrow \underset{CompareTrap.}{\underbrace{(1)p.7-4}}$ $\underset{SimpleTrap.}{\underbrace{(2)p.7-4}}\Rightarrow \underset{CompareTrap.}{\underbrace{(4)p.7-4}}$