{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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maple Input" -1 259 "Courier" 0 0 255 0 0 1 0 1 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 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 } {CSTYLE "" -1 266 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Purple Emphasis" -1 267 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "Red Emphasis" -1 268 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 269 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{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 0 1 0 0 0 0 0 0 0 }{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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{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 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 278 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Headi ng 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "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 "Time s" 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 "Bullet Item" -1 15 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 15 2 }{PSTYLE "Normal" -1 256 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 "Normal" -1 257 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 55 "Introduction to iterative methods for solving equations" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3. 2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load root-finding procedure s" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 278 7 "roo ts.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 10 "fixedpoint" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command si milar to the one that follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/proc drs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 43 "Hero's formula for calculating square roots" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We consider the problem of calculating \+ " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 0 "" } {XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 27 " must lie b etween 1 and 2." }}{PARA 15 "" 0 "" {TEXT -1 16 "Try the number " } {XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 35 " whic h is mid-way between 1 and 2." }}{PARA 15 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "(3/2)^2;" "6#*$*&\"\"$\"\"\"\"\"#!\"\"F'" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "9/4;" "6#*&\"\"*\"\"\"\"\"%!\"\"" }{TEXT -1 33 " is greater than 2, we see that " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$ \"\"\"\"\"#!\"\"" }{TEXT -1 13 " is too big." }}{PARA 15 "" 0 "" {TEXT -1 14 "Also dividing " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 15 " into 2 gives " }{XPPEDIT 18 0 "4/3;" "6#*&\" \"%\"\"\"\"\"$!\"\"" }{TEXT -1 25 " , which is smaller than " } {XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 9 " itsel f." }}{PARA 0 "" 0 "" {TEXT -1 8 " Since " }{XPPEDIT 18 0 "4/3;" "6#* &\"\"%\"\"\"\"\"$!\"\"" }{TEXT -1 5 " < " }{XPPEDIT 18 0 "sqrt(2)" " 6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " < " }{XPPEDIT 18 0 "(3/2)" "6#*&\" \"$\"\"\"\"\"#!\"\"" }{TEXT -1 61 ", we could now try the number mid-w ay between these, namely " }{XPPEDIT 18 0 "(3/2+4/3)/2;" "6#*&,&*&\" \"$\"\"\"\"\"#!\"\"F'*&\"\"%F'F&F)F'F'F(F)" }{TEXT -1 5 " = " } {XPPEDIT 18 0 "17/12;" "6#*&\"#<\"\"\"\"#7!\"\"" }{TEXT -1 3 " . " }} {PARA 15 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "17/12" "6#*&\"#<\"\"\" \"#7!\"\"" }{TEXT -1 27 " is a bit too large since " }{XPPEDIT 18 0 " (17/12)^2;" "6#*$*&\"#<\"\"\"\"#7!\"\"\"\"#" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "289/144;" "6#*&\"$*G\"\"\"\"$W\"!\"\"" }{TEXT -1 17 ", \+ which is 2 + " }{XPPEDIT 18 0 "1/144;" "6#*&\"\"\"F$\"$W\"!\"\"" } {TEXT -1 2 " ." }}{PARA 15 "" 0 "" {TEXT -1 9 "Dividing " }{XPPEDIT 18 0 "17/12" "6#*&\"#<\"\"\"\"#7!\"\"" }{TEXT -1 14 " into 2 gives " } {XPPEDIT 18 0 "24/17;" "6#*&\"#C\"\"\"\"#F*7%7$$\"#9F*$F* F*7$$\"#:F*F.7$F5F(7%7$$!#;F*F77$$!#:F*F.7$F>F(-%)POLYGONSG6$7&7$F)F)7 $F)F*7$F*F*7$F*F)-%&COLORG6&%$RGBG$\"\"(F*FP$\"\"*F*-F$6#7(7$F)F.7$\" \"#F.7$FYF/7$F/F/7$F/F.7$F*F.-%%TEXTG6$7$F.F.%\"gG-%*AXESSTYLEG6#%%NON EG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g := 'g':\ng := x -> (x + 2/ x)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operato rG%&arrowGF(,&9$#\"\"\"\"\"#*&F/F/F-!\"\"F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We can use the symbols " }{TEXT 259 1 "%" }{TEXT -1 35 " to refer to the last Maple result." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 190 "r[1] := 3/2;\nr[2] := g(r[1]);\nevalf(%,2);\nr[3] := g(r[2]);\nev alf(%,5);\nr[4] := g(r[3]);\nevalf(%,12);\nr[5] := g(r[4]);\nevalf(%,2 4);\nr[6] := g(r[5]);\nevalf(%,49);\nr[7] := g(r[6]);\nevalf(%,98);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"##\"#<\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6# \"\"$#\"$x&\"$3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&UT\"!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%#\"'dem\"'K3Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-PiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&#\"-(*))3Jn))\"-[gc8qi" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"9p,)[]4tBc8UT\"!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%\"rG6#\"\"'#\":6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Rx`(=np&y!)p4Us)o,)[]4tBc8UT\"!#[" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"(#\"Rx\")\\o*=91^Ld(['3v(y=?biHcq?&=&3-s6%zvb#zt\\$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"]q;kFV`(Q])Qq5i%yCt!*ztzm " 0 "" {MPLTEXT 1 0 131 "g := x -> (x + 2/x)/2;\nr[1 ] := 1.3;\nr[2] := evalf(g(r[1]));\nr[3] := evalf(g(r[2]));\nr[4] := e valf(g(r[3]));\nr[5] := evalf(g(r[4]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$#\"\"\"\"\"#*&F/F/F -!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"$\"#8 !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+p2B>9!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"+JCA99!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "We can write this as a loop using: " }{TEXT 0 39 " for . . from . . to . . do . . end do; " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 0 2 "do" }{TEXT -1 5 " an d " }{TEXT 0 6 "end do" }{TEXT -1 36 " serve to bracket repeated comma nds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "g := x -> (x + 2/x)/2;\nr[1] := 1.3;\nfor i from 1 to 4 do\n r[i+1] := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$#\"\"\"\" \"#*&F/F/F-!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6# \"\"\"$\"#8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+ p2B>9!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"+JCA99! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"+iN@99!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Iterative formulas for solvin g equations" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 39 "If we were given the iterative formula " } {XPPEDIT 18 0 "r[i+1] = (2/r[i]+r[i])/2;" "6#/&%\"rG6#,&%\"iG\"\"\"F)F )*&,&*&\"\"#F)&F%6#F(!\"\"F)&F%6#F(F)F)F-F0" }{TEXT -1 117 ", but we d id not know how it had been constructed, would it be possible to disco ver the number to which it converges?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 105 "The assumption here is, of course, t hat it does converge, but, with this assumption, the answer is 'yes'. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Suppo se that a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3];" "6%&%\"rG6#\"\" \"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 72 " . . . obtained by using this it erative formula converges to the number " }{XPPEDIT 18 0 "alpha;" "6#% &alphaG" }{TEXT -1 35 ". Then, if we apply the formula to " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 110 ", we would expect the formul a to give no further change in value, since it has already reached its objective. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 " This leads to the equation " }{XPPEDIT 18 0 "alpha = (2/alpha +alpha)/2;" "6#/%&alphaG*&,&*&\"\"#\"\"\"F$!\"\"F)F$F)F)F(F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 54 "A more formal way of deriving \+ this equation is to say " }{XPPEDIT 18 0 "Limit(r[i+1],i = infinity) = Limit((2/r[i]+r[i])/2,i = infinity);" "6#/-%&LimitG6$&%\"rG6#,&%\"iG \"\"\"F,F,/F+%)infinityG-F%6$*&,&*&\"\"#F,&F(6#F+!\"\"F,&F(6#F+F,F,F4F 7/F+F." }{TEXT -1 35 " , and then use the fact that both " }{XPPEDIT 18 0 "Limit(r[i+1],i = infinity);" "6#-%&LimitG6$&%\"rG6#,&%\"iG\"\"\" F+F+/F*%)infinityG" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Limit(r[i],i \+ = infinity);" "6#-%&LimitG6$&%\"rG6#%\"iG/F)%)infinityG" }{TEXT -1 15 " are equal to " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 "alpha \+ = (2/alpha+alpha)/2;" "6#/%&alphaG*&,&*&\"\"#\"\"\"F$!\"\"F)F$F)F)F(F* " }{TEXT -1 27 " can be simplified to give " }{XPPEDIT 18 0 "2*alpha = 2/alpha+alpha;" "6#/*&\"\"#\"\"\"%&alphaGF&,&*&F%F&F'!\"\"F&F'F&" } {TEXT -1 8 " or " }{XPPEDIT 18 0 "alpha = 2/alpha;" "6#/%&alphaG*& \"\"#\"\"\"F$!\"\"" }{TEXT -1 7 " or " }{XPPEDIT 18 0 "alpha^2 = 2; " "6#/*$%&alphaG\"\"#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 33 "This tells us that if a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3]; " "6%&%\"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$" }{TEXT -1 132 " . . . is obta ined by using the iterative formula, then, if this sequence converges, it must converge to a solution of the equation " }{XPPEDIT 18 0 "x^2 \+ = 2;" "6#/*$%\"xG\"\"#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "It turns out that if we start with any positive number, the result ing sequence converges to " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\" \"#" }{TEXT -1 100 " while, if we start with a negative number, the se quence obtained by using the formula converges to " }{XPPEDIT 18 0 "-s qrt(2);" "6#,$-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "g := x -> ( 2/x+x)/2;\nr[1] := -10;\nr[2] := evalf(g(r[1]));\nr[3] := evalf(g(r[2] ));\nr[4] := evalf(g(r[3]));\nr[5] := evalf(g(r[4]));\nr[6] := evalf(g (r[5]));\nr[7] := evalf(g(r[6]));\nr[8] := evalf(g(r[7]));\nr[9] := ev alf(g(r[8]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&9$#\"\"\"\"\"#*&F/F/F-!\"\"F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$!+++++^!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$!+J%ygu#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$!+v[>P&%\"rG6#\"\"&$!+&4QUW\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$!+bc_99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($!+(f8UT\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$!+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$!+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 27 "We can use a loop for this." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "g := x -> (x + 2/x)/2;\nr[1] := -10;\nfor i from 1 to 8 do\n r [i+1] := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$#\"\"\"\"\"#*&F/F/F-!\"\"F /F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$!+++++^!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$!+J%ygu#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$!+v[>P&%\"rG6#\"\"&$!+&4QUW\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$!+bc_99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($!+(f8UT\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$!+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$!+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 32 "Let's try the iterative formula " }{XPPEDIT 18 0 "r[i+1] = (r[i ]^3+1)/5;" "6#/&%\"rG6#,&%\"iG\"\"\"F)F)*&,&*$&F%6#F(\"\"$F)F)F)F)\"\" &!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "g := x -> (x^3+1)/5;\nr[1] := 0;\n r[2] := evalf(g(r[1]));\nr[3] := evalf(g(r[2]));\nr[4] := evalf(g(r[3] ));\nr[5] := evalf(g(r[4]));\nr[6] := evalf(g(r[5]));\nr[7] := evalf(g (r[6]));\nr[8] := evalf(g(r[7]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"#F1\"\"&F2F1 F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+++++?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"++++;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"+!3(Q;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+@lR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+^nR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+dnR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+dnR;?!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Again we can use a loop." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "g := x -> (x^3+1)/5;\nr[1] := 0;\nfor i from 1 to 7 do\n r[i+1] \+ := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf *6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\"\"\"#F1\"\"&F2F1F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+++++?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"++++;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"+!3(Q;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+@lR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+^nR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+dnR;?!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+dnR;?!#5" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&% \"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 40 "converges to a solution of the equation " }{XPPEDIT 18 0 "x = (x^3+1)/5;" "6#/%\"x G*&,&*$F$\"\"$\"\"\"F)F)F)\"\"&!\"\"" }{TEXT -1 6 " or " }{XPPEDIT 18 0 "x^3-5*x+1 = 0;" "6#/,(*$%\"xG\"\"$\"\"\"*&\"\"&F(F&F(!\"\"F(F(\" \"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "subs(x=r[8],x^3-5*x+1);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "A procedure for finding a fixed point of a real function: " }{TEXT 259 10 "fixedpoint" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 267 11 "fixed point" }{TEXT -1 29 " o f a function f is a number " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "f(a) = a;" "6#/-%\"fG6#%\"aGF'" } {TEXT -1 38 ", that is, such that the output value " }{XPPEDIT 18 0 "f (a);" "6#-%\"fG6#%\"aG" }{TEXT -1 33 " is the same as the input number " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 249 171 171 {PLOTDATA 2 "6*-%'CURVESG6$7%7$$ \"#9!\"\"$F*F*7$$\"#:F*\"\"!7$F($\"\"\"F*7%7$$!#;F*F+7$$!#:F*F/7$F5F1- %)POLYGONSG6$7&7$F2F27$F2F*7$F*F*7$F*F2-%&COLORG6&%$RGBG$\"\"(F*FG$\" \"*F*-F$6$7$7$F2F/7$\"\"#F/7$7$!\"#F/7$F*F/-%%TEXTG6$7$FS$\"\"$F*%\"aG -FV6$7$F/F/%\"fG-FV6$7$FP$\"#NFS%)f(a)~=~aG-%*AXESSTYLEG6#%%NONEG-%(SC ALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 1 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "For example, 0 and 1 are fixed points of the function " }{XPPEDIT 18 0 "f(x) = x^2;" "6# /-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 17 "fixedpoint: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " " }{TEXT -1 33 " fixedpoint( expr, s tartvalue ) " }{TEXT 262 1 "\n" }{TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 12 " expr - " }{TEXT -1 54 " an expression \+ involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{TEXT 256 2 "OR" }{TEXT -1 35 " a procedure of the form x -> f(x)," }}{PARA 0 "" 0 "" {TEXT -1 94 " \+ where f(x) evaluates to a real or complex floating poi nt number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 15 "startvalue - " }{TEXT 263 65 "an initial value for the iteration, which may be real or complex:" }}{PARA 0 "" 0 "" {TEXT 266 103 " in the form of a real con stant a when the1st argument is a procedure, and" }}{PARA 0 "" 0 "" {TEXT 265 111 " in the form of an equatio n x=a when the1st argument is an expression or equation." }}{PARA 0 " " 0 "" {TEXT -1 10 " " }}{PARA 257 "" 0 "" {TEXT -1 12 "Descr iption:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Given a starting value for x, the procedure " }{TEXT 0 10 "fixedpo int" }{TEXT -1 72 " attempts to find a fixed point of a function f, th at is, a solution of " }{XPPEDIT 18 0 "f(x) = x;" "6#/-%\"fG6#%\"xGF' " }{TEXT -1 28 ", by iterating the commands:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "newx := f(x);" "6#>%%newxG-%\"fG6#%\"x G" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x := newx;" "6# >%\"xG%%newxG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "until " }{XPPEDIT 18 0 "newx = x;" "6#/%%newxG%\"xG" }{TEXT -1 108 " as floati ng point numbers in the current precision, that is, until there is no \+ further change in the value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 29 "maxiterations=n or maxiter=n " }}{PARA 0 "" 0 "" {TEXT -1 119 "This option can be used to override the default value of Digit s*5 for the maximum number of iterations to be performed." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "info=true/false\n The option info=true allows the progress of the procedure to be monito red by printing the result of each iteration as it occurs. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 17 "How to activate:\n" }{TEXT -1 154 "To make the proc edure active open the subsection, place the cursor anywhere after the \+ prompt [ > and press [Enter].\nYou can then close up the subsection. " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "fixedpoint: implementation" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "fixedpoint" {MPLTEXT 1 0 4771 "fixedpoint := proc(ff,startvalue)\n local Options ,x,newx,prevx,h,k,i,maxit,prntflg,complexround, \n fx,fn,x0,x x,eps,diverg,zero,tiny,huge,proctype;\n \n if nargs<2 then\n err or \"at least 2 arguments are required; the basic syntax is: 'fixedpoi nt(f(x),x=a)'.\"\n end if; \n\n if type(ff,procedure) then\n \+ if nops([op(1,eval(ff))])<>1 then\n error \"the 1st argument, %1, is invalid .. it should be a procedure with a single argument\",f f;\n end if;\n proctype := true;\n if type(startvalue,' complexcons') then\n x0 := startvalue;\n else\n e rror \"the 2nd argument, %1, is invalid .. when the 1st argument is a \+ procedure, the 2nd argument should be a complex constant, which gives \+ the starting value\",startvalue;\n end if;\n elif type(ff,algeb raic) then\n proctype := false;\n if type(startvalue,name=co mplexcons) then\n x := op(1,startvalue);\n x0 := op(2, startvalue);\n else\n error \"the 2nd argument, %1, is in valid .. it should be be an equation of the form 'x=a', where a is the starting value\",startvalue;\n end if;\n if not type(indets (ff,name) minus \{x\},set(complexcons)) then\n error \"the 1st argument, %1, is invalid .. it should be an algebraic expression whic h depends only on the variable %2\",ff,x;\n end if;\n else\n \+ error \"the 1st argument, %1, is invalid .. it should be an algebra ic expression in a single variable, or a procedure with a single real \+ argument\",ff;\n end if;\n \n # Get the options \"maxiterations \" and \"info\".\n # Set the default values to start with.\n maxit := Digits*5;\n prntflg := false;\n if nargs>2 then\n Options :=[args[3..nargs]];\n if not type(Options,list(equation)) then\n \+ error \"each optional argument must be an equation\"\n en d if;\n if hasoption(Options,'maxiterations','maxit','Options') t hen\n if not type(maxit,posint) then\n error \"\\\" maxiterations\\\" must be a positive integer\"\n end if;\n \+ elif hasoption(Options,'maxiter','maxit','Options') then\n i f not type(maxit,posint) then\n error \"\\\"maxiter\\\" mus t be a positive integer\"\n end if;\n end if;\n if h asoption(Options,'info','prntflg','Options') then\n if prntflg <>true then prntflg := false end if;\n end if;\n if nops(Opt ions)>0 then\n error \"%1 is not a valid option for %2 .. t he recognised options are \\\"maxiterations\\\",(or \\\"maxiter\\\") a nd \\\"info\\\"\",op(1,Options),procname;\n end if;\n end if;\n \n if proctype then\n fn := ff;\n else\n # Evaluate any \+ real constants in f\n fn := unapply(evalf(ff),x);\n end if;\n\n # local procedure\n complexround := proc(zz)\n local re,im,e ps;\n re := Re(zz);\n im := Im(zz);\n if im=0 then retu rn Re(zz) end if;\n if re=0 then return Im(zz) end if;\n if \+ not type(re,float) or not type(im,float) then\n return zz\n \+ end if;\n eps := Float(1,-Digits);\n if abs(re)<=eps*abs( im) then return im*I\n elif abs(im)<=eps*abs(re) then return re\n else return zz end if;\n end proc: # of complexround\n \n \+ xx := evalf(x0);\n if prntflg then print(`starting value ->`,xx) end if;\n eps := Float(5,-Digits);\n diverg := 0;\n zero := 0;\n \+ tiny := Float(1,-100);\n huge := Float(1,100);\n\n for i from 1 to maxit do\n newx := traperror(evalf(fn(xx)));\n if newx=last error or not type(Re(newx),numeric) or not type(Im(newx),numeric) then \n error \"evaluation failed at %1\",evalf(xx,saveDigits);\n \+ end if;\n if prntflg then print(`iteration `,i,` value ->`,ne wx) end if;\n h := abs(newx-xx);\n if newx = xx or abs(h)<=e ps*abs(newx) then\n return complexround(newx) \n end if; \n k := abs(xx-prevx);\n if i > 6 then\n if h>10*k t hen \n diverg := diverg + 1\n else\n div erg := 0;\n end if;\n if 10*h3 and abs(newx)>huge then\n if newx>0 \+ and xx>0 then\n WARNING(\"the values appear to be diverg ing to infinity\");\n elif newx<0 and xx<0 then\n \+ WARNING(\"the values appear to be diverging to minus infinity\"); \n else\n WARNING(\"the values appear to be d iverging\");\n end if;\n return complexround(new x);\n end if;\n if zero>3 and abs(newx)`,newx);\n error \"no fixed point found after %1 iterations\", i;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 10 "fixedpoint" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{PARA 0 "" 0 "" {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 21 " is a fixed point of " }{XPPEDIT 18 0 "f(x) = (2/x+x)/2;" "6#/- %\"fG6#%\"xG*&,&*&\"\"#\"\"\"F'!\"\"F,F'F,F,F+F-" }{TEXT -1 3 " .\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "fixedpoint((2/x+x)/2,x=1.5, info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2starting~value~->G$\" #:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\"\"%*~value ~->G$\"+nmm;9!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\" #%*~value~->G$\"+'o:UT\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iter ation~G\"\"$%*~value~->G$\"+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\"%%*~value~->G$\"+iN@99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 37 "We can have a complex starting value." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "evalf(fixedpoint((2/x+x)/2,x=1+0.9*I,info=true),20);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%2starting~value~->G^$$\"\"\"\"\"!$\"\"*!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\"\"%*~value~->G^$$\"5 u'QIXy='[_5!#>$!4m![t21pvBZ!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+it eration~G\"\"#%*~value~->G^$$\"5Z8?S(fCkWZ\"!#>$\"5)4$zmwos\"R*=!#@" } }{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\"$%*~value~->G^$$\"5& =<^K$)oK`T\"!#>$\"4vq+0d-?`f(!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+ iteration~G\"\"%%*~value~->G^$$\"5CY))3N,Q@99!#>$\"2)*y#)RIdW+'!#B" }} {PARA 11 "" 1 "" {XPPMATH 20 "6&%+iteration~G\"\"&%*~value~->G^$$\"5?x ()HPiN@99!#>$\"/)f9S]Y,\"!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%+iter ation~G\"\"'%*~value~->G^$$\"5)[]4tBc8UT\"!#>$!([np(!#L" }}{PARA 11 " " 1 "" {XPPMATH 20 "6&%+iteration~G\"\"(%*~value~->G^$$\"5)[]4tBc8UT\" !#>$\"\"!F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5)[]4tBc8UT\"!#>" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 " " 0 "" {TEXT -1 14 "Starting with " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG \"\"!" }{TEXT -1 29 ", and iterating the function " }{XPPEDIT 18 0 "f( x) = (x^3+1)/5;" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"$\"\"\"F,F,F,\"\"&!\"\" " }{TEXT -1 76 " gives a sequence which converges to the middle root \+ of the cubic equation " }{XPPEDIT 18 0 "x^3-5*x+1 = 0;" "6#/,(*$%\"xG \"\"$\"\"\"*&\"\"&F(F&F(!\"\"F(F(\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "fixedpo int((x^3+1)/5,x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+dnR;?!#5" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 0 6 "fsolve" } {TEXT -1 45 " gives all three roots of the cubic equation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fsol ve(x^3-5*x+1=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%$!+S(e+L#!\"*$\" +dnR;?!#5$\"+k!>%G@F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 14 "Starting with " }{XPPEDIT 18 0 "x = 4;" "6#/%\"xG\"\"%" }{TEXT -1 29 ", and iterating \+ the function " }{XPPEDIT 18 0 "f(x) = 5-2/x;" "6#/-%\"fG6#%\"xG,&\"\"& \"\"\"*&\"\"#F*F'!\"\"F-" }{TEXT -1 80 " gives a sequence which conve rges to the larger root of the quadratic equation " }{XPPEDIT 18 0 "x^ 2-5*x+2 = 0;" "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"&F(F&F(!\"\"F'F(\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "fixedpoint(5-2/x,x=4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+8GbhX!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(x^2-5*x+2=0);\nevalf(% );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"&\"\"#\"\"\"*&F&!\"\"\"#< #F'F&F',&F$F'*&F&F)F*F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+8GbhX !\"*$\"*(=Z%Q%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Exa mple 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "A fixed point of " }{XPPEDIT 18 0 "f(x) = sin(x)+1;" "6#/-%\"fG6#%\"x G,&-%$sinG6#F'\"\"\"F,F," }{TEXT -1 31 " is a solution of the equation " }{XPPEDIT 18 0 "x = sin(x)+1;" "6#/%\"xG,&-%$sinG6#F$\"\"\"F)F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "fixedpoint(sin(x)+1,x=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6KcM>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 0 6 "fsolve" }{TEXT -1 50 " gives the same root \+ with the same starting value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "fsolve(x=sin(x)+1,x=2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+6KcM>!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 0 "" 0 "" {TEXT -1 26 "The cube root of a number " }{TEXT 270 1 "a" }{TEXT -1 19 " is fix ed point of " }{XPPEDIT 18 0 "f(x) = (2*x^3+a)/(3*x^2);" "6#/-%\"fG6#% \"xG*&,&*&\"\"#\"\"\"*$F'\"\"$F,F,%\"aGF,F,*&F.F,*$F'F+F,!\"\"" } {TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 52 "Thus the cube root of 1 0 can be computed as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "g := x -> (2*x^3+10)/(3*x^2) ;\nc := fixedpoint(g,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6# %\"xG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"\"$F/*&,&*&\"\"#F/)9$F0F/F /\"#5F/F/F6!\"#F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\" +!pMW:#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Check . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 4 "c^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++ +++5!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "Graphical illustration of an ite rative solution . . Ex 1" }}{PARA 0 "" 0 "" {TEXT -1 48 "Provided that we take a suitable starting value " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6 #\"\"\"" }{TEXT -1 25 ", the iterative formula " }{XPPEDIT 18 0 "r[i+ 1] = 5*r[i]/3-1/4-r[i]^2/7;" "6#/&%\"rG6#,&%\"iG\"\"\"F)F),(*(\"\"&F)& F%6#F(F)\"\"$!\"\"F)*&F)F)\"\"%F0F0*&&F%6#F(\"\"#\"\"(F0F0" }{TEXT -1 19 " gives a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6 &&%\"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 48 ", which c onverges to a solution of the equation " }{XPPEDIT 18 0 "x = 5*x/3-1/4 -x^2/7;" "6#/%\"xG,(*(\"\"&\"\"\"F$F(\"\"$!\"\"F(*&F(F(\"\"%F*F**&F$\" \"#\"\"(F*F*" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 64 "Note tha t this equation is equivalent to the quadratic equation " }{XPPEDIT 18 0 "12*x^2-56*x+21 = 0;" "6#/,(*&\"#7\"\"\"*$%\"xG\"\"#F'F'*&\"#cF'F )F'!\"\"\"#@F'\"\"!" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Le t's try taking " }{XPPEDIT 18 0 "r[1] = 2;" "6#/&%\"rG6#\"\"\"\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "g := x -> 5*x/3-1/4-x^2/7;\nr[1] := 2;\nfor i fr om 1 to 30 do\n r[i+1] := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$#\"\"& \"\"$#!\"\"\"\"%\"\"\"*$)F-\"\"#F4#F2\"\"(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"rG6#\"\"#$\"+iZ!>^#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\" rG6#\"\"$$\"+Zr7NI!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\" \"%$\"+-da#\\$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$ \"+jeNGQ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+KZ$o .%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+PH0]T!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+%)\\L1U!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+$f]HB%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$\"+Q=AXU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$\"+!Q63D%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#7$\"+\\JM`U!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#8$\"+rp[aU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$\"+MJ+bU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$\"+PfBbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$\"+74MbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$\"+U#)QbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#=$\"+\"e4aD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#>$\"+-#>aD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#?$\"+SNUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#@$\"+'\\DaD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#A$\"+xjUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#B$\"+unUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#C$\"+`pUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#D$\"+MqUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#E$\"+qqUbU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#F$\"+'3FaD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#G$\"+%4FaD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#H$\"+(4FaD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#I$\"+)4FaD%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#J$\"+)4FaD%!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 64 "The following graph illustrates the convergence of the \+ sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\" \"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The graph shows the line " }{XPPEDIT 18 0 "y = x;" "6#/% \"yG%\"xG" }{TEXT -1 16 " and the curve " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "g(x) = \+ 5*x/3-1/4-x^2/7;" "6#/-%\"gG6#%\"xG,(*(\"\"&\"\"\"F'F+\"\"$!\"\"F+*&F+ F+\"\"%F-F-*&F'\"\"#\"\"(F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 52 "The points of intersection of these two graphs have " } {TEXT 273 1 "x" }{TEXT -1 18 " coordinates (and " }{TEXT 272 1 "y" } {TEXT -1 50 " coordinates) which are solutions of the equation " } {XPPEDIT 18 0 "g(x) = x;" "6#/-%\"gG6#%\"xGF'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 40 "The blue path shown connects the pointss " }{XPPEDIT 18 0 " ``(r[1],r[1]),``(r[1],r[2]),``(r[2],r[2]),``(r[2],r [3]),``(r[3],r[3]),` . . . `" "6(-%!G6$&%\"rG6#\"\"\"&F'6#F)-F$6$&F'6# F)&F'6#\"\"#-F$6$&F'6#F2&F'6#F2-F$6$&F'6#F2&F'6#\"\"$-F$6$&F'6#F?&F'6# F?%(~.~.~.~G" }{TEXT -1 12 ", and so on." }}{PARA 0 "" 0 "" {TEXT -1 18 "Points of the form" }{XPPEDIT 18 0 "``(r[i],r[i]);" "6#-%!G6$&%\"r G6#%\"iG&F'6#F)" }{TEXT -1 12 " have equal " }{TEXT 274 1 "x" }{TEXT -1 5 " and " }{TEXT 275 1 "y" }{TEXT -1 36 " coordinates and so lie on the line " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 26 ", wh ile points of the form" }{XPPEDIT 18 0 "``(r[i],r[i+1]);" "6#-%!G6$&% \"rG6#%\"iG&F'6#,&F)\"\"\"F-F-" }{TEXT -1 15 " are such that " } {XPPEDIT 18 0 "r[i+1] = g(r[i]);" "6#/&%\"rG6#,&%\"iG\"\"\"F)F)-%\"gG6 #&F%6#F(" }{TEXT -1 26 ", and so lie on the curve " }{XPPEDIT 18 0 "y \+ = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 75 "This path provides a route towards the right-hand point o f intersection of " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 49 " and illustrates the convergence of the sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$ %(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "path := NULL:\nfor i from 1 to 8 d o\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do: \nplot([g(x),x,[path]],x=0..5,y=0..5,color=[red,green,blue]);" }} {PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$ \"\"!$!1+++++++D!#;7$$\"1LLL3x&)*3\"F+$!1B;u@\"*Q0q!#<7$$\"1nm\"H2P\"Q ?F+$\"13`8aA_v$)F17$$\"1LL$eRwX5$F+$\"1j V$RtF+7$$\"1***\\il'pisF+$\"1w*e`Cp4&))F+7$$\"1MLe*)>VB$)F+$\"1r;&>#zE Q5!#:7$$\"1++DJbw!Q*F+$\"1..![S[x=\"FU7$$\"1nm;/j$o/\"FU$\"1l!3D&[jU6FU$\"1*ef**=ryY\"FU7$$\"1++]i^Z]7FU$\"1\"*>(>DT2h\"FU7$ $\"1++](=h(e8FU$\"1AVkbX&3v\"FU7$$\"1++]P[6j9FU$\"1FU7$$\"1nm;a/cq;FU$\"1\\FU$ \"14TO)[9[[#FU7$$\"1+]i!f#=$3#FU$\"1.A.,q =HFU7$$\"1LL3_?`(\\#FU$\"1iE6F#e9-$FU7$$\"1M$e*)>pxg#FU$\"1\\to%o(yCJF U7$$\"1+]Pf4t.FFU$\"1@g&p`4>@$FU7$$\"1MLe*Gst!GFU$\"1ck#*>#[II$FU7$$\" 1+++DRW9HFU$\"1&*Q\"GM!)RR$FU7$$\"1++DJE>>IFU$\"1ez!Q)*p(zMFU7$$\"1+]i !RU07$FU$\"1I'>%o=zfNFU7$$\"1++v=S2LKFU$\"1c+5#=/_k$FU7$$\"1mmm\"p)=ML FU$\"1nB(R$[')=PFU7$$\"1++](=]@W$FU$\"1)>Peq)G%z$FU7$$\"1L$e*[$z*RNFU$ \"1*Hhr*yvfQFU7$$\"1,+]iC$pk$FU$\"1=nDWT?GRFU7$$\"1m;H2qcZPFU$\"1v(*)H PA'*)RFU7$$\"1+]7.\"fF&QFU$\"1kg\"42H20%FU7$$\"1mm;/OgbRFU$\"1%\\a)[bT 2TFU7$$\"1+]ilAFjSFU$\"1;(3\\3BN;%FU7$$\"1MLL$)*pp;%FU$\"1^1lP,V9UFU7$ $\"1ML3xe,tUFU$\"1I82fCJjUFU7$$\"1n;HdO=yVFU$\"1-[eUnh3VFU7$$\"1,++D># [Z%FU$\"1Q^JZ1YZVFU7$$\"1nmT&G!e&e%FU$\"1$\\Fqe)p)Q%FU7$$\"1MLL$)Qk%o% FU$\"17_g.HhAWFU7$$\"1+]iSjE!z%FU$\"1#)GkYTobWFU7$$\"1,]P40O\"*[FU$\"1 ?)G&R@N%[%FU7$$\"\"&F($\"1jZ!>w/>^%FU-%'COLOURG6&%$RGBG$\"*++++\"!\")F (F(-F$6$7S7$F(F(7$F-F-7$F3F37$F8F87$F=F=7$FBFB7$FGFG7$FLFL7$FQFQ7$FWFW 7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoFjo7$F_pF_p7$FdpFdp7$FipFip7$F^qF ^q7$FcqFcq7$FhqFhq7$F]rF]r7$FbrFbr7$FgrFgr7$F\\sF\\s7$FasFas7$FfsFfs7$ F[tF[t7$F`tF`t7$FetFet7$FjtFjt7$F_uF_u7$FduFdu7$FiuFiu7$F^vF^v7$FcvFcv 7$FhvFhv7$F]wF]w7$FbwFbw7$FgwFgw7$F\\xF\\x7$FaxFax7$FfxFfx7$F[yF[y7$F` yF`y7$FeyFey7$FjyFjy7$F_zF_z7$FdzFdz-Fiz6&F[[lF(F\\[lF(-F$6$7:7$$\"\"# F(Fi^l7$Fi^l$\"1+++iZ!>^#FU7$F\\_lF\\_lF^_l7$F\\_l$\"1+++Zr7NIFU7$F`_l F`_lFb_l7$F`_l$\"1+++-da#\\$FU7$Fd_lFd_lFf_l7$Fd_l$\"1+++jeNGQFU7$Fh_l Fh_lFj_l7$Fh_l$\"1+++KZ$o.%FU7$F\\`lF\\`lF^`l7$F\\`l$\"1+++PH0]TFU7$F` `lF``lFb`l7$F``l$\"1+++%)\\L1UFU7$Fd`lFd`lFf`l7$Fd`l$\"1+++$f]HB%FU7$F h`lFh`l-Fiz6&F[[lF(F(F\\[l-%+AXESLABELSG6$Q\"x6\"Q\"yFaal-%%VIEWG6$;F( FdzFfal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curv e 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 54 "We can zoom in closer to the point of intersection . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "path := NULL:\nfor i from 7 to 13 do\n path := pat h,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do:\nplot([g(x),x,[p ath]],x=4.2..4.27,y=4.2..4.27,color=[red,green,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"1+++++++U !#:$\"1,+++++IUF*7$$\"1n;z+e_,UF*$\"1%H)y2R`G?%F*$\"1v ow>/LJUF*7$$\"1nT&pSYV?%F*$\"1\"z%\\Cc-KUF*7$$\"1n\"z'=$\\e?%F*$\"1?/E %zCFB%F*7$$\"1$3Ft3Xt?%F*$\"1sCB++ULUF*7$$\"15uR1MUF* 7$$\"1](=`xn,@%F*$\"1b=s#>IZB%F*7$$\"1UF*$ \"1BLEXb#)QUF*7$$\"1+]s2O[?UF*$\"1ev>x!*\\RUF*7$$\"1[UF*$\"1hnTfp:_UF*7$$\"1kUF*$\"1xyA`.PfUF*7$$\"1nmO9]elUF*$\"1'*\\;b=**fUF*7$$\"1] (o(GP1nUF*$\"1bN^+RlgUF*7$$\"1\\78Z!z%oUF*$\"1NvCxpGhUF*7$$\"1++++++qU F*$\"1nmmmm'>E%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")\"\"!F^[l-F$6$7S7$F(F (7$F.F.7$F3F37$F8F87$F=F=7$FBFB7$FGFG7$FLFL7$FQFQ7$FVFV7$FenFen7$FjnFj n7$F_oF_o7$FdoFdo7$FioFio7$F^pF^p7$FcpFcp7$FhpFhp7$F]qF]q7$FbqFbq7$Fgq Fgq7$F\\rF\\r7$FarFar7$FfrFfr7$F[sF[s7$F`sF`s7$FesFes7$FjsFjs7$F_tF_t7 $FdtFdt7$FitFit7$F^uF^u7$FcuFcu7$FhuFhu7$F]vF]v7$FbvFbv7$FgvFgv7$F\\wF \\w7$FawFaw7$FfwFfw7$F[xF[x7$F`xF`x7$FexFex7$FjxFjx7$F_yF_y7$FdyFdy7$F iyFiy7$F^zF^z7$FczFcz-Fhz6&FjzF^[lF[[lF^[l-F$6$777$$\"1+++PH0]TF*Fi^l7 $Fi^l$\"1+++%)\\L1UF*7$F\\_lF\\_lF^_l7$F\\_l$\"1+++$f]HB%F*7$F`_lF`_lF b_l7$F`_l$\"1+++Q=AXUF*7$Fd_lFd_lFf_l7$Fd_l$\"1+++!Q63D%F*7$Fh_lFh_lFj _l7$Fh_l$\"1+++\\JM`UF*7$F\\`lF\\`lF^`l7$F\\`l$\"1+++rp[aUF*7$F``lF``l Fb`l7$F``l$\"1+++MJ+bUF*7$Fd`lFd`l-Fhz6&FjzF^[lF^[lF[[l-%+AXESLABELSG6 $Q\"x6\"Q\"yF]al-%%VIEWG6$;$\"#U!\"\"$\"$F%!\"#Fbal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " . . a nd closer . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "path := NULL:\nfor i from 13 to 20 do\n path \+ := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do:\nplot([g(x ),x,[path]],x=4.2545..4.2556,y=4.2545..4.2556,\ncolor=[red,green,blue] );" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG 6$7S7$$\"1+++++]aU!#:$\"1jZ!>,4]D%F*7$$\"1$epo(R_aUF*$\"1HVnF)>]D%F*7$ $\"10c,R[aaUF*$\"1jRGQ#H]D%F*7$$\"142o+$oXD%F*$\"1j,N@)R]D%F*7$$\"1f\\ 'y\">faUF*$\"1c*3WZ]]D%F*7$$\"1a3!GU:YD%F*$\"1#Rxm2h]D%F*7$$\"1r_)[@PY D%F*$\"1+k>142bUF*7$$\"1PkKz(fYD%F*$\"1-5&R3\"3bUF*7$$\"1rP]:JoaUF*$\" 1^jk4;4bUF*7$$\"1(oToP1ZD%F*$\"1:3V,@5bUF*7$$\"1a^D%F*7$$\"1 r([JtU[D%F*$\"1__T+O;bUF*7$$\"1<**HBv'[D%F*$\"1Z!>1yu^D%F*7$$\"1n'4Q_) )[D%F*$\"1'z!e^U=bUF*7$$\"1Q\\R_H\"\\D%F*$\"1'H-%o_>bUF*7$$\"1n@$edM\\ D%F*$\"1$z8+-0_D%F*7$$\"1Q*p,Ie\\D%F*$\"1hM#*=d@bUF*7$$\"17)\\8*3)\\D% F*$\"1igm1fAbUF*7$$\"1Ug(GY/]D%F*$\"1c7NOlBbUF*7$$\"1a`*)3h-bUF*$\"1m1 e(HY_D%F*7$$\"1f90d%\\]D%F*$\"11_BEoDbUF*7$$\"1rPA4P2bUF*$\"1KLQixEbUF *7$$\"1i5\"3#[4bUF*$\"15c>#Gx_D%F*7$$\"13P!>i<^D%F*$\"1P1rjvGbUF*7$$\" 1+Nmx69bUF*$\"17dO&=)HbUF*7$$\"1v)yBAk^D%F*$\"1r&Hkd3`D%F*7$$\"1QfK>l= bUF*$\"1EP RRbUF*$\"1t!e297aD%F*7$$\"1LjRLnTbUF*$\"1#HV#Q%HVbUF*7$$\"10YS+KYbUF*$\"1zT[uLWbUF*7$$\"1+N#3Y%[bUF*$\"1hC/fH XbUF*7$$\"1#ziw#)3bD%F*$\"1T!pQ%RYbUF*7$$\"1MVl@1`bUF*$\"1o^ooPZbUF*7$ $\"1Q\\feQbbUF*$\"1f)QQC%[bUF*7$$\"1j?J*4wbD%F*$\"1ZIvpU\\bUF*7$$\"1++ +++gbUF*$\"1B&4Q/0bD%F*-%'COLOURG6&%$RGBG$\"*++++\"!\")\"\"!F^[l-F$6$7 S7$F(F(7$F.F.7$F3F37$F8F87$F=F=7$FBFB7$FGFG7$FLFL7$FQFQ7$FVFV7$FenFen7 $FjnFjn7$F_oF_o7$FdoFdo7$FioFio7$F^pF^p7$FcpFcp7$FhpFhp7$F]qF]q7$FbqFb q7$FgqFgq7$F\\rF\\r7$FarFar7$FfrFfr7$F[sF[s7$F`sF`s7$FesFes7$FjsFjs7$F _tF_t7$FdtFdt7$FitFit7$F^uF^u7$FcuFcu7$FhuFhu7$F]vF]v7$FbvFbv7$FgvFgv7 $F\\wF\\w7$FawFaw7$FfwFfw7$F[xF[x7$F`xF`x7$FexFex7$FjxFjx7$F_yF_y7$Fdy Fdy7$FiyFiy7$F^zF^z7$FczFcz-Fhz6&FjzF^[lF[[lF^[l-F$6$7:7$$\"1+++rp[aUF *Fi^l7$Fi^l$\"1+++MJ+bUF*7$F\\_lF\\_lF^_l7$F\\_l$\"1+++PfBbUF*7$F`_lF` _lFb_l7$F`_l$\"1+++74MbUF*7$Fd_lFd_lFf_l7$Fd_l$\"1+++U#)QbUF*7$Fh_lFh_ lFj_l7$Fh_l$\"1+++\"e4aD%F*7$F\\`lF\\`lF^`l7$F\\`l$\"1+++-#>aD%F*7$F`` lF``lFb`l7$F``l$\"1+++SNUbUF*7$Fd`lFd`lFf`l7$Fd`l$\"1+++'\\DaD%F*7$Fh` lFh`l-Fhz6&FjzF^[lF^[lF[[l-%+AXESLABELSG6$Q\"x6\"Q\"yFaal-%%VIEWG6$;$ \"&XD%!\"%$\"&cD%FialFfal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "We can check the solution by su bstitution in various ways." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "subs(x=4.255427098,g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+)4FaD%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "subs(x=4.255427098,x= g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/$\"+)4FaD%!\"*F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "su bs(x=4.255427098,x-g(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Mapl e procedure " }{TEXT 259 5 "solve" }{TEXT -1 80 " can be used to obtai n analytical expressions for the solutions of the equation " } {XPPEDIT 18 0 "g(x) = x;" "6#/-%\"gG6#%\"xGF'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(g(x)=x,x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&# \"\"(\"\"$\"\"\"*$-%%sqrtG6#\"$L\"F'#F'\"\"',&F$F'F(#!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+)4FaD%!\"*$\"*o&R7TF%" }}}{PARA 0 "" 0 " " {TEXT -1 40 "As we observed previously, the equation " }{XPPEDIT 18 0 "g(x) = x;" "6#/-%\"gG6#%\"xGF'" }{TEXT -1 41 " is equivalent to the quadratic equation " }{XPPEDIT 18 0 "12*x^2-56*x+21 = 0;" "6#/,(*&\"# 7\"\"\"*$%\"xG\"\"#F'F'*&\"#cF'F)F'!\"\"\"#@F'\"\"!" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(12*x^2-56*x+21=0,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,& #\"\"(\"\"$\"\"\"*$-%%sqrtG6#\"$L\"F'#F'\"\"',&F$F'F(#!\"\"F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Take anot her look at the graphs of " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "g := x -> 5*x/3-1/4-x^2/7;\nplot([g(x),x],x=0. .5,y=0..5,color=[red,green]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"g Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$#\"\"&\"\"$#!\"\"\"\"%\"\"\"*$ )F-\"\"#F4#F2\"\"(F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6&-%'CURVESG6$7S7$\"\"!$!1+++++++D!#;7$$\"1LLL3x&)*3\"F+$ !1B;u@\"*Q0q!#<7$$\"1nm\"H2P\"Q?F+$\"13`8aA_v$)F17$$\"1LL$eRwX5$F+$\"1 jV$RtF+7$$\"1***\\il'pisF+$\"1w*e`Cp4&)) F+7$$\"1MLe*)>VB$)F+$\"1r;&>#zEQ5!#:7$$\"1++DJbw!Q*F+$\"1..![S[x=\"FU7 $$\"1nm;/j$o/\"FU$\"1l!3D&[jU6FU$\"1*ef**=ryY\"FU7$$\" 1++]i^Z]7FU$\"1\"*>(>DT2h\"FU7$$\"1++](=h(e8FU$\"1AVkbX&3v\"FU7$$\"1++ ]P[6j9FU$\"1FU7$$\"1nm;a/c q;FU$\"1\\FU$\"14TO)[9[[#FU7$$\"1+]i!f#=$3#FU$\"1.< RE5--EFU7$$\"1+](=xpe=#FU$\"1oIP\\5a5FFU7$$\"1nm\"H28IH#FU$\"1OW\\/'e0 #GFU7$$\"1n;zpSS\"R#FU$\"1+>A.,q=HFU7$$\"1LL3_?`(\\#FU$\"1iE6F#e9-$FU7 $$\"1M$e*)>pxg#FU$\"1\\to%o(yCJFU7$$\"1+]Pf4t.FFU$\"1@g&p`4>@$FU7$$\"1 MLe*Gst!GFU$\"1ck#*>#[II$FU7$$\"1+++DRW9HFU$\"1&*Q\"GM!)RR$FU7$$\"1++D JE>>IFU$\"1ez!Q)*p(zMFU7$$\"1+]i!RU07$FU$\"1I'>%o=zfNFU7$$\"1++v=S2LKF U$\"1c+5#=/_k$FU7$$\"1mmm\"p)=MLFU$\"1nB(R$[')=PFU7$$\"1++](=]@W$FU$\" 1)>Peq)G%z$FU7$$\"1L$e*[$z*RNFU$\"1*Hhr*yvfQFU7$$\"1,+]iC$pk$FU$\"1=nD WT?GRFU7$$\"1m;H2qcZPFU$\"1v(*)HPA'*)RFU7$$\"1+]7.\"fF&QFU$\"1kg\"42H2 0%FU7$$\"1mm;/OgbRFU$\"1%\\a)[bT2TFU7$$\"1+]ilAFjSFU$\"1;(3\\3BN;%FU7$ $\"1MLL$)*pp;%FU$\"1^1lP,V9UFU7$$\"1ML3xe,tUFU$\"1I82fCJjUFU7$$\"1n;Hd O=yVFU$\"1-[eUnh3VFU7$$\"1,++D>#[Z%FU$\"1Q^JZ1YZVFU7$$\"1nmT&G!e&e%FU$ \"1$\\Fqe)p)Q%FU7$$\"1MLL$)Qk%o%FU$\"17_g.HhAWFU7$$\"1+]iSjE!z%FU$\"1# )GkYTobWFU7$$\"1,]P40O\"*[FU$\"1?)G&R@N%[%FU7$$\"\"&F($\"1jZ!>w/>^%FU- %'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7S7$F(F(7$F-F-7$F3F37$F8F87$F= F=7$FBFB7$FGFG7$FLFL7$FQFQ7$FWFW7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoF jo7$F_pF_p7$FdpFdp7$FipFip7$F^qF^q7$FcqFcq7$FhqFhq7$F]rF]r7$FbrFbr7$Fg rFgr7$F\\sF\\s7$FasFas7$FfsFfs7$F[tF[t7$F`tF`t7$FetFet7$FjtFjt7$F_uF_u 7$FduFdu7$FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F]wF]w7$FbwFbw7$FgwFgw7$F\\x F\\x7$FaxFax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFey7$FjyFjy7$F_zF_z7$FdzFdz-F iz6&F[[lF(F\\[lF(-%+AXESLABELSG6$Q\"x6\"Q\"yFi^l-%%VIEWG6$;F(FdzF^_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We can see from these graphs that the iterative formula " }{XPPEDIT 18 0 "r[i+1] = 5*r[i]/3-1/4-r[i]^2/7;" "6#/&%\"rG6#,&%\"iG\"\"\"F)F),( *(\"\"&F)&F%6#F(F)\"\"$!\"\"F)*&F)F)\"\"%F0F0*&&F%6#F(\"\"#\"\"(F0F0" }{TEXT -1 23 " will give a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3], ` . . . `;" "6&&%\"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 40 ", which converges to the larger solution" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "7/3+sqrt(133)/6;" "6#,&*&\"\"(\"\"\"\" \"$!\"\"F&*&-%%sqrtG6#\"$L\"F&\"\"'F(F&" }{TEXT -1 2 " " }{TEXT 279 1 "~" }{TEXT -1 13 " 4.255427098 " }}{PARA 0 "" 0 "" {TEXT -1 26 "of t he quadratic equation " }{XPPEDIT 18 0 "12*x^2-56*x+21 = 0;" "6#/,(*& \"#7\"\"\"*$%\"xG\"\"#F'F'*&\"#cF'F)F'!\"\"\"#@F'\"\"!" }{TEXT -1 25 " , for any starting value " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"\"" } {TEXT -1 43 " which is greater than the smaller solution" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "7/3-sqrt(133)/6;" "6#,&*&\"\"( \"\"\"\"\"$!\"\"F&*&-%%sqrtG6#\"$L\"F&\"\"'F(F(" }{TEXT -1 2 " " } {TEXT 271 1 "~" }{TEXT -1 13 " 0.411239568." }}{PARA 0 "" 0 "" {TEXT -1 139 "On the other hand, if we take a starting value less than the s maller solution, the resulting sequence \"runs away\" to infinity, tha t is, it " }{TEXT 267 8 "diverges" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "To give an example of thi s divergence, we perform a few iterations starting with " }{XPPEDIT 18 0 "r[1] = 0;" "6#/&%\"rG6#\"\"\"\"\"!" }{TEXT -1 3 ".4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "g := x -> 5*x/3-1/4-x^2/7;\nr[1] := 0.4;\nfor i from 1 to 20 do\n r[i+1] := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG f*6#%\"xG6\"6$%)operatorG%&arrowGF(,(9$#\"\"&\"\"$#!\"\"\"\"%\"\"\"*$) F-\"\"#F4#F2\"\"(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\" \"\"$\"\"%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+Q _4QR!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"+>2%>%Q!#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"+D,P#p$!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+NX=fM!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+1`O%4$!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+:%)[?D!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+c)e+h\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+'>()RY\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$!+9$3jD#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$!+*QTKL'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"rG6#\"#7$!+c,%GO\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6# \"#8$!+(eLny#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$!+! *z'R+'!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$!+)\\F1a\" !\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$!+^SZ$)f!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$!+N;J9h!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#=$!+wohUa!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#>$!+*fKEB%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#?$!+<8JfD\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#@$!+n\"\\sN*\"#A" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "path := NULL:\nfo r i from 1 to 10 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1] ,r[i+1]];\nend do:\nplot([g(x),x,[path]],x=-1..1,y=-1..1,color=[red,gr een,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'- %'CURVESG6$7S7$$!\"\"\"\"!$!15Q_4Q_f?!#:7$$!1nmm;p0k&*!#;$!1'f=rg#ou>F -7$$!1LL$3F-7$$!1nmmT%p\"e()F1$!1]'G$QUF>=F-7$$!1m mm\"4m(G$)F1$!1a65<`APxXpWqM\"F-7$$!1LLL$yaE\"eF1$!1[RpwG/n7F-7 $$!1nmm\">s%HaF1$!1=u[H^-(>\"F-7$$!1+++]$*4)*\\F1$!1Ma1(p.(=6F-7$$!1++ +]_&\\c%F1$!1vNw7cfS5F-7$$!1+++]1aZTF1$!11'[?%>Je'*F17$$!1nm;/#)[oPF1$ !1)eCGH#p$)*)F17$$!1MLL$=exJ$F1$!1%eP)[s%o=)F17$$!1MLLL2$f$HF1$!1]]/2j N;vF17$$!1++]PYx\"\\#F1$!19A3%)olTnF17$$!1MLLL7i)4#F1$!1kErc$>11'F17$$ !1++]P'psm\"F1$!1W*Rp)R\\=`F17$$!1++]74_c7F1$!1J$)fYkv;YF17$$!1JLL$3x% z#)!#<$!19FLncq*)QF17$$!1MLL3s$QM%Fer$!1UtHG%omA$F17$$!1^omm;zr)*!#>$! 12kGyVX;DF17$$\"1%y\"F17$$\"1!****\\PQ#\\\")Fer$! 1]FKf2G^6F17$$\"1KLLe\"*[H7F1$!13\"H@6jWs%Fer7$$\"1*******pvxl\"F1$\"1 @*pS5#*pB#Fer7$$\"1)****\\_qn2#F1$\"1wMS\"o-n**)Fer7$$\"1)***\\i&p@[#F 1$\"1'fZuhK*[:F17$$\"1)****\\2'HKHF1$\"1yi$oSEVE#F17$$\"1lmmmZvOLF1$\" 1=\"*onTb%[F17$$\"1lm;H!o-*\\F1$\"1c7L`%f8Y& F17$$\"1****\\7k.6aF1$\"1\"*=W\"4=,5'F17$$\"1mmm;WTAeF1$\"1%4HH.J(>nF1 7$$\"1****\\i!*3`iF1$\"12(e_jFKO(F17$$\"1MLLL*zym'F1$\"1\"4gyY\")z(zF1 7$$\"1LLL3N1#4(F1$\"1KgFk?d,')F17$$\"1mm;HYt7vF1$\"1W/a(y@\\@*F17$$\"1 *******p(G**yF1$\"1ChYS)oSx*F17$$\"1mmmT6KU$)F1$\"1*e+pRm4/\"F-7$$\"1L LLLbdQ()F1$\"15tw!oRt4\"F-7$$\"1++]i`1h\"*F1$\"1&>Y0?^p:\"F-7$$\"1++]P ?Wl&*F1$\"1fO>m$HN@\"F-7$$\"\"\"F*$\"1C&4Q_4QF\"F--%'COLOURG6&%$RGBG$ \"*++++\"!\")F*F*-F$6$7S7$F(F(7$F/F/7$F5F57$F:F:7$F?F?7$FDFD7$FIFI7$FN FN7$FSFS7$FXFX7$FgnFgn7$F\\oF\\o7$FaoFao7$FfoFfo7$F[pF[p7$F`pF`p7$FepF ep7$FjpFjp7$F_qF_q7$FdqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FirFir7$F^sF^s7$Fd sFds7$FisFis7$F^tF^t7$FctFct7$FhtFht7$F]uF]u7$FbuFbu7$FguFgu7$F\\vF\\v 7$FavFav7$FfvFfv7$F[wF[w7$F`wF`w7$FewFew7$FjwFjw7$F_xF_x7$FdxFdx7$FixF ix7$F^yF^y7$FcyFcy7$FhyFhy7$F]zF]z7$FbzFbz7$FgzFgz-F\\[l6&F^[lF*F_[lF* -F$6$7@7$$\"1+++++++SF1F\\_l7$F\\_l$\"1+++Q_4QRF17$F__lF__lFa_l7$F__l$ \"1+++>2%>%QF17$Fc_lFc_lFe_l7$Fc_l$\"1+++D,P#p$F17$Fg_lFg_lFi_l7$Fg_l$ \"1+++NX=fMF17$F[`lF[`lF]`l7$F[`l$\"1+++1`O%4$F17$F_`lF_`lFa`l7$F_`l$ \"1+++:%)[?DF17$Fc`lFc`lFe`l7$Fc`l$\"1+++c)e+h\"F17$Fg`lFg`lFi`l7$Fg`l $\"1+++'>()RY\"Fer7$F[alF[alF]al7$F[al$!1+++9$3jD#F17$F_alF_alFaal7$F_ al$!1******)QTKL'F17$FcalFcal-F\\[l6&F^[lF*F*F_[l-%+AXESLABELSG6$Q\"x6 \"Q\"yF\\bl-%%VIEWG6$;F(FgzFabl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "Graphical illustration of an iter ative solution . . Ex 2" }}{PARA 0 "" 0 "" {TEXT -1 23 "The iterative \+ formula " }{XPPEDIT 18 0 "r[i+1] = 21/5-3/r[i];" "6#/&%\"rG6#,&%\"iG \"\"\"F)F),&*&\"#@F)\"\"&!\"\"F)*&\"\"$F)&F%6#F(F.F." }{TEXT -1 19 " \+ gives a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"r G6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 58 ", which converg es to a solution of the quadratic equation " }{XPPEDIT 18 0 "5*x^2-21* x-15 = 0;" "6#/,(*&\"\"&\"\"\"*$%\"xG\"\"#F'F'*&\"#@F'F)F'!\"\"\"#:F- \"\"!" }{TEXT -1 50 ", provided that we take a suitable starting value " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Let's try taking " }{XPPEDIT 18 0 "r[1] = 2;" " 6#/&%\"rG6#\"\"\"\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "g := x -> 21/5-3/x;\nr[1 ] := 2;\nfor i from 1 to 18 do\n r[i+1] := evalf(g(r[i]));\nend do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,&#\"#@\"\"&\"\"\"*&\"\"$F09$!\"\"F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"rG6#\"\"#$\"+++++F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"r G6#\"\"$$\"+*))))))3$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6# \"\"%$\"+ypxGK!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$ \"+:c&3F$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+'z3 GG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+A%[hG$!\" *" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+Xr2(G$!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+!3NtG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$\"+\"p1uG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$\"+qlU(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#7$\"+)3KuG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#8$\"+?OV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$\"+XSV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$\"+jTV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$\"+'>MuG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$\"+0UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#=$\"+3UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#>$\"+3UV(G$!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 64 "The following graph illustrates the convergence of the \+ sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\" \"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "The graph shows the line " }{XPPEDIT 18 0 "y = x;" "6#/% \"yG%\"xG" }{TEXT -1 16 " and the curve " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "g(x) = \+ 21/5-3/x;" "6#/-%\"gG6#%\"xG,&*&\"#@\"\"\"\"\"&!\"\"F+*&\"\"$F+F'F-F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 122 "The points of inter section of these two graphs have x coordinates (and y coordinates) whi ch are solutions of the equation " }{XPPEDIT 18 0 "g(x) = x;" "6#/-%\" gG6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "The blue p ath shown connects the points" }{XPPEDIT 18 0 " ``(r[1],r[1]),``(r[1], r[2]),``(r[2],r[2]),``(r[2],r[3]),``(r[3],r[3]),` . . . `" "6(-%!G6$&% \"rG6#\"\"\"&F'6#F)-F$6$&F'6#F)&F'6#\"\"#-F$6$&F'6#F2&F'6#F2-F$6$&F'6# F2&F'6#\"\"$-F$6$&F'6#F?&F'6#F?%(~.~.~.~G" }{TEXT -1 12 ", and so on. " }}{PARA 0 "" 0 "" {TEXT -1 75 "This path provides a route towards th e right-hand point of intersection of " }{XPPEDIT 18 0 "y = x;" "6#/% \"yG%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-% \"gG6#%\"xG" }{TEXT -1 49 " and illustrates the convergence of the seq uence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\"\"&F $6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "path := NUL L:\nfor i from 1 to 5 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r [i+1],r[i+1]];\nend do:\nplot([g(x),x,[path]],x=0..4,y=0..4,color=[red ,green,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 " 6'-%'CURVESG6$7gn7$$\"1+++m;')=()!#<$!1![gl9;3-$!#97$$\"1+++Z8+$>*F*$! 1$y#ps?NVGF-7$$\"1*****z-Trm*F*$!1A#\\8*fH$o#F-7$$\"1+++r!GT,\"!#;$!1b &Q?D1#QDF-7$$\"1+++R?ah5F;$!1/gVKu21CF-7$$\"1+++w*pj:\"F;$!1X!Q&*RDV<# F-7$$\"1+++7z>^7F;$!19#**4@-x(>F-7$$\"1+++&y`3W\"F;$!1ec\"R%))4i;F-7$$ \"1+++e'40j\"F;$!1-a]/b\"*>9F-7$$\"1+++BvzV=F;$!1]B'eqwq?\"F-7$$\"1+++ )Q&3d?F;$!1R'Q]2u$Q5F-7$$\"1+++`KPqAF;$!1E,WAjo8!*!#:7$$\"1+++<6m$[#F; $!1'fx(>%F;$!1^g7dXjZHF`o7$$\"1+++\">K'*)\\F;$!1HgNAsY7=F `o7$$\"1+++Dt:5eF;$!1S6?APrL'*F;7$$\"1+++\"fX(emF;$!1&p`S'3``IF;7$$\"1 +++DCh/vF;$\"1h,z![%eC?F;7$$\"1+++L/pu$)F;$\"1nz)GP!yxhF;7$$\"1+++;c0T \"*F;$\"1u0;&GL5=*F;7$$\"1+++I,Q+5F`o$\"1s#om&*R6?\"F`o7$$\"1+++]*3q3 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UYoOF`o$\"1j1+%4>AQ$F`o7$$\"1+++2^rZPF`o$\"1aq(*eA^*R$F`o7$$\"1+++sI@K QF`o$\"1i2HED;(>%F;Fe_l7$$\"1mmm\">K'*)\\F;Fh_l7$$\"1***** \\Kd,\"eF;F[`l7$$\"1mmm\"fX(emF;F^`l7$$\"1*****\\U7Y](F;Fa`l7$$\"1MLLL /pu$)F;Fd`l7$$\"1nmm;c0T\"*F;Fg`l7$FdrFdr7$FirFir7$F^sF^s7$$\"1nm;fBIY 7F`oF]al7$$\"1LLLj$[kL\"F`oF`al7$$\"1LLL`Q\"GT\"F`oFcal7$$\"1++]s]k,:F `oFfal7$$\"1LLL`dF!e\"F`oFial7$$\"1++]sgam;F`oF\\bl7$$\"1++]F`oFebl7$$\"1nmmTc-)*>F`oFhbl7$ $\"1mm;f`@'3#F`oF[cl7$$\"1++]nZ)H;#F`oF^cl7$$\"1mmmJy*eC#F`oFacl7$FdwF dw7$FiwFiw7$$\"1++]7RV'\\#F`oFfcl7$FcxFcx7$$\"1LLL`4NnEF`oFjcl7$F]yF]y 7$$\"1mm;zM)>$GF`oF^dl7$FgyFgy7$$\"1LL$eg`!)*HF`oFbdl7$$\"1++]#G2A3$F` oFedl7$$\"1LLL$)G[kJF`oFhdl7$$\"1++]7yh]KF`oF[el7$$\"1nmm')fdLLF`oF^el 7$$\"1nmm,FT=MF`oFael7$$\"1LL$e#pa-NF`oFdel7$F_\\lF_\\l7$$\"1LLLGUYoOF `oFhel7$$\"1nmm1^rZPF`oF[fl7$$\"1++]sI@KQF`oF^fl7$$\"1++]2%)38RF`oFafl 7$Fh]lFh]l-F^^l6&F`^lFj]lFa^lFj]l-F$6$717$$\"\"#Fj]lFjfl7$Fjfl$\"1++++ +++FF`o7$F]glF]glF_gl7$F]gl$\"1+++*))))))3$F`o7$FaglFaglFcgl7$Fagl$\"1 +++ypxGKF`o7$FeglFeglFggl7$Fegl$\"1+++:c&3F$F`o7$FiglFiglF[hl7$Figl$\" 1+++'z3GG$F`o7$F]hlF]hl-F^^l6&F`^lFj]lFj]lFa^l-%+AXESLABELSG6$Q\"x6\"Q \"yFfhl-%%VIEWG6$;Fj]lFh]lF[il" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "We can zoom in close r to the point of intersection . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "path := NULL:\nfor i from 2 to 5 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\n end do:\nplot([g(x),x,[path]],x=3..3.4,y=3..3.4,color=[red,green,blue] );" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG 6$7S7$$\"\"$\"\"!$\"1+++++++K!#:7$$\"1mmmh)=(3IF-$\"1RO>ly*G?$F-7$$\"1 LLe'40j,$F-$\"1h,I_cS0KF-7$$\"1nm;6m$[-$F-$\"11hY$*3@3KF-7$$\"1mm;yYUL IF-$\"18R&Gz=5@$F-7$$\"1LLeF>(>/$F-$\"1fh;0wz8KF-7$$\"1nm\">K'*)\\IF-$ 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2oa&3H$F-7$$\"1++DG2A3LF-$\"14tyE\"oJH$F-7$$\"1LLL)G[kJ$F-$\"1W(*yEyT& H$F-7$$\"1++D\"yh]K$F-$\"1hnCD6w(H$F-7$$\"1nmm)fdLL$F-$\"12Bf^l++LF-7$ $\"1mm;q7%=M$F-$\"1fyo%H\"H-LF-7$$\"1LLe#pa-N$F-$\"1s_@\"pXXI$F-7$$\"1 +++ad)zN$F-$\"1dy)yG2mI$F-7$$\"1LL$GUYoO$F-$\"1N\"*Rm%e*3LF-7$$\"1nmm5 :xuLF-$\"1.hxG406LF-7$$\"1++D28A$Q$F-$\"14:*>8rKJ$F-7$$\"1++vS)38R$F-$ \"1ZQN(y&Q:LF-7$$\"1+++++++MF-$\"1IN#)eqkF>7$FCFC7$FHFH7$FMFM7$FR FR7$FWFW7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoFjo7$F_pF_p7$FdpFdp7$FipF ip7$F^qF^q7$FcqFcq7$FhqFhq7$F]rF]r7$FbrFbr7$FgrFgr7$F\\sF\\s7$FasFas7$ FfsFfs7$F[tF[t7$F`tF`t7$FetFet7$FjtFjt7$F_uF_u7$FduFdu7$FiuFiu7$F^vF^v 7$FcvFcv7$FhvFhv7$F]wF]w7$FbwFbw7$FgwFgw7$F\\xF\\x7$FaxFax7$FfxFfx7$F[ yF[y7$F`yF`y7$FeyFey7$FjyFjy7$F_zF_z7$FdzFdz-Fiz6&F[[lF*F\\[lF*-F$6$7. 7$$\"1+++++++FF-Fi^l7$Fi^l$\"1+++*))))))3$F-7$F\\_lF\\_lF^_l7$F\\_l$\" 1+++ypxGKF-7$F`_lF`_lFb_l7$F`_l$\"1+++:c&3F$F-7$Fd_lFd_lFf_l7$Fd_l$\"1 +++'z3GG$F-7$Fh_lFh_l-Fiz6&F[[lF*F*F\\[l-%+AXESLABELSG6$Q\"x6\"Q\"yFa` l-%%VIEWG6$;F($\"#M!\"\"Ff`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 " . . and closer . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "path := NULL:\nfor i from 7 to 10 do\n path := path,[r[i],r [i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do:\nplot([g(x),x,[path]],x=3 .287..3.2875,y=3.287..3.2875,color=[red,green,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"1++++++(G $!#:$\"1CA()fOJ(G$F*7$$\"1$3x&)*3,(G$F*$\"1@0-'o;tG$F*7$$\"1H2P\"Q?qG$ F*$\"1Y'R!>$>tG$F*7$$\"1eRwX5.(G$F*$\"1#)o7!GAtG$F*7$$\"13x%3yTqG$F*$ \"1@F#3EDtG$F*7$$\"1z%4\\Y_qG$F*$\"1.DLF#GtG$F*7$$\"1'R-/PiqG$F*$\"1$e \\w(4L(G$F*7$$\"1il'pisqG$F*$\"1ZhUDQL(G$F*7$$\"1'*)>VB$3(G$F*$\"1\\?e qnL(G$F*7$$\"17`l2Q4(G$F*$\"1#)QF1(RtG$F*7$$\"1jU6(G$F*$\"1*36dQXtG$F*7$$\"1]i^Z]7(G$F*$\"1mk#*z$[tG$F*7$ $\"1](=h(e8(G$F*$\"15mT'Q^tG$F*7$$\"1]P[6j9(G$F*$\"16#3PGatG$F*7$$\"1' *[z(ybrG$F*$\"181r9pN(G$F*7$$\"1;a/cq;(G$F*$\"1,s?V+O(G$F*7$$\"1n;t,m< (G$F*$\"1(RPMpitG$F*7$$\"1jSj0x=(G$F*$\"1%ylixltG$F*7$$\"1n\"pW`(>(G$F *$\"1'*Q20&otG$F*7$$\"1i!f#=$3sG$F*$\"18,**)\\rtG$F*7$$\"1(=xpe=sG$F*$ \"1$4)))\\VP(G$F*7$$\"1#H28IHsG$F*$\"1p?[CtP(G$F*7$$\"1zpSS\"RsG$F*$\" 1F32c+Q(G$F*7$$\"14_?`(\\sG$F*$\"1(**RC+$Q(G$F*7$$\"1'*)>pxgsG$F*$\"1e $pG1'Q(G$F*7$$\"1Qf4t.F(G$F*$\"1.a'ps)Q(G$F*7$$\"1e*Gst!G(G$F*$\"1XYC/ ;R(G$F*7$$\"1+DRW9H(G$F*$\"1\"fOnd%R(G$F*7$$\"1DJE>>I(G$F*$\"138s%[(R( G$F*7$$\"1j!RU07tG$F*$\"12%G$)H+uG$F*7$$\"1v=S2LK(G$F*$\"1e!RBU.uG$F*7 $$\"1n\"p)=ML(G$F*$\"1JLQHiS(G$F*7$$\"1](=]@WtG$F*$\"1#=$[E#4uG$F*7$$ \"1'*[$z*RN(G$F*$\"1-TGU>T(G$F*7$$\"1]iC$pktG$F*$\"1\\LN6\\T(G$F*7$$\" 1H2qcZP(G$F*$\"1&[)*\\q#[ZuG$F*$\"1BYI$*yV(G$F*7$$\"1U&G!e&euG$F*$\"1 5T$y'4W(G$F*7$$\"1M$)Qk%ouG$F*$\"1$Q7xrVuG$F*7$$\"1jSjE!zuG$F*$\"1:zj \\mW(G$F*7$$\"1Q40O\"*[(G$F*$\"14r%eX\\uG$F*7$$\"1+++++](G$F*$\"1!*G[r CX(G$F*-%'COLOURG6&%$RGBG$\"*++++\"!\")\"\"!F^[l-F$6$7S7$F(F(7$F.F.7$F 3F37$F8F87$F=F=7$FBFB7$FGFG7$FLFL7$FQFQ7$FVFV7$FenFen7$FjnFjn7$F_oF_o7 $FdoFdo7$FioFio7$F^pF^p7$FcpFcp7$FhpFhp7$F]qF]q7$FbqFbq7$FgqFgq7$F\\rF \\r7$FarFar7$FfrFfr7$F[sF[s7$F`sF`s7$FesFes7$FjsFjs7$F_tF_t7$FdtFdt7$F itFit7$F^uF^u7$FcuFcu7$FhuFhu7$F]vF]v7$FbvFbv7$FgvFgv7$F\\wF\\w7$FawFa w7$FfwFfw7$F[xF[x7$F`xF`x7$FexFex7$FjxFjx7$F_yF_y7$FdyFdy7$FiyFiy7$F^z F^z7$FczFcz-Fhz6&FjzF^[lF[[lF^[l-F$6$7.7$$\"1+++A%[hG$F*Fi^l7$Fi^l$\"1 +++Xr2(G$F*7$F\\_lF\\_lF^_l7$F\\_l$\"1+++!3NtG$F*7$F`_lF`_lFb_l7$F`_l$ \"1+++\"p1uG$F*7$Fd_lFd_lFf_l7$Fd_l$\"1+++qlU(G$F*7$Fh_lFh_l-Fhz6&FjzF ^[lF^[lF[[l-%+AXESLABELSG6$Q\"x6\"Q\"yFa`l-%%VIEWG6$;$\"%(G$!\"$$\"&vG $!\"%Ff`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The Maple procedure " }{TEXT 259 5 "solve" } {TEXT -1 80 " can be used to obtain analytical expressions for the sol utions of the equation " }{XPPEDIT 18 0 "g(x) = x;" "6#/-%\"gG6#%\"xGF '" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve(21/5-3/x=x,x);\nevalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$,&#\"#@\"#5\"\"\"*$-%%sqrtG6#\"$T\"F'#F'F&,&F$F' F(#!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+4UV(G$!\"*$\"*\"zlD \"*F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 57 "We can see from th ese graphs that the iterative formula " }{XPPEDIT 18 0 "r[i+1] = 21/5 -3/r[i];" "6#/&%\"rG6#,&%\"iG\"\"\"F)F),&*&\"#@F)\"\"&!\"\"F)*&\"\"$F) &F%6#F(F.F." }{TEXT -1 23 " will give a sequence " }{XPPEDIT 18 0 "r[ 1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~.~.~. ~G" }{TEXT -1 41 " , which converges to the larger solution" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "21/10+sqrt(141)/10;" "6#,&* &\"#@\"\"\"\"#5!\"\"F&*&-%%sqrtG6#\"$T\"F&F'F(F&" }{TEXT -1 2 " " } {TEXT 277 1 "~" }{TEXT -1 13 " 3.287434209 " }}{PARA 0 "" 0 "" {TEXT -1 26 "of the quadratic equation " }{XPPEDIT 18 0 "5*x^2-21*x-15 = 0; " "6#/,(*&\"\"&\"\"\"*$%\"xG\"\"#F'F'*&\"#@F'F)F'!\"\"\"#:F-\"\"!" } {TEXT -1 25 ", for any starting value " }{XPPEDIT 18 0 "r[1];" "6#&%\" rG6#\"\"\"" }{TEXT -1 44 " which is greater than the smaller solution \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "21/10-sqrt(141)/1 0;" "6#,&*&\"#@\"\"\"\"#5!\"\"F&*&-%%sqrtG6#\"$T\"F&F'F(F(" }{TEXT -1 2 " " }{TEXT 276 1 "~" }{TEXT -1 13 " 0.912565791." }}{PARA 0 "" 0 " " {TEXT -1 221 "On the other hand, if we take a starting value less th an the smaller solution, but greater than 0, the resulting sequence in itially diverges away from the smaller solution, but eventually conver ges to the larger solution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "As an example of this phenomenon we perform a f ew iterations starting with " }{XPPEDIT 18 0 "r[1] = 0;" "6#/&%\"rG6# \"\"\"\"\"!" }{TEXT -1 3 ".9." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "g := x -> 21/5-3/x;\nr[1] := 0.9;\nfor i from 1 to 24 do\n r[i+1] := evalf(g(r[i]));\nend do;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,&#\"#@\"\"&\"\"\"*&F0F09$!\"\"!\"$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"$\"\"*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"*nmmm)!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"*T:YQ(!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"*9+]P\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$!+gz\"=w\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+f'y-P%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+<\\a8N!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+y9;YL!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+@.X.L!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$\"++'e=H$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$\"+M2m)G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#7$\"+bXx(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#8$\"+w'GvG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$\"+J/Y(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$\"+)[TuG$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$\"+IiV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$\"+qZV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#=$\"+kVV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#>$\"+_UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#?$\"+@UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#@$\"+7UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#A$\"+5UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#B$\"+4UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#C$\"+4UV(G$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#D$\"+4UV(G$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 173 "If you execute the following c omands you will see that the path for the sequence of iterates tempora rily switches over to the left-hand branch of the rectangular hyperbol a " }{XPPEDIT 18 0 "y = 21/5-3/x;" "6#/%\"yG,&*&\"#@\"\"\"\"\"&!\"\"F (*&\"\"$F(%\"xGF*F*" }{TEXT -1 105 ", and then comes back to the right -hand branch before converging to the larger solution of the quadratic ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 225 "g := x -> 21/5-3/x;\nr[1] := 0.9;\npath := NULL:\nfo r i from 1 to 10 do\n r[i+1] := evalf(g(r[i]));\npath := path,[r[i], r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do:\nplot([g(x),x,[path]],x= -20..8,y=-20..8,color=[red,green,blue]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Oscillating values from an iterative formula" } }{PARA 0 "" 0 "" {TEXT -1 32 "Consider the iterative formula " } {XPPEDIT 18 0 "r[i+1] = 8*r[i]/3-r[i]^2;" "6#/&%\"rG6#,&%\"iG\"\"\"F)F ),&*(\"\")F)&F%6#F(F)\"\"$!\"\"F)*$&F%6#F(\"\"#F0" }{TEXT -1 2 " ." }} {PARA 0 "" 0 "" {TEXT -1 51 "We might expect that this formula gives a sequence " }{XPPEDIT 18 0 "r[1],r[2],r[3],` . . . `;" "6&&%\"rG6#\"\" \"&F$6#\"\"#&F$6#\"\"$%(~.~.~.~G" }{TEXT -1 32 ", which converges to a solution " }{XPPEDIT 18 0 "x = 5/3;" "6#/%\"xG*&\"\"&\"\"\"\"\"$!\"\" " }{TEXT -1 27 " of the quadratic equation " }{XPPEDIT 18 0 "x^2-5/3*x = 0;" "6#/,&*$%\"xG\"\"#\"\"\"*(\"\"&F(\"\"$!\"\"F&F(F,\"\"!" }{TEXT -1 50 ", provided that we take a suitable starting value " }{XPPEDIT 18 0 "r[1];" "6#&%\"rG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "Let's try taking " }{XPPEDIT 18 0 "r[1] = 1/2;" "6#/&%\"r G6#\"\"\"*&F'F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "g := x -> 8/3*x-x^2; \nr[1] := 0.5;\nfor i from 1 to 18 do\n r[i+1] := evalf(g(r[i]));\ne nd do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operat orG%&arrowGF(,&9$#\"\")\"\"$*$)F-\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"\"$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"+LLL$3\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"+yxF:&%\"rG6#\"\"%$\"+>i*=j\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+\")zj)o\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+&QO:l\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+'fCln\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+Nv**f;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+x#o5n\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$\"+')Grj;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$\"+Hrio;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#7$\"+[eNl;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#8$\"+G)Qvm\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$\"+lW3m;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$\"+iW0n;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$\"+')zSm;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$\"+`!Rom\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#=$\"+7&%\"rG6#\"#>$\"+!HVnm\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "The values seem to oscillate ei ther side of " }{XPPEDIT 18 0 "5/3 = 1.66666;" "6#/*&\"\"&\"\"\"\"\"$ !\"\"-%&FloatG6$\"'mm;!\"&" }{TEXT -1 6 " . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "path := NUL L:\nfor i from 1 to 18 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[ r[i+1],r[i+1]];\nend do:\nplot([g(x),x,[path]],x=0..2.8,y=0..2,color=[ red,green,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$\"\"!F(7$$\"1mmmmJ?.h!#<$\"1M3bbjt!RL%F/7$$\"1nm mrusRBF/$\"141h$zS=p&F/7$$\"1LL$3$\\.QHF/$\"1se2!*[brpF/7$$\"1mm;MDu# \\$F/$\"16M6]]0%4)F/7$$\"1****\\F,6nSF/$\"1e`uF&)[\">*F/7$$\"1mm;9>7hY F/$\"1I0zo_qD5!#:7$$\"1++](pGKD&F/$\"1Cw^oo*[7\"FS7$$\"1LLL.LGieF/$\"1 [,F#*=h>7FS7$$\"1nmmJ*Q()R'F/$\"16:i#y\"*oH\"FS7$$\"1******44m-qF/$\"1 YAYWO+x8FS7$$\"1******\\E14wF/$\"1$3<#H]5]9FS7$$\"1+++!4VM>)F/$\"1YEkg If8:FS7$$\"1lm;9l6C()F/$\"1*)o\"4)*G`c\"FS7$$\"1KLLV&Q^N*F/$\"1Yf!*Ru^ >;FS7$$\"1LLLtpp*))*F/$\"1P'yRY\">f;FS7$$\"1++v]::^5FS$\"1qeN>7:)p\"FS 7$$\"1LLLFI>16FS$\"1/!o\"f]=EFS$\"1#[?x-?YU\"FS7$$\"1 mmTN%)Q#)>FS$\"1stB__]c8FS7$$\"1+++z@GU?FS$\"1zc+7$p^F\"FS7$$\"1LL3Cvj )4#FS$\"1m#)Rts3#>\"FS7$$\"1++v(4Xv:#FS$\"1V\\+5FS7$$\"1++voCVvAFS$\"1s%\\#o*pA!*)F/7$$\"1nmm!>.NL#FS $\"1qffU.QuxF/7$$\"1mm;\"*)))GR#FS$\"1AFUwz>^lF/7$$\"1LL3[Gy^CFS$\"1xE L2Y[o_F/7$$\"1+++y-!f]#FS$\"1d&G0QX'GSF/7$$\"1LL$)f\\#zc#FS$\"1;'Qb$Hh NDF/7$$\"1mmmu0SBEFS$\"1:Q;1H/N6F/7$$\"1++v]\"\\Do#FS$!1wY&*=WbgUF,7$$ \"1++D&)=;RFFS$!1PL1'Rhd)>F/7$$\"1+++++++GFS$!1JLLLLLLPF/-%'COLOURG6&% $RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$F*F*7$F1F17$F6F67$F;F;7$F@F@7$FEFE7$ FJFJ7$FOFO7$FUFU7$FZFZ7$FinFin7$F^oF^o7$FcoFco7$FhoFho7$F]pF]p7$FbpFbp 7$FgpFgp7$F\\qF\\q7$FaqFaq7$FfqFfq7$F[rF[r7$F`rF`r7$FerFer7$FjrFjr7$F_ sF_s7$FdsFds7$FisFis7$F^tF^t7$FctFct7$FhtFht7$F]uF]u7$FbuFbu7$FguFgu7$ F\\vF\\v7$FavFav7$FfvFfv7$F[wF[w7$F`wF`w7$FewFew7$FjwFjw7$F_xF_x7$FdxF dx7$FixFix7$F^yF^y7$FcyFcy7$FhyFhy7$F]zF]z7$FbzFbz-Fgz6&FizF(FjzF(-F$6 $7X7$$\"1+++++++]F/Ff^l7$Ff^l$\"1+++LLL$3\"FS7$Fi^lFi^lF[_l7$Fi^l$\"1+ ++yxF:i*=j\"FS7$Fa_lFa_lFc_l7$Fa_l$\"1+ ++\")zj)o\"FS7$Fe_lFe_lFg_l7$Fe_l$\"1+++&QO:l\"FS7$Fi_lFi_lF[`l7$Fi_l$ \"1+++'fCln\"FS7$F]`lF]`lF_`l7$F]`l$\"1+++Nv**f;FS7$Fa`lFa`lFc`l7$Fa`l $\"1+++x#o5n\"FS7$Fe`lFe`lFg`l7$Fe`l$\"1+++')Grj;FS7$Fi`lFi`lF[al7$Fi` l$\"1+++Hrio;FS7$F]alF]alF_al7$F]al$\"1+++[eNl;FS7$FaalFaalFcal7$Faal$ \"1+++G)Qvm\"FS7$FealFealFgal7$Feal$\"1+++lW3m;FS7$FialFialF[bl7$Fial$ \"1+++iW0n;FS7$F]blF]blF_bl7$F]bl$\"1+++')zSm;FS7$FablFablFcbl7$Fabl$ \"1+++`!Rom\"FS7$FeblFeblFgbl7$Febl$\"1+++7 " 0 "" {MPLTEXT 1 0 172 "path := NULL:\nfor i from 1 to 18 \+ do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]];\nend do :\nplot([g(x),x,[path]],x=1.6..1.75,y=1.6..1.75,color=[red,green,blue] );" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG 6$7S7$$\"1+++++++;!#:$\"1mmmmmm1)))f7;q\"F*7 $$\"1+DJaU`7;F*$\"1j@]BY#)*p\"F*7$$\"1]P%GZRdh\"F*$\"1,Y9]X-)p\"F*7$$ \"1](=276(=;F*$\"1#yWiHPjp\"F*7$$\"1](o**3)y@;F*$\"1!=_6jrXp\"F*7$$\"1 ](ofHq\\i\"F*$\"1mD!)fcs#p\"F*7$$\"1]Pf'HU\"G;F*$\"1'4fHXl3p\"F*7$$\"1 +]7*309j\"F*$\"1^q<15$*)o\"F*7$$\"1+Dce*yUj\"F*$\"1/KV3&4so\"F*7$$\"1+ ]([D9vj\"F*$\"1RyuWI8o\"F*7$$\"1](o%QjtY;F*$\"1n)\\TLc&z;F*7$$\"1+]i8o6 ];F*$\"1x?R/gUx;F*7$$\"1++]>0)Hl\"F*$\"15N:aMgv;F*7$$\"1](=-p6jl\"F*$ \"1u8weFYt;F*7$$\"1++vS.Ef;F*$\"1$>9OO\\:n\"F*7$$\"1](=xZ&\\i;F*$\"1s8 [g+Vp;F*7$$\"1]i:$4wbm\"F*$\"1HIKEDRn;F*7$$\"1+v=#R!zo;F*$\"1*oU%RjCl; F*7$$\"1]P4A@ur;F*$\"1&[n%psDj;F*7$$\"1+Dchf#\\n\"F*$\"1os6aA4h;F*7$$ \"1](of2L#y;F*$\"1,yIX>#)e;F*7$$\"1]7yG>6\"o\"F*$\"1_gKEy#ol\"F*7$$\"1 +voo6A%o\"F*$\"1X#yZ]bYl\"F*7$$\"1++v1DkXjj\"F*7$$\"1](=-,FCr\"F*$\"1tK#>wlSj\"F*7$$\"1]P4tFe:(yOC;F*7$$\"1+DJw/>G;F*7$$\"1++vdYCMh\"F*7$$\"1](= -*zqVi*=j\"F*7$Fe_lFe_lFg_l7$Fe_l$\"1+++\")zj)o\"F*7$Fi_lFi_lF[` l7$Fi_l$\"1+++&QO:l\"F*7$F]`lF]`lF_`l7$F]`l$\"1+++'fCln\"F*7$Fa`lFa`lF c`l7$Fa`l$\"1+++Nv**f;F*7$Fe`lFe`lFg`l7$Fe`l$\"1+++x#o5n\"F*7$Fi`lFi`l F[al7$Fi`l$\"1+++')Grj;F*7$F]alF]alF_al7$F]al$\"1+++Hrio;F*7$FaalFaalF cal7$Faal$\"1+++[eNl;F*7$FealFealFgal7$Feal$\"1+++G)Qvm\"F*7$FialFialF [bl7$Fial$\"1+++lW3m;F*7$F]blF]blF_bl7$F]bl$\"1+++iW0n;F*7$FablFablFcb l7$Fabl$\"1+++')zSm;F*7$FeblFeblFgbl7$Febl$\"1+++`!Rom\"F*7$FiblFiblF[ cl7$Fibl$\"1+++7 " 0 "" {MPLTEXT 1 0 90 "g := x -> 3*x-x^2;\nr[1] := 0.5;\nfor i from 1 to 16 \+ do\n r[i+1] := evalf(g(r[i]));\nend do;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"$*$)F- \"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\" \"\"$\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"#$\"$D \"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"$$\"&v=#!\"%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"%$\"*vVtx\"!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"&$\"+W/3t@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"'$\"+;F'pz\"!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"($\"+XJ\"=;#!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"rG6#\"\")$\"+hL+7=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"\"*$\"+.Rl_@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#5$\"+$)G/C=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#6$\"+5i*\\9#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#7$\"+*))zR$=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#8$\"+MuXQ@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#9$\"+0@PU=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#:$\"+U8yK@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#;$\"+txe\\=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"rG6#\"#<$\"+*Q)yF@!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "The values given by the sequenc e again oscillate above and below the solution " }{XPPEDIT 18 0 "x = 2 ;" "6#/%\"xG\"\"#" }{TEXT -1 51 ", but this time the convergence is in credibly slow." }}{PARA 0 "" 0 "" {TEXT -1 51 "If we look at a graph, \+ we can see why this happens." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "path := NULL:\nfor i from 1 to 18 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]]; \nend do:\nplot([g(x),x,[path]],x=0..3,y=0..3,color=[red,green,blue]); " }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$ 7S7$\"\"!F(7$$\"1+++]i9Rl!#<$\"166KTM)*=>!#;7$$\"1++vVA)GA\"F/$\"1(4UI j-\">NF/7$$\"1++]Peui=F/$\"1Vw)pIb7C&F/7$$\"1++]i3&o]#F/$\"1,$=GYA@*oF /7$$\"1++voX*y9$F/$\"1CMl%ofFX)F/7$$\"1++vVTAUPF/$\"1eM/x#[i#)*F/7$$\" 1++v$*zhdVF/$\"1i/L_qR<6!#:7$$\"1++v$>fS*\\F/$\"16#=e[6)[7FN7$$\"1++v= $f%GcF/$\"1jnf_Aur8FN7$$\"1+++Dy,\"G'F/$\"1B.K)\\$z*[\"FN7$$\"1++]7FN7$$\"1++]siL-5FN$\"113L93L-?FN7$$\"1+++!R5'f5FN$\"1[SS \"*p0c?FN7$$\"1+]P/QBE6FN$\"19!35$))H5@FN7$$\"1+++:o?&=\"FN$\"1cvn]_!4 :#FN7$$\"1+]Pa&4*\\7FN$\"1*32!pZX(=#FN7$$\"1+]7j=_68FN$\"1H,#>*fZ9AFN7 $$\"1++vVy!eP\"FN$\"1wfK3jdMAFN7$$\"1+](=WU[V\"FN$\"1(=>E\\adC#FN7$$\" 1++DJ#>&)\\\"FN$\"14RK2y**\\AFN7$$\"1+]P>:mk:FN$\"1<@\"z))=eC#FN7$$\"1 +]iv&QAi\"FN$\"1s\"H1td]B#FN7$$\"1++vtLU%o\"FN$\"1nY@>!))f@#FN7$$\"1++ +bjm[W!eH:#FN7$$\"1+]PMaKs= FN$\"1ou\"4xt86#FN7$$\"1++D6W%)R>FN$\"1T@*Q*o`c?FN7$$\"1+++:K^+?FN$\"1 k.h@l[**>FN7$$\"1++]7,Hl?FN$\"1P(4()3Z/$>FN7$$\"1+]P4w)R7#FN$\"1skMj%R 1'=FN7$$\"1++]x%f\")=#FN$\"1Fp_LlVw=+DFN$\"1S'zo)fj\\7FN7$$\"1++D E&4Qc#FN$\"1^(f=H4$=6FN7$$\"1+]P%>5pi#FN$\"1(*p8QTt+)*F/7$$\"1+++bJ*[o #FN$\"1W9M7@Gg%)F/7$$\"1++Dr\"[8v#FN$\"1#z#4VvFToF/7$$\"1+++Ijy5GFN$\" 1>8&3(>R=`F/7$$\"1+]P/)fT(GFN$\"1.1S?$[oh$F/7$$\"1+]i0j\"[$HFN$\"1ki7 \"p@I\">F/7$$\"\"$F(F(-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6$7SF'7$F *F*7$F1F17$F6F67$F;F;7$F@F@7$FEFE7$FJFJ7$FPFP7$FUFU7$FZFZ7$FinFin7$F^o F^o7$FcoFco7$FhoFho7$F]pF]p7$FbpFbp7$FgpFgp7$F\\qF\\q7$FaqFaq7$FfqFfq7 $F[rF[r7$F`rF`r7$FerFer7$FjrFjr7$F_sF_s7$FdsFds7$FisFis7$F^tF^t7$FctFc t7$FhtFht7$F]uF]u7$FbuFbu7$FguFgu7$F\\vF\\v7$FavFav7$FfvFfv7$F[wF[w7$F `wF`w7$FewFew7$FjwFjw7$F_xF_x7$FdxFdx7$FixFix7$F^yF^y7$FcyFcy7$FhyFhy7 $F]zF]z7$FbzFbz-Fez6&FgzF(FhzF(-F$6$7X7$$\"1+++++++]F/Fd^l7$Fd^l$\"1++ ++++]7FN7$Fg^lFg^lFi^l7$Fg^l$\"1+++++](=#FN7$F[_lF[_lF]_l7$F[_l$\"1+++ ]PMx " 0 "" {MPLTEXT 1 0 261 "g := x -> 3*x-x^2;\nr[1] := 0.5;\nfor i from 1 to 30 do\n r[i+1] := evalf(g(r[i]));\nend do:\npath := NULL:\nfor i from \+ 4 to 32 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r[i+1],r[i+1]]; \nend do:\nplot([g(x),x,[path]],x=1.8..2.2,y=1.8..2.2,color=[red,green ,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(,&9$\"\"$*$)F-\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>&%\"rG6#\"\"\"$\"\"&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"1+++++++=!#:$ \"1++++++g@F*7$$\"1nmmh)=(3=F*$\"1P7XkEpa@F*7$$\"1LLe'40j\"=F*$\"13a(e 3^*\\@F*7$$\"1nm;6m$[#=F*$\"1cd/w6[W@F*7$$\"1nm;yYUL=F*$\"1b%*)R)z#)Q@ F*7$$\"1LLeF>(>%=F*$\"1R\\u+_0L@F*7$$\"1nm\">K'*)\\=F*$\"1)=4Rcsv7#F*7 $$\"1++Dt:5e=F*$\"1veBjKw@@F*7$$\"1nm\"fX(em=F*$\"1YX;POh:@F*7$$\"1++D Ch/v=F*$\"1a]o//M4@F*7$$\"1LLL/pu$)=F*$\"1$e9MJQF5#F*7$$\"1nm;c0T\"*=F *$\"1+8_wxz'4#F*7$$\"1+++8!Q+!>F*$\"17]:)ep**3#F*7$$\"1+++&*3q3>F*$\"1 *>E%yN'H3#F*7$$\"1+++(=\\q\">F*$\"1REiv*pg2#F*7$$\"1nm\"fBIY#>F*$\"1d` ui\"*op?F*7$$\"1LLLO[kL>F*$\"1O9#f:_>1#F*7$$\"1LLL&Q\"GT>F*$\"1%He*Q2F b?F*7$$\"1++D2X;]>F*$\"1lBT;>NZ?F*7$$\"1LLLvv-e>F*$\"1oUBS2@S?F*7$$\"1 ++D2Ylm>F*$\"1\\B`xMBK?F*7$$\"1++v\"ep[(>F*$\"1!eG.)))\\C?F*7$$\"1ML$e /TM)>F*$\"1tN?bZG;?F*7$$\"1LLeDBJ\"*>F*$\"1(*)H()>7'3?F*7$$\"1nm;kD!)* *>F*$\"1DC&oR(>+?F*7$$\"1nm\"f`@'3?F*$\"1qF*7$$\"1++vw%)H;?F* $\"1Ic!H)eV$)>F*7$$\"1nm;$y*eC?F*$\"1]rRfb![(>F*7$$\"1+++9b:L?F*$\"1&R \"*y>Xd'>F*7$$\"1++]5a`T?F*$\"1'fu\"*RRn&>F*7$$\"1++D\"RV'\\?F*$\"1*>b C9#*y%>F*7$$\"1++]@fke?F*$\"1:9RMZ\"z$>F*7$$\"1MLL&4Nn1#F*$\"1F*7$$\"1+++:?Pv?F*$\"1S[NWq%*=>F*7$$\"1nm\"zM)>$3#F*$\"1qs)poz)4>F*7 $$\"1+++(fa<4#F*$\"1o$Hp\\E)**=F*7$$\"1LLeg`!)*4#F*$\"1(\\:%HNB!*=F*7$ $\"1++DG2A3@F*$\"1SqsXv1!)=F*7$$\"1LLL)G[k6#F*$\"1.4\"y]\"**p=F*7$$\"1 ++D\"yh]7#F*$\"1x0ipxHf=F*7$$\"1nmm)fdL8#F*$\"1%>6A:e)[=F*7$$\"1nm;q7% =9#F*$\"1S3\"RyR!Q=F*7$$\"1MLe#pa-:#F*$\"1bLxM)or#=F*7$$\"1+++ad)z:#F* $\"1^I`ZZ0<=F*7$$\"1ML$GUYo;#F*$\"1'Qa$[eJ0=F*7$$\"1nmm5:xu@F*$\"1CER3 Mo%z\"F*7$$\"1++D28A$=#F*$\"1,'>`k3Ky\"F*7$$\"1++vS)38>#F*$\"1\"*3q'3# 4s@F^_l7$F]clF]clF_cl7$F]cl$\"+U\"*Qm=F^_l7$FaclFaclFcc l7$Facl$\"+(**ed6#F^_l7$FeclFeclFgcl7$Fecl$\"+e3%3(=F^_l7$FiclFiclF[dl 7$Ficl$\"+eqZ7@F^_l7$F]dlF]dlF_dl7$F]dl$\"+`=([(=F^_l7$FadlFadlFcdl7$F adl$\"+%4r%4@F^_l7$FedlFedlFgdl7$Fedl$\"+')\\ay=F^_l7$FidlFidlF[el7$Fi dl$\"+%p.n5#F^_l7$F]elF]elF_el7$F]el$\"+G1\">)=F^_l7$FaelFaelFcel7$Fae l$\"+sU9/@F^_l7$FeelFeelFgel7$%%FAILGFiel-Fhz6&FjzF^[lF^[lF[[l-%+AXESL ABELSG6$Q\"x6\"Q\"yF`fl-%%VIEWG6$;$\"#=!\"\"$\"#AFhflFefl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " . . and another . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 299 "g := x -> 3*x-x^2;\nr[1] := 0.5;\nfor i \+ from 1 to 50 do\n r[i+1] := evalf(g(r[i]));\nend do:\npath := NULL: \nfor i from 32 to 50 do\n path := path,[r[i],r[i]],[r[i],r[i+1]],[r [i+1],r[i+1]];\nend do:\nplot([g(x),x,[path]],x=1.884..2.116,y=1.884.. 2.116,\n color=[red,green,blue]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9 $\"\"$*$)F-\"\"#\"\"\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"rG6#\"\"\"$\"\"&!\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 360 270 270 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"1++++++%)=!#:$\"1++++Sa-@F*7$$\"1nmw Rp0*)=F*$\"1(R>wpM')4#F*7$$\"1L$=g&pX$*=F*$\"1(ygPi\">&4#F*7$$\"1nmZM_ S)*=F*$\"1#fT\"pKF\"4#F*7$$\"1nmL8jQ.>F*$\"1`E@#[zs3#F*7$$\"1L$)*zrV$3 >F*$\"1)HrzSbK3#F*7$$\"1n;rm)RH\">F*$\"1^,[m1[z?F*7$$\"1+][7*)p<>F*$\" 1[)3&=w_v?F*7$$\"1n;VC2iA>F*$\"1`3GBF*$\"1_(oi(3An ?F*7$$\"1LL^/KdK>F*$\"1mqtA/)G1#F*7$$\"1nmdA\"=q$>F*$\"1l&R0;:!f?F*7$$ 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+u0r$4#F*7$F\\alF\\alF^al7$F\\al$\"1+++ax](*=F*7$F`alF`alFbal7$F`al$\" 1+++'e()>4#F*7$FdalFdalFfal7$Fdal$\"1+++)p]&**=F*7$FhalFhalFjal7$Fhal$ \"1+++S#f.4#F*7$F\\blF\\blF^bl7$F\\bl$\"1+++ofZ,>F*7$F`blF`blFbbl7$F`b l$\"1+++Zq\"))3#F*7$FdblFdblFfbl7$Fdbl$\"1+++&[%H.>F*7$FhblFhblFjbl7$F hbl$\"1+++eNN(3#F*7$F\\clF\\clF^cl7$F\\cl$\"1+++)z:]!>F*7$F`clF`clFbcl 7$F`cl$\"1+++.A'f3#F*7$FdclFdcl-Fhz6&FjzF^[lF^[lF[[l-%+AXESLABELSG6$Q \"x6\"Q\"yF]dl-%%VIEWG6$;$\"%%)=!\"$$\"%;@FedlFbdl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 90 "Calculate the square root of 5 correct to 30 decimal places, by using an iterative method." }}{PARA 0 "" 0 "" {TEXT -1 88 "You can simply i terate an appropriate function yourself, as in the first calculation o f " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 82 " above , or construct an appropriate loop to do the job. You may use the proc edure " }{TEXT 259 10 "fixedpoint" }{TEXT -1 5 " . . " }{HYPERLNK 17 " fixedpoint" 1 "" "fixedpoint" }{TEXT -1 14 " if you wish." }}{PARA 0 "" 0 "" {TEXT -1 39 " ______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "____________ __________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 31 "(a) Use the iterative formula " }{XPPEDIT 18 0 "r[i+1] = 2*r[i]/3+4/(r[i]^2);" "6#/&%\"rG6#,&%\"iG\"\"\"F)F),&*(\"\"#F)&F%6#F( F)\"\"$!\"\"F)*&\"\"%F)*$&F%6#F(F,F0F)" }{TEXT -1 17 " , starting with " }{XPPEDIT 18 0 "r[1] = 2;" "6#/&%\"rG6#\"\"\"\"\"#" }{TEXT -1 10 ", to find " }{XPPEDIT 18 0 "r[2],r[3],r[4],r[5],r[6];" "6'&%\"rG6#\"\"# &F$6#\"\"$&F$6#\"\"%&F$6#\"\"&&F$6#\"\"'" }{TEXT -1 63 ", giving your \+ answers to10 digits. You can apply the function " }{XPPEDIT 18 0 "g(x ) = 2*x/3+4/(x^2);" "6#/-%\"gG6#%\"xG,&*(\"\"#\"\"\"F'F+\"\"$!\"\"F+*& \"\"%F+*$F'F*F-F+" }{TEXT -1 28 " repeatedly to achieve this." }} {PARA 0 "" 0 "" {TEXT -1 26 "You may use the procedure " }{TEXT 259 10 "fixedpoint" }{TEXT -1 5 " . . " }{HYPERLNK 17 "fixedpoint" 1 "" "f ixedpoint" }{TEXT -1 24 " by setting the option \"" }{TEXT 278 13 "max iterations" }{TEXT -1 12 "\" to 5 and \"" }{TEXT 278 4 "info" }{TEXT -1 6 "\" to \"" }{TEXT 278 4 "true" }{TEXT -1 2 "\"." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "(b) Which cubic equatio n does part (a) find a solution for? Check your answer by using " } {TEXT 259 4 "subs" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "____ __________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "____________________ __________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 19 "The cubic equation " }{XPPEDIT 18 0 "x^3-10*x+1 = 0;" "6#/,(*$%\"xG\" \"$\"\"\"*&\"#5F(F&F(!\"\"F(F(\"\"!" }{TEXT -1 31 " can be rearranged \+ in the form " }{XPPEDIT 18 0 "x = (x^3+1)/10;" "6#/%\"xG*&,&*$F$\"\"$ \"\"\"F)F)F)\"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 134 "Use this rearrangement to form an iterative formula and use it to fin d the solution of this cubic equation which lies between 0 and 1." }} {PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "__ ____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {PARA 0 "" 0 "" {TEXT -1 22 "(a) Use the procedure " }{TEXT 259 10 "fi xedpoint" }{TEXT -1 5 " . . " }{HYPERLNK 17 "fixedpoint" 1 "" "fixedpo int" }{TEXT -1 36 " to find a solution of the equation " }{XPPEDIT 18 0 "cos(x) = x;" "6#/-%$cosG6#%\"xGF'" }{TEXT -1 54 " correct to10 digi ts. You may need to use the option \"" }{TEXT 278 13 "maxiterations" } {TEXT -1 49 "\" to increase the number of iterations performed." }} {PARA 0 "" 0 "" {TEXT -1 65 "You can find a suitable starting approxim ation for the procedure " }{TEXT 0 10 "fixedpoint" }{TEXT -1 24 " by p lotting the graphs " }{XPPEDIT 18 0 "y = cos(x);" "6#/%\"yG-%$cosG6#% \"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" } {TEXT -1 21 " in the same diagram." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "(b) Why is the convergence so slow in (a) ?" }}{PARA 0 "" 0 "" {TEXT -1 83 " You could give a graphical illustra tion of the convergence to explain your answer." }}{PARA 0 "" 0 "" {TEXT -1 38 "______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 38 "__ ____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "Code for 1st picture " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 281 "PLOT(CURVES([[0.1,-2.1],[0,-2],[0.1,-1.9]],\n[[1.4,-0.1],[1.5 ,0],[1.4,0.1]],[[-1.6,-0.1],[-1.5,0],[-1.6,0.1]]),\nPOLYGONS([[1,1],[1 ,-1],[-1,-1],[-1,1]],COLOR(RGB,0.7,0.7,0.9)),\nCURVES([[1,0],[2,0],[2, -2],[-2,-2],[-2,0],[-1,0]]),\nTEXT([0,0],`f`),AXESSTYLE(NONE),SCALING( CONSTRAINED));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 20 "Cod e for 2nd picture" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "PLOT(CURVES([[1.4,-0.1],[1.5,0],[1.4,0.1]], \n[[-1.6,-0.1],[-1.5,0],[-1.6,0.1]]),\nPOLYGONS([[1,1],[1,-1],[-1,-1], [-1,1]],COLOR(RGB,0.7,0.7,0.9)),\nCURVES([[1,0],[2,0]],[[-2,0],[-1,0]] ),TEXT([-2,0.3],'a'),\nTEXT([0,0],`f`),TEXT([2,0.35],`f(a) = a`),\nAXE SSTYLE(NONE),SCALING(CONSTRAINED));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 \+ 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }