{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List \+ Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "AC - Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 1 2 258 1 }{PSTYLE "AC - Author" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 259 1 }{PSTYLE "AC - Note" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Normal Text" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Section Heading" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 260 1 }{PSTYLE "AC - Disclaimer" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 9 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "AC - Author" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 259 1 }{PSTYLE "Normal" -1 264 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 261 65 "Bisection Method of Solving a \+ Nonlinear Equation -- Convergence. " }{TEXT 257 0 "" }}{PARA 263 "" 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 144 "Nathan Collier, Autar Kaw, \+ Jai Paul , Michael Keteltas, University of South Florida , kaw@eng.usf .edu , http://numericalmethods.eng.usf.edu/mws " }}{PARA 258 "" 0 "" {TEXT -1 162 "NOTE: This worksheet demonstrates the use of Maple to il lustrate the convergence of the roots using the bisection method of fi nding roots of a nonlinear equation." }}{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introduction" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 259 18 "Bisection method [" }{URLLINK 17 "click here for textbook note s" 4 "http://numericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_tx t_bisection.pdf" "" }{TEXT 273 2 "][" }{URLLINK 17 "Click here for Pow er Point presentation" 4 "http://numericalmethods.eng.usf.edu/mws/gen/ 03nle/mws_gen_nle_ppt_bisection.ppt" "" }{TEXT 274 268 "] is one of th e first numerical methods developed to find the root of a nonlinear eq uation f(x)=0 (also called Binary-Search method). The method is based on the theorem that \"An equation f(x)=0, where f(x) is a real contin uous function, has at least one root between " }{OLE 1 4100 1 "[xm]Br= WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy;:::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JZhVIDjr

::::::;K;HYLkNG>::::::::NZ:nF>:nY>;V:;Jyk:[B::a: c:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::j:b:B:::\\[Z:bNWS>wE?pn^HNcxnctqRbDHZ>ruwwKUscqehWCn=G?s:?jF DvHK:Jj`nD^\\Ci<;J:VZ;FZ=N[DFZLVjrs:YR:hj:Dk;m@]J?Dj:=\\:;P:<:TnEj``pkDqqHqqTPtZoBFyyyxy;B:CJ:F:vYxY;<:;:u g>ViD:;B:E:?R:=J:\\:B:;xyyQMyyyyYZ=;GY:V[:JMJ@fc[_hb_ds?h_Kj< JHB:qi:;fyB:>l;J:@CB:f _;J>JSdJ:AJvD_mlVH[C:[;;B:::::::JFNZ;B:NZ:>:yayA:;B::::: :^:np=Z:j::<::::::wqy[:::::::::::::vYxI:;Z:: ::::::::::::::::::yay=J:B::::::::b:=FSDj:=\\LVjr;ZokBN;b:``kjb:?fE;N@>qIF:;JR `Q>JSJZqj:jN`Q>JSJfij:<:KSXK:_;kY;N@naBF:>_h>;N@nt@F:>^h>;N@Nr ?NZ:jnTP>JSJtPj:JJTP>JSJaIj:>:;k>>Z:N@^o;V:;B:=eUCUSK:_;kT:=J:>^h_=>O@;N`DJCuj:>:<:UK:^:>X=J><:_KjAJGpj:B:eeUK:_;;V<= :SeUK:_;;I;=:e=N@Nx:FZ:Jp^=VY;><:[V:>:Cb::::jIJJ:B:f?=J;N@;Ntlj :ih?;N@N\\EF:nfh?;N@NuBF:>IJSJvXj:>:G=N@Na>F:>IJSJ`F:>IO:GkAN;yayI :>:s:qQBv:pbQ::::WTJWTLB::::kR:OY:;J:wgi;>:::::::::::::::::::::::::::::1: " }{TEXT -1 125 " Since the method is based on finding the root betwee n two points, the method falls under the category of bracketing method s." }}{PARA 0 "" 0 "" {TEXT -1 130 "The following simulation illustrat es the convergence of the roots using bisection method of finding root s of a nonlinear equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}}{PARA 260 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 17 "Section I : Data ." }}{PARA 0 "" 0 "" {TEXT -1 71 "The following is the data that is us ed to solve the nonlinear equation " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Function in f(x)=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f:=x->x^3-0.165*x^2+3.993*10^(-4):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Lower initial guess" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "xl:=0.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Upper initial guess" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xu:=0.11: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Upper bound of range of " } {TEXT 276 2 "x " }{TEXT -1 28 "that is desired for plotting" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "uxrange:=0.12:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Lower bound of range of " }{TEXT 275 1 "x " }{TEXT -1 29 " that is desired for plotting" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "lxrange:=-0.02:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Maximum number of iterations" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "nmax:=30:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 36 "Section II: Validity of the Guesses." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Check if the lower and upper guess bracket the ro ot of the equation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(xl) *f(xu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!++m$H1\"!#;" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{OLE 1 4100 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :fyyyyya:nYf::G:jy;::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::J:]wJDjrZ:j:vCSmlJ::::::::::OJ;@jyyyyyy;jysy;Z::: ::::^<>:fB]mtFFcmnvGWMJnC==nHE=;:::::JJNZ:vyyuy:>:<::::::=J:fG>:F:Alqf G[maNFO=;::::::::_J;Zy=J:B::::::N:;B:G=;:wAK:AJ:nYf:n:v:JyK?j?J@j@> :W:YJ:>\\:B:]:wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::j:b:B:::;DkS:;::::::JHJ;F]AFUF:;P:<:TnEj``p kDqqHqqTPtZ::::: ::yayY:^:;j:jysy?B:>:vqDjwD:?h:F:MZ=Ff;B:?JHJ:f:;j>D:;B:E:?R:=Z:f:FZ=f:V[b<>rCN\\:JMTjAN=yyyxY:ryyYGwyyyy;Vl;J:@CB:f_;J>>:_c<;V:?H<=Z:>`sO:YLpJbNHEms>@[C:>Z:::::::: kJ;@:NZ:>:yayA:;B::::::^:r::<:::::::::::::::::::vYxI:;Z::::::::ZAfSNjbZ r;ZokBN;?N:DJ=m]<>kjJ::::>;N@Ns@F:F_s?;N@Nv>F:>_s?;N@NZ=N:;jNtL>JSR:KT :=J:>_h^=VYZ:JBC:DJ:DZJVdscRYEUXQZB:gd;ORYgrJKpIJJNSJ<<:Uk:^:>x;J>JSd: sv;=Z:^ah^=VY;syB:>L=J:^ZSy:;;Z:jP>:C:[q:>;N@;Y:Cc;=Z:fhh>;N@N kL>:Cb:^D:::y:;;Z:jPF:C:[Q:>;N@;NyPj:B:keEK:_;sB;=B:nf h>;N@^v;F:;Jv:_;?n:<:GM?JMJ?vYxy:>:;JHjw?:sg:B:=b:?bBaTXaEWEUUB:O jJNk;Z::[Gt_jNDN^YQ<>:UK:^:>X=j;B:;:::Ja@Na`>;B::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::5:" }{TEXT -1 36 " , there is atleast one root between " }{OLE 1 3588 1 "[xm]Br=WfoRrB::: wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::wyyyqy;::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::N DYmq^H;C:ELq^H_mvJ::::::::gjjRI^r= B:<:=ja^GE=;:::::::::N;?R:yyyyyyA:yayA:<::::::JDJ:j\\FHemj^HMmqnG;KaFF JufF>::::::;K;:;:=j[vGUMrvC?MoJ::::::::JCN:ry:>: <::::::?J:j;Jyk:;P:<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV`:^xsNpkK:<::LZiqqca_nAiAg_sXiNZqS UBU@pyyiyY::=J:vYxY;<:;:uf;:EJ:F[Z:f:FZ=f:V[b\\:B:;xyyQM yyyyYZQ:GY:V[:JMJ@fc[_hb_ds?h_Kj^=Z:^ZcTTUUSaEBWTSiEB _tUUURWMEHN^YqIHN^YQJ;;JSdJ:AJtPj:B:WSX?ja^GEMbNHEms>@[; ;B:::::::JFNZ;:?B:;jysy;J:<::::::C:we:B:FZ:vCJ:: <::::::wqy[:::::::::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::b:=F SDj:=\\LVjr;ZokBN;?fEJSJaIj:>:;SXMZ:N@^o;V:;B:=eUCUSK:_;kT: =J:>^h_=VY;syB:>L=:Cb::::>Sy:;;J<<:Uk:^:>x;B:K:_KjAJjMj:B:eeUK:_;_I:=B :^gh_=VY;><:[>;J>:<:UK:^:>x;F:K:_K:Oa>F:JSJ`F:>IO:G;Ojy sy=J:>:s:qQBv::Uk:^ :>X;j;::::WTJWTLB:yay=::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 259 "" 0 "" {TEXT -1 226 "We now plot the data. The followin g function determines the upper and lower ranges on the Y-axis. This i s done using the upper and lower ranges of the X-axis specified, and t he value of the original functional at these values." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "yranger:= proc(uxrange,lxrange)\nlocal i,maxi,mini,tot;\nmaxi:=f(lxrange);\nmini :=f(lxrange);\nfor i from lxrange by (uxrange-lxrange)/10 to uxrange d o\nif f(i)maxi then maxi:=f(i) \+ end if;\nend do;\ntot:=maxi-mini;\n-0.1*tot+mini..0.1*tot+maxi;\nend p roc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "yrange:=yranger(uxr ange,lxrange):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xrange:=l xrange..uxrange:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "The following calls are needed to use the plot function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "with(plottools):" }} {PARA 7 "" 1 "" {TEXT -1 43 "Warning, the name arrow has been redefine d\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot(f(x),x=xrange, y=yrange,title=\"Entered function on given interval\",legend=[\"Functi on\"],thickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 549 549 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!3/+++++++?!#>$\"3%************HD$!#@7 $$!3kmmmT)R[p\"F*$\"3CeRW%[c.Z$F-7$$!3JLLe>;KH9F*$\"3,!f\\'y5rEOF-7$$! 3xmm;4'=28\"F*$\"3u.\\;IpenPF-7$$!3NnmmTEO,$)!#?$\"3cVra([tN(QF-7$$!3e OL$eMD)4`F@$\"3zhnBIC)\\%RF-7$$!3Frm;HtGODF@$\"3QSOv.GA#)RF-7$$\"3ow** *\\P1bN$F-$\"3q=1L(f9G*RF-7$$\"3aHL$3d4cI$F@$\"3yAkhl:LvRF-7$$\"3/(*** \\([VhE'F@$\"3ssk')eQnIRF-7$$\"3wjmm;lT6$*F@$\"3!*\\B'399!eQF-7$$\"3!G LLeYp$*>\"F*$\"3?LX;,C!Hx$F-7$$\"3')*****\\XI8]\"F*$\"3q*y(4=5$\\l$F-7 $$\"3n*****\\KJX!=F*$\"3u=y^flY9NF-7$$\"3k*****\\a@n4#F*$\"3ypG/LrzfLF -7$$\"3'HL$3d#e?O#F*$\"3Aj#f/5*>/KF-7$$\"3Qmmmr#pvn#F*$\"3G)*[C-y,-IF- 7$$\"3Dmmm'[[[%HF*$\"3yIbN&Gzu\"GF-7$$\"3K***\\PvddD$F*$\"3)[z*[6e6*e# F-7$$\"3GlmmO^'4`$F*$\"3A%p*R@x0wBF-7$$\"39++v`7\"H$QF*$\"3%ol=(e:0K@F -7$$\"3V***\\7ON/7%F*$\"3S`%y[0*>\"*=F-7$$\"3^mm;/mV?WF*$\"3i!R'>L>iK; F-7$$\"3]mmT&RJfp%F*$\"3))QG\\VP***Q\"F-7$$\"3wJL$eu*3$*\\F*$\"3$R0w\" 4&=U7\"F-7$$\"3YJL3dPv,`F*$\"3'pK`b^0LX)!#A7$$\"3C)**\\ioY/d&F*$\"3\\f !4JGJd,'Ffs7$$\"3)=LL3TU1'eF*$\"3!e#z@V2'oQ$Ffs7$$\"3'z******)HWghF*$ \"33>YZXR(G!p!#B7$$\"3])***\\n$RPX'F*$!3D8a'f$3V8>Ffs7$$\"3S)**\\Pp=vt 'F*$!33Hj*Q.ifQ%Ffs7$$\"3K)***\\_sg_qF*$!3?eZYw4kgqFfs7$$\"3pjmmO$GdL( F*$!3#f^bO$Gh&Q*Ffs7$$\"3c)****\\_?!QwF*$!3)f8q5*=-x6F-7$$\"3#4L$3x@%> \"zF*$!3m&)*[@jBIQ\"F-7$$\"3/+++&*3T6#)F*$!3oTvsi#pdf\"F-7$$\"3akmT?w= $\\)F*$!3]Y2TMHl#y\"F-7$$\"3A***\\()[Dxy)F*$!3g2-mmgti>F-7$$\"3okmm\"4 !pv!*F*$!3r,ZFpnEA@F-7$$\"3#z**\\PMirP*F*$!3_qAF-7$$\"3CLLL`f^n' *F*$!3y4x/r')p#R#F-7$$\"3GKL$eXWW'**F*$!3'))o%o>J;'\\#F-7$$\"3`m;/C9*e -\"!#=$!3G)zPgw]ad#F-7$$\"3)*******Q,&H0\"F[y$!3!)pJOic^EEF-7$$\"3_mm \"*zC'R3\"F[y$!3I2&)*\\T(zdEF-7$$\"3HLLL(G+<6\"F[y$!3w>#\\n=D(fEF-7$$ \"3\"***\\PvXFT6F[y$!3iS2y0w=LEF-7$$\"3#***\\iU4ep6F[y$!3gCtPejuyDF-7$ $\"3%**************>\"F[y$!3c.+++++([#F--%'COLOURG6&%$RGBG$\"#5!\"\"$ \"\"!Fd[lFc[l-%'LEGENDG6#Q)Function6\"-%*THICKNESSG6#\"\"$-%&TITLEG6#Q CEntered~function~on~given~intervalFi[l-%+AXESLABELSG6$Q\"xFi[lQ\"yFi[ l-%%VIEWG6$;$!\"#F\\]l$\"#7F\\]l;$!++!3IH$!#8$\"++!3#)e%Fb]l" 1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Function" }}}}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } {TEXT 264 24 "Section III: True Value." }}{PARA 259 "" 0 "" {TEXT -1 479 "The \"true\" solution is taken as the solution that Maple's numer ical root solver obtains. This is a decent assumption because their s ubroutines have been professionally written. You must take caution, h owever, because Maple's \"RootOf\" function might be finding another o f the function's roots. For the rest of the sheet to be correct, you \+ need to ensure that xrtrue is the root that you are attempting to find . This can be altered by changing the value of 'rootnumber' above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xrtrue:=fsolve(f(x),x,x=xl..xu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xrtrueG$\"+^\"exB'!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 54 "Section IV: Value of root as a function of iterations. " }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 296 "Here the bisection method a lgorithm is applied to generate the values of the roots, true error, a bsolute relative true error, approximate error, absolute relative appr oximate error, and the number of significant digits at least correct i n the estimated root as a function of number of iterations." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "xr:=proc(n)\n local p, \+ i, l, u;\n u:=xu;\n l:=xl;\n\n for i from 1 to n do\n \+ p:=(u+l)/2;\n if f(u)*f(p)<=0 then\n l:=p; \n else\n u:=p;\n end if;\n end do ;\n \n p; \nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nrange:=1..nmax:" }}}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 19 "Absolute true error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Et:=proc(n)\n abs(xrtrue-xr(n)); \nend proc:" }}}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 28 "Absolute relative true error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "et:=proc(n)\n abs(Et(n)/xrtrue)*100;\nend proc:" }} }{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 26 "Absolu te approximate error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Ea:= proc(n)\nlocal p:\nif n<2 then\n p:=0;\nelse\n p:=abs(xr(n)-xr(n-1)) ;\nend if;\np;\nend proc:" }}}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 35 "Absolute relative approximate error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ea:=proc(n)\n abs(Ea(n)/xr(n))*10 0;\nend proc: " }}}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 " " {TEXT -1 35 "Significant digits at least correct" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 168 "sigdigits:=proc(n)\nlocal p;\n if n < 2 th en\n p:=0;\n else\n p:=floor((2-log10(ea(n)/0.5)));\n if \+ p<0 then\n p:=0:\n end if:\n end if;\np;\nend proc:" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 266 0 "" }{TEXT 267 29 "S ection V: Graphs of Results." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot(xr,nrange,title=\"Estimated 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