{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 273 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{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 Out put" -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 "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 "Tim es" 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 }{PSTYLE "AC - No rmal Text" -1 265 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 }} {SECT 0 {PARA 256 "" 0 "" {TEXT 261 49 "Root Jumping to Another Locati on--Secant Method. " }{TEXT 257 0 "" }}{PARA 263 "" 0 "" {TEXT 256 7 " \251 2003 " }{TEXT -1 144 "Nathan Collier, Autar Kaw, Jai Paul , Micha el Keteltas, University of South Florida , kaw@eng.usf.edu , http://nu mericalmethods.eng.usf.edu/mws " }}{PARA 258 "" 0 "" {TEXT -1 210 "NOT E: This worksheet demonstrates the use of Maple to illustrate how, in \+ the Secant method, an initial guess close to one root can jump to a lo cation several roots away when a function is oscillatory in nature." } }{SECT 0 {PARA 260 "" 0 "" {TEXT 258 12 "Introduction" }{TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 259 15 "Secant method [" }{URLLINK 17 "text \+ notes" 4 "http://numericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nl e_txt_secant.pdf" "" }{TEXT 271 2 "][" }{URLLINK 17 "PPT" 4 "http://nu mericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_ppt_secant.ppt" " " }{TEXT 272 45 "] is derived from the Newton-Raphson Method. " } {TEXT -1 162 "The Secant method may or may not converge, but it conver ges slower then Newton-Raphson method.The same drawbacks can be seen i n this method as in Newton-Raphson. " }{TEXT 274 223 "One of them is R oot jumping. In some cases where the function f(x) is oscillating and \+ has a number of roots, one may choose an initial guess close to a root . However, the guesses may jump and converge to some other root. " } {TEXT -1 182 "The following simulation illustrates how, in the Secant \+ method, an initial guess close to one root can jump to a location seve ral roots away when a function is oscillatory in nature." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restar t;" }}}}{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 13 "f(x) :=sin(x):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Initial guess 1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xguess1:=8.5:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Initial guess 2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xguess2:=7.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Upper bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "uxrange:=-10.0:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Lower bound of range of 'x' that is desired" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lxrange:=10.0:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 263 30 "Section II: Plotting the Data." }} {PARA 0 "" 0 "" {TEXT -1 226 "We now plot the data. The following func tion determines the upper and lower ranges on the Y-axis. This is done using the upper and lower ranges of the X-axis specified, and the val ue of the original functional at these values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 343 "yranger:=pr oc(uxrange,lxrange)\nlocal i,maxi,mini,tot;\nmaxi:=eval(f(x),x=lxrange );\nmini:=eval(f(x),x=lxrange);\nfor i from lxrange by (uxrange-lxrang e)/10 to uxrange do\nif eval(f(x),x=i)maxi then maxi:=eval(f(x),x=i) end if;\nend do;\ntot:=maxi-mini;\n-0.1*tot+mini..0.1*tot+maxi;\nend proc:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "yrange:=yranger(uxrange,lxra nge):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "xrange:=lxrange..u xrange:" }}}{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 redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "plot(f(x),x=xrange,y=yrange ,title=\"Entered function on given interval\",legend=[\"Function\"],th ickness=3);" }}{PARA 13 "" 1 "" {GLPLOT2D 549 549 549 {PLOTDATA 2 "6'- %'CURVESG6%7ct7$$!#5\"\"!$\"3&))p$*)36@Sa!#=7$$!3Gmm;HU,\"*)*!#<$\"3m: %H%=YF&\\%F-7$$!3cKLLe%G?y*F1$\"3U'flOa'*p\\$F-7$$!3i++](oUIn*F1$\"3Kn a%yfAsX#F-7$$!3!pmmm\"p0k&*F1$\"3182F!*3H)Q\"F-7$$!3#)***\\P/,WP*F1$!3 @iOGo6cN]!#>7$$!3uKL$3c6 `F-7$$!3WmmmT%p\"e()F1$!3-%4Yf/TK='F-7$$!3&emmmwnMa)F1$!3-kS.)3Idr(F-7 $$!3/nmm\"4m(G$)F1$!3sg;\\nF\"R*))F-7$$!3?++Dc[3:\")F1$!3!))GV5td5m*F- 7$$!3OLL$3i.9!zF1$!3AoM'G)yv))**F-7$$!3M++Dch(=&yF1$!3/Yf#[$y(*****F-7 $$!3Xmmm\"p[B!yF1$!3]gWZdJn')**F-7$$!3aKL3F7#Gv(F1$!35_$4,[w)[**F-7$$! 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Find the absolute relativ e approximate error." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "epsi lon:=abs((x1-xguess2)/x1)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(e psilonG$\"+7=mh>!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "Although it is not necesary for the method, it is hel pful to define the equation for the secant line passing through the tw o guesses. 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1 2 0 1 10 3 2 9 1 4 2 1.000000 45.000000 45.000000 0 1 "Funct ion" "xguess1, First guess" "xguess2, Second guess" "x1, New guess" "S ecant line" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 265 24 "S ection IV: Iteration 2." }}{PARA 259 "" 0 "" {TEXT -1 63 "Using the sa me method, calculate the next estimate of the root." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x0:=xguess2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Estimate of the root" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "x2:=x1-(eval(f(x),x=x1)*(x0-x1))/(eval(f(x),x=x0)-eva l(f(x),x=x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$!+qVkAO!#5" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Absolute relative approximate er ror" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "epsilon:=abs((x2-x1) /x2)*100;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(epsilonG$\"+BJ+)4\"!\" (" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Secant line for the graph" } }}{EXCHG {PARA 0 "> " 0 "" 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]o7$$\"*9@BM)F^eo$\"+ydwx\")Fa]o7$$\"*`v&Q()F^eo$\"+a3 " 0 "" {MPLTEXT 1 0 0 "" }}}{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 23 "Section VI: Conclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 258 "Maple \+ helped us to apply our knowledge of numerical methods of finding roots of a nonlinear equation to simulate how, in the Secant method, an ini tial guess close to one root can jump to a location several roots away when a function is oscillatory in nature." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT 268 10 "References" }}{PARA 0 "" 0 " " {TEXT -1 4 "[1] " }{TEXT 270 174 "Nathan Collier, Autar Kaw, Jai Pau l , Michael Keteltas, Holistic Numerical Methods Institute, See http:/ /numericalmethods.eng.usf.edu/mws/gen/03nle/mws_gen_nle_txt_secant.pdf " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 0 "" } {TEXT 260 10 "Disclaimer" }{TEXT 269 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 "0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }