{VERSION 4 0 "IBM INTEL NT" "4.0" } {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 "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "Helvetica" 0 1 128 0 0 1 0 0 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 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 255 0 0 1 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 255 0 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 14 0 0 1 1 0 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }1 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 40 "Fixed Point Iteration: \+ Logistic Function" }}{PARA 0 "" 0 "" {TEXT 256 29 "Mth 351 June 30 200 2 Maple 6" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 31 "Filename: 351u2002_logistic.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 262 31 "Assignment 1. Due J uly 8, 2002." }{TEXT 263 1 " " }{TEXT 264 50 "See the two problems at \+ the end of this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "We will investigate the f ixed point iteration for the logistic function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=(x,c)->c *x*(1-x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"cG6\"6$% )operatorG%&arrowGF)*(9%\"\"\"9$F/,&F/F/F0!\"\"F/F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 474 "Here c \+ is a parameter which satisfies 0 <= c <= 4. For these values of c \+ we have 0 <= f(x,c) <= 1 for each x satisfying 0 <= x <= 1, tha t is, as a function of x, f(x,c) maps the interval [0,1] into its elf. Thus by the Intermediate Value Theorem (IVT) f(x,c) is guarante ed to have a fixed point in [0,1]. This fact is not particularly use ful here though since the origin is an obvious fixed point, and the IV T doesn't give us any additional fixed points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 335 "It is not difficult to s ee that f(x,c) is a contraction map for 0 <= c < 1. In this case w e have a unique fixed point and the fixed point iterations will conver ge very rapidly to the unique fixed point, no matter what initial \"gu ess\" we start with. For other values of c it is not clear how the \+ fixed point iterates will behave." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 332 "Our goal here is to plot some iterates f or various values of c and attempt to describe what we observe. Even in the case of convergence to a fixed point, the first few iterates m ay depend heavily on the initial point and so may be all over the plac e. Therefore we discard the first M0 iterates and plot only subseque nt iterates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Here are the values of the parameter c that we will work wit h. We omit 0 because it is not interesting.:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "N0:=30; # N0 valu es of c" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#N0G\"#I" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Cvals:=[seq(4*k/N0, k=1..N0)];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&CvalsG7@#\"\"#\"#:#\"\"%F(#F'\"\"&# \"\")F(#F'\"\"$#F*F,#\"#9F(#\"#;F(#\"\"'F,#F*F0#\"#AF(#F.F,#\"#EF(#\"# GF(F'#\"#KF(#\"#MF(#\"#7F,#\"#QF(#F.F0#F3F,#\"#WF(#\"#YF(#F5F,#\"#5F0# \"#_F(#\"#=F,#\"#cF(#\"#eF(F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 99 "Here are the initial points. We omit \+ 0 because it is a fixed point and so it's not very exciting." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "M0:=16; # M0 values of x0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# M0G\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Inits:=[seq(k/M0 , k=1..M0)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&InitsG72#\"\"\"\"#; #F'\"\")#\"\"$F(#F'\"\"%#\"\"&F(#F,F*#\"\"(F(#F'\"\"##\"\"*F(#F0F*#\"# 6F(#F,F.#\"#8F(#F3F*#\"#:F(F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Q0:=3; # Number of iterates to discard" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#Q0G\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "R0:=25; # R0 is number of iterates to plot" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#R0G\"#D" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 239 "We loop over the parameter values, c, an d the initial points, x0, and for each initial point x0 we do Q0+R0 iterates and discard the first Q0+1 iterates (including x0). The r emaining iterates are saved in a list of M0*R0*N0 points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "L :=[]: # list of all points" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "prec:=6: # precision" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "for k from 1 to nops(Cvals) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " c:=evalf(Cvals[k],prec):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f or j from 1 to nops(Inits) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " \+ x0:=evalf(Inits[j],prec):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " for \+ h from 1 to Q0 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 " x0:=evalf( f(x0,c),prec):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od:" }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 23 " for h from 1 to R0 do" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 50 " x0:=evalf(f(x0,c),prec): # compute new point \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 " L:=[op(L),[c,x0]]: # push o n L" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "od: od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "Let's plot the points in the list L Her e't the number of points we will be plotting:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "nops(L); M0 *R0*N0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&+?\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"&+?\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "PLOT(POINTS(op(L)),SYMBOL(DIAMOND,12));" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6$-%'POINTSG6\\iim7$$\"'LL8!\"'$\"' @N=!#57$F'$\"'!pW#!#67$F'$\"'_iK!#77$F'$\"'-]V!#87$F'$\"',+e!#97$F'$\" 'LLx!#:7$F'$\"'6J5F@7$F'$\"'\"[P\"!#;7$F'$\"'3L=!#<7$F'$\"'5WC!#=7$F'$ \"'zeK!#>7$F'$\"'/XV!#?7$F'$\"'P$z&!#@7$F'$\"'ZCx!#A7$F'$\"'$*H5Fin7$F '$\"'Ct8!#B7$F'$\"')4$=!#C7$F'$\"'ITC!#D7$F'$\"'1bK!#E7$F'$\"'2SV!#F7$ F'$\"'v'y&!#G7$F'$\"'l:x!#H7$F'$\"'vG5Fhp7$F'$\"'mr8!#I7$F'$\"'))G=!#J 7$F'$\"''*)R$F,7$F'$\"'zJXF07$F'$\"'MUgF47$F'$\"'Uc!)F87$F'$\"'>u5F87$ F'$\"'DK9F<7$F'$\"'m4>F@7$F'$\"'@YDFG7$F'$\"'%\\R$FK7$F'$\"'eEXFO7$F'$ \"'UNgFS7$F'$\"'@Z!)FW7$F'$\"''H2\"FW7$F'$\"'hI9Fen7$F'$\"'[2>Fin7$F'$ \"'IVDF`o7$F'$\"'1\"R$Fdo7$F'$\"'S@XFho7$F'$\"'_GgF\\p7$F'$\"',Q!)F`p7 $F'$\"'tr5F`p7$F'$\"'(*G9Fdp7$F'$\"'H0>Fhp7$F'$\"'QSDF_q7$F'$\"';(Q$Fc q7$F'$\"'x-ZF,7$F'$\"'0qiF07$F'$\"'+g$)F47$F'$\"'m96F47$F'$\"'@'[\"F87 $F'$\"'h\")>F<7$F'$\"'9UEF@7$F'$\"'%G_$FG7$F'$\"'6(p%FK7$F'$\"'!GE'FO7 $F'$\"'Q]$)FS7$F'$\"'Q86FS7$F'$\"']%[\"FW7$F'$\"'Lz>Fen7$F'$\"'5REFin7 $F'$\"'z=NF`o7$F'$\"'r\"p%Fdo7$F'$\"'gbiFho7$F'$\"'yS$)F\\p7$F'$\"'576 F\\p7$F'$\"'!G[\"F`p7$F'$\"'1x>Fdp7$F'$\"'2OEFhp7$F'$\"'v9NF_q7$F'$\"' K'o%Fcq7$F'$\"'YcdF,7$F'$\"'\"[n(F07$F'$\"'IB5F07$F'$\"'Sk8F47$F'$\"'? >=F87$F'$\"'fDCF<7$F'$\"'6MKF@7$F'$\"'97VFG7$F'$\"'^\\dFK7$F'$\"'*fm(F O7$F'$\"'8A5FO7$F'$\"'%GO\"FS7$F'$\"'7<=FW7$F'$\"'#GU#Fen7$F'$\"'UIKFi n7$F'$\"'A2VF`o7$F'$\"'&Hu&Fdo7$F'$\"'DdwFho7$F'$\"''4-\"Fho7$F'$\"'Gh 8F\\p7$F'$\"'/:=F`p7$F'$\"'0?CFdp7$F'$\"'tEKFhp7$F'$\"'I-VF_q7$F'$\"'R OdFcq7$F'$\"'&yc'F,7$F'$\"'`c()F07$F'$\"'_n6F07$F'$\"'pc:F47$F'$\"'ev? 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bnFc[bnFc[bnFc[bnFc[bnFc[bnFc[bnFc[bnFc[bn-%'SYMBOLG6$%(DIAMONDG\"#7" 1 2 2 1 12 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "W e can see another fixed point (other than 0) emerging when c > 1. Wh en c > 3 (or so) the behavior of the iterates changes dramatically. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "Here \+ are some problems for you. You can do them \"by hand.\"" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 10 "Problem 1." }{TEXT -1 78 " When 1 < c <= 4 there are two fixed points. What are they ( in terms of c)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 9 "Problem 2" }{TEXT -1 126 ". Use initial values 1/4, 1/2, 3/4, compute a few iterates and comment on their apparent behavior. D o this for c = 2, 3, 3.9." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 6 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }