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" }{TEXT 257 0 "" }}{PARA 263 " " 0 "" {TEXT 256 7 "\251 2003 " }{TEXT -1 144 "Nathan Collier, Autar K aw, Jai Paul , Michael Keteltas, University of South Florida , kaw@eng .usf.edu , http://numericalmethods.eng.usf.edu/mws " }}{PARA 258 "" 0 "" {TEXT -1 151 "NOTE: This worksheet demonstrates the use of Maple to illustrate the slow convergence of the Bisection method of finding ro ots of a nonlinear equation." }}{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introduction" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 259 18 "Bis ection method [" }{URLLINK 17 "click here for textbook notes" 4 "http: //numericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_txt_bisection .pdf" "" }{TEXT 273 1 "]" }{TEXT 274 174 " of solving nonlinear equati ons always converges to the root. However, the convergence process may be slow. The following simulation illustrates the slow convergence o f the " }{TEXT -1 58 "Bisection method of finding roots of a nonlinear equation." }}{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." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Function \+ in f(x)=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f(x):=x^2-1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Lower initial guess" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "xl:=-1.25:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Upper initial guess" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "xu:=-0.5:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Uppe r bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "uxrange:=2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "L ower bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 12 "lxrange:=-2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Maximum number of iterations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nmax:=30:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Ente r the number of the root desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "rootnumber:=2:" }}}{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 16 "eval( f(x),x=xl);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%Dc!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(f(x),x=xu);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$!#v!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "%*%%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!'v=U!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{OLE 1 4098 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j:: >:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::fyyyyya:nYf::G:jy;:::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm= ;:::::::n:;`:Z@[::JRLxhmK^Am[Aj;J:M:<:=ja^GE=;::::::: ::N;?R:yyyyyyA:yayA:<::::::JDJ:j\\FHemj^HMmqnG;KaFFJufF>::::::;K;HYLkNG>::::::::NZ:nF>:nY>;V :;Jykb>a;>:::::::s:?jFD vb>Z:>:>uJgAFXHjA@ZEFZLVjfkBN[R;B:bKi:UTTAeVYuVYeScEBETVeURcUTYeU;sFWCF;B=BKaDBETV:;r:=:yayQZ:J:jYe:qe:Nr :Mb:>Z:f:NZ;F:^;UTRcETcTX[US>FsZ:Vy<>jx]:JBA:;B:Cb:j;NrBF:`:J:<:::::::>=?R:J;Z::::::JZ:>:::::::::J ?>Z:vYxI:;Z::::::JywYB:::::::::::::yay=J:B:::::::::::::::::::jysy:>:<: :::::::DJ;u\\LVj`K;]<\\A:PA]J?NJ;b:GFIDJ>==;:::J>JSJ_Uj:jN`Q>JSJkMj:JN `Q>JSJ;HJ;>:McEK:_[;>c;F:;JNtLHjw;<:[^:b:;b:>:<:UK: ^:>X=J>JS>v;^^@F:JSJ?Ej:>:SeEs:qQ:[:JBK:^Z>:<:Uk:^:>X;J >JS>JwU;=Z:>ih>;N@^]>F:nfh>;N@^v;F:;Jv:_;?n:<:GM?JMJ?vYxy:>:;JHjw?:sg:B:=b:?bBaTXaEWEUUB:OjJNk;Z::[Gt_jNDN^YQ<>:UK:^:>X=j;B:;:::Ja@Na`> ;B:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::3:" }{TEXT -1 36 ", there is atleast one root between " }{OLE 1 3586 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyy ya:nYf::wyyyqy;::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gjtdwuV>skVr;V:>r=B:<:=ja^GE=;:::::::::N;?R:yyyyyyA:yayA:<:::: ::JDJ:j\\FHemj^HMmqnG;KaFFJufF>::::::;K;:;:=j[vG UMrvC?MoJ::::::::JCN:ry:>:<::::::?J:j;Jyk:; 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This is a decent assumption because their subroutines hav e been professionally written. You must take caution, however, becaus e Maple's \"RootOf\" function might be finding another of 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 40 "xrtrue: =RootOf(f(x),x,index=rootnumber):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "xrtrue:=evalf(xrtrue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xrtrueG$!\"\"\"\"!" }}}{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 0 285 "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 eval(f(x),x=u)*eval(f(x),x=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 tru e error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Et:=proc(n)\n a bs(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)*1 00;\nend proc:" }}}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 26 "Absolute 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 approx imate error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ea:=proc(n)\n abs(Ea(n)/xr(n))*100;\nend proc: " }}}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 35 "Significant digits at least corre ct" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "sigdigits:=proc(n)\nl ocal p;\n if n < 2 then\n p:=0;\n else\n p:=floor((2-log10(ea( n)/0.5)));\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 "Section V: Graphs of Results." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "plot(xr,nrange,title=\"Estim ated root as 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 4 "" 0 "" {TEXT 270 10 "References" }}{PARA 0 "" 0 "" {TEXT -1 4 "[1] " }{TEXT 272 177 "Nathan Collier, Autar Kaw, Jai Paul , Michael Keteltas, Holistic Numerical Methods Institute, See http://numericalm ethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_txt_bisection.pdf" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 0 "" } {TEXT 260 10 "Disclaimer" }{TEXT 271 1 ":" }{TEXT -1 248 " While every effort has been made to validate the solutions in this worksheet, Uni versity of South Florida and the contributors are not responsible for \+ any errors contained and are not liable for any damages resulting from the use of this material." }}}{MARK "9 1 0" 346 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }