University of Florida/Egm6341/s11.team1.Gong/Mtg30

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page30-1 $\underset{\_}{ApplofHOTRp.29-2:}\underset{\_}{CorrectedTrap.rules}$ $C{T}_1(n)=\underset{C{T}_0(n)}{\underbrace{T_0(n)}}+a_1^0{h}^2+Q(h^4)$ $I=C{T}_1(n)+Q(h^{(4)})$ $C{T}_2(n)=C{T}_1(n)+a_2^0{h}^4+Q(h^6)$ $I=C{T}_2(n)+Q(h^6)$ $C{T}_k(n)=C{T}_{k-1}(n)+a_k^0{h}^{2k}+Q(h^{2(k+1)})$ Template:FontTemplate:FontTemplate:FontTemplate:FontTemplate:Font 8,16,... until error I-In=Q(10-6). Template:FontTemplate:FontTemplate:Font Template:FontTemplate:FontTemplate:FontTemplate:FontTemplate:FontTemplate:FontTemplate:FontTemplate:Font Template:FontTemplate:FontTemplate:FontTemplate:Font   page30-2 $E_n^T=I-I_n=\underset{a}{\overset{b}{\int }}f(x)dx-T_0(n)$ $=\sum _{k=0}^{n-1}[\underset{x_k}{\overset{x_{k+1}}{\int }}f(x)dx-\frac{h}{2}\left\{f(x_k)+f(x_{kH})\right\}](1)$ $x_k:=a+kh,h:=(b-a)/n(2)$ $Transfer\underset{x_k}{\overset{x_{kH}}{\int }}-dxto\underset{-1}{\overset{+1}{\int }}-dt$ $x(t):=\frac{x_k+x_{k+1}}{2}+t\frac{h}{2},t\in [-1,+1](3)$ $\{\begin{array}{cc}& x(-1)=x_k\\ & x(0)=(x_k+x_{k+1})/2(4)\\ & x(+1)=x_{k+1}\end{array}$ $E_n^T=\frac{h}{2}\sum _{k=0}^{n-1}[\underset{-1}{\overset{+1}{\int }}{g}_k(t)dt-\left\{g_k^{(-1)}+g_k^{(+1)}\right\}](5)$ $g_k^{(t)}:=f(x(t))stx\in [x_k,x_{k+1}](6)$ page30-3 $\underset{-1}{\overset{+1}{\int }}\underset{P_1(t)}{\underbrace{(-t)}}{g}^{(1)}(t)dt\stackrel{H{W}^{*}5.8}{=}\underset{:=E}{\underbrace{\underset{-1}{\overset{+1}{\int }}g(t)dt-[g(-1)+g(+1)]}}(1)$ $\underset{\frac{d^i}{d{t}^i}{g}_k^{(t)}}{\underbrace{g_k^{(i)}(t)}}=(\frac{h}{2}{)}^i{f}^{(i)}(x(t))(2)$ $x\in [x_k,x_{k+1}]$ $Toobtain\underset{\_}{higherpowersofh}\Rightarrow Takehigher$ $derivationof{g}_k(t)\Rightarrow \underset{\_}{successiveint.byparts}$ $(recallpfofTaylorseriesMtg5andMtg6)$ $(1)\Rightarrow E=\underset{-1}{\overset{+1}{\int }}\underset{P_1(t)}{\underbrace{(-t)}}{g}^{(1)}dt$ $P_2(t)\stackrel{(1)}{:=}{c}_1\frac{t^2}{2!}+c_3,c_2\stackrel{(1)}{=}0$ page30-4 $E=[P_2(t)g^{(1)}(t){]}_{-1}^{+1}-\stackrel{A\stackrel{2b}{=}}{\overbrace{[P_3{g}^{(2)}(t){]}_{-1}^{+1}}}+\underset{-1}{\overset{+1}{\int }}{P}_3{g}^{(3)}(t)dt(3)$ $P_3\stackrel{(4)}{=}\int P_2(t)dt=-\frac{t^3}{6}+\alpha t+\beta (newintergrationconstant)=c_1\frac{t^3}{3!}+C_{3}t+C_4(C_2\stackrel{(2)}{=}0)$ $i.e.,make$ $A_{2b}\stackrel{(5)}{=}0$ $\Rightarrow$ $want{P}_3(t)oddfunctions$ (recall PTemplate:FontTemplate:FontTemplate:Font) $Consider{P}_{3}st{P}_3=0,P_3(+1and-1)=0(6)$ page30-5 File:Nm1.s11.Mtg30.pg5.fig1.svg $P_3(0)=0\Rightarrow C_4=0=\beta (1)$ $P_3(-1)=0=-\frac{(-1{)}^3}{6}+\alpha (-1)\Rightarrow C_3=\alpha =\frac{1}{6}(2)$ $P_3(+1)\stackrel{P_{3}odd}{=}-P_3(-1)=0(3)$ Template:Font $P_2=C_1\frac{t^2}{2!}+C_3$ $P_3=C_1\frac{t^3}{3!}+C_{3}t$ $C_1=-1,C_3=\frac{1}{6}$ $(4)$ $\underset{\_}{steps3ab:}(P_4,P_5)$ Template:Font Template:Font Template:Font ; intergration by parts $E=[P_2(t)g^{(1)}(t){]}_{-1}^{+1}+\underset{-1}{\overset{+1}{\int }}{P}_3(t)g^{(3)}(t)dt$ $=[P_2(t)g^{(1)}(t){]}_{-1}^{+1}+[P_4(t)g^{(3)}(t){]}_{-1}^{+1}-\underset{-1}{\overset{+1}{\int }}{P}_4(t)g^{(4)}(t)dt(5)$ page30-6 $P_4(t)=\int P_3(t)dt=-\frac{t^4}{4!}+\frac{t^2}{12}+\alpha$ $=C_1\frac{t^4}{4!}+C_3\frac{t^2}{2!}+\underset{=\alpha }{C_5}(1)$ $C_{1}and{C}_{3}in(4)p.30-5$ $(5)p.30-5\Rightarrow E=[P_2{g}^{(1)}+P_4{g}^{(3)}{]}_{-1}^{+1}$ $[P_5{g}^{(4)}{]}_{-1}^{+1}+\underset{-1}{\overset{+1}{\int }}{P}_5{g}^{(5)}dt(2)$ $P_5(t)=\int P_4(t)dt=\frac{t^5}{120}+\frac{t^3}{36}+\alpha t+\underset{C_6}{\underbrace{\beta }}(3)$ $Wanteliminate{g}^{(4)}(thus{h}^5);$ $Somake$ $A_{3b}\stackrel{(4)}{=}0$ $\Rightarrow$ $want{P}_{5}oddfunctions$ (recall PTemplate:FontTemplate:FontTemplate:FontTemplate:FontTemplate:FontTemplate:Font) page30-7 Similar to Template:FontTemplate:Font $Consider{P}_5(t)st{P}_5(0)=0,P_5(+1and-1)=0(1)$ $\Rightarrow C_6=0,C_5=-\frac{7}{360}H{W}^{*}5.8(2)$ $P_4(t)=C_1\frac{t^4}{4!}+C_3\frac{t^2}{2!}+C_5$ $P_5(t)=C_1\frac{t^5}{5!}+C_3\frac{t^3}{3!}+C_{5}t$ $C_1=-1,C_3=\frac{1}{6},C_5=\frac{-7}{360}$ $(3)$ $E=[P_{g^{(1)}}+P_4{g}^{(3)}{]}_{-1}^{+1}(+)\underset{-1}{\overset{+1}{\int }}{P}_5{g}^{(5)}dt$ $(4)$ Template:FontTemplate:FontTemplate:Font $E=[P_2(t)g^{(1)}(t){]}_{-1}^{+1}+P_3(t)g^{(3)}(t)$