{VERSION 6 1 "Windows XP" "6.1" } {USTYLETAB {PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 3 }{PSTYLE "Wa rning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 2 }{PSTYLE "Normal258" -1 200 1 {CSTYLE "" -1 -1 "Tim es" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 } {PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 5 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 0 0 0 2 2 1 0 0 0 1 }1 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }{PSTYLE "R3 Font 2" -1 201 1 {CSTYLE "" -1 -1 "Couri er" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 0 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "R3 Font 0" -1 202 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 255 1 2 1 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "MS Serif" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle259" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "LaTe X" -1 32 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Maple Comment" -1 21 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold " -1 5 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlin ed" -1 44 "Times" 1 12 0 0 0 0 0 0 1 2 2 2 0 0 0 1 }{CSTYLE "Output La bels" -1 29 "Times" 1 8 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Maple In put" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Page N umber" -1 33 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Maple I nput Placeholder" -1 200 "Courier" 1 12 200 0 200 1 0 1 0 2 1 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "2D Math Italic Small" -1 202 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 0 1 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 0 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "" 1 10 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 0 1 255 0 0 1 0 0 2 2 1 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 "" 0 1 104 64 92 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Italic" -1 42 "Times" 1 12 0 0 0 0 1 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Bold" -1 41 "Times" 1 12 0 0 0 0 0 1 1 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "" 0 1 147 0 15 1 2 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 0 1 0 0 0 0 1 0 2 2 2 2 0 0 0 1 }{CSTYLE "Popup" -1 31 "" 0 1 0 128 128 1 1 0 1 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 0 1 0 0 0 0 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Text" -1 28 "" 1 8 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 0 0 1 0 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "" 0 1 0 0 0 1 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 } {CSTYLE "Help Variable" -1 25 "Courier" 0 1 0 0 0 1 2 2 0 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 0 1 0 0 0 0 0 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 203 "" 0 1 0 0 0 0 1 2 0 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 0 1 0 0 0 1 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle1" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle1 " -1 204 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle2" -1 205 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle2" -1 205 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{PSTYLE "_pstyle3" -1 206 1 {CSTYLE "" -1 -1 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 1 0 0 1 }3 3 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle3" -1 206 "Times" 0 1 0 0 255 1 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle4" -1 207 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 0 0 0 1 }{PSTYLE "_pstyle4" -1 207 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyl e5" -1 208 "Times" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }{PSTYLE "_pstyle5" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "_cstyle6" -1 209 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle6" -1 209 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 204 "" 0 "" {TEXT 204 23 "# Iteration of mappings " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 4 "Iter" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 106 "Iter is a package of simple tools for studying the sequences a, f (a), f(f(a)), f(f(f(a))), ... obtained by" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 101 "repeatedly applying a function f to an init ial value a. To iterate something is to do it repeatedly. " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 5 "iter " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 95 " U se iter( f, n, a ) to compute the sequence which begins with a, f(a), \+ f(f(a)) ... and has" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 27 " a total of n+1 terms. " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 10 "iterprint " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 97 " Use iterprint( f, n , \+ a ) to apply f to a n times and print the results in vertical format. \+ " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 9 "iterplot " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 93 " Use iterplot( f, n, a ) instead of iterprint to plot the \+ sequence a, f(a), f(f(a)), ... " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 10 "iterplot2 " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 97 " Use iterplot( f, g, n, a ) to plot the sequences a, f(a), f(f(a)), ... and a, g(a), g(g(a)), " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 9 " ... " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 7 "cobweb " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 89 " \+ Use cobweb( f, n, s, a..b ) to show how the sequence of iterates s, f (s), f(f(s)) is" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 42 " \+ generated by the \"cobweb\" procedure. " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 11 "printarra y " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 61 " Use printar ray(y) to print a list y in vertical format. " }{TEXT 204 0 "" }} {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }} {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 14 "# Jim Carlson " }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 36 "# http://www .math.utah.edu/~carlson)" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 16 "# August 9, 1993" }{MPLTEXT 1 205 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 31 "# Last revised: October \+ 2, 1993" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 23 "printarray := proc( L )" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 10 " local i;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 40 " f or i from 0 to op(2,op(2,eval(L))) do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 18 " print(i, L[i])" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 6 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 " " }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 20 "iter := proc(f,n,a)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " local x, i;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 19 " x := ar ray(0..n);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " x[0] := a;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " for i from 1 to n do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 28 " x[i] := evalf(f(x[i-1]))" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " RETURN(x);" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 24 "iterprint := proc(f,n ,a)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " l ocal y, z;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " y := iter(f,n,a);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 18 " printarray(y);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 23 "iterplot := proc(f,n,a)" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " local y, z;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " y := iter2(f,n,a);" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 23 " z := co nvert(y,list);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 29 " plot(z, 0..n, style=POINT);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 25 "iterplotna := proc(f,n,a)" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " local y, z;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " y := iter2na(f,n,a);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 23 " z := c onvert(y,list);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 29 " plot(z, 0..n, style=POINT);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 29 "iterplot2 := proc(f,g,n,a,b)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " local y, z, y2, z2;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " y := it er2(f,n,a);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 23 " z := convert(y,list);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " y2 := iter2(g,n,b);" }{MPLTEXT 1 205 0 "" } }{PARA 205 "> " 0 "" {MPLTEXT 1 205 25 " z2 := convert(y2,list);" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 37 " plot(\{ z, z2 \}, 0..n, style=POINT);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 24 "iterplotP := proc(f,n,a)" }{MPLTEXT 1 205 0 "" }}{PARA 205 " > " 0 "" {MPLTEXT 1 205 22 " local y, u, z, i, j;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " u := array(0..2*n);" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " y := it er2(f,n,a);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 9 " j := 0;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 24 " for i from 0 to n-1 do" }{MPLTEXT 1 205 0 "" }}{PARA 205 " > " 0 "" {MPLTEXT 1 205 31 " u[j] := [y[i][2],y[i][1]] ;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " j := \+ j+1;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 33 " \+ u[j] := [y[i+1][2],y[i][1]] ;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " j := j+1;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 31 " u[2*n] := [y[n][2],y[n][1]] ;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 23 " z := co nvert(u,list);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 34 " plot(z, 0..y[n][2], style=LINE);" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 20 "iter2 := proc(f,n,a)" }{MPLTEXT 1 205 0 "" } }{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " local x, i;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 19 " x := array(0..n);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " x[0] \+ := [0, a];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " for i from 1 to n do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 31 " x[i] := [ i, f(x[i-1][2]) ]" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " RETURN(x);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 22 "iter2na := proc(f,n,a)" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 13 " local x , i;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 19 " \+ x := array(0..n);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " x[0] := [0, a];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " for i from 1 to n do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 33 " x[i] := [ i, f(i ,x[i-1][2]) ]" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " RETURN(x);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 29 "makethread := proc( f, n, a )" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " local T, i, x, y;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " T := array(0..n);" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " x := e valf(a);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " y := 0.0;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " T[0] := [x,y];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 " " {MPLTEXT 1 205 28 " for i from 1 by 2 to n do" }{MPLTEXT 1 205 0 " " }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " x := T[i-1][1];" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 25 " y : = evalf( f(x) );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 23 " T[i] := [ x, y ];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 25 " T[i+1] := [ y, y ];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 6 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " convert( T, list );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 31 "discretize := proc( f , n, a, b)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " local T, i, x, y, h;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 19 " T := array(0..n);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " x := evalf(a);" }{MPLTEXT 1 205 0 " " }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 12 " y := f(x);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 24 " h := evalf( (b-a)/ n );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " \+ T[0] := [x,y];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 22 " for i from 1 to n do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 15 " x := x + h;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 14 " y := f(x);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 18 " T[i] := [x,y];" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " od;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 21 " convert ( T, list );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 21 "xminoflist := proc(L) " }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " loca l i,x, xm;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " xm:= L[1][1];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 30 " for i from 2 to nops(L) do" }{MPLTEXT 1 205 0 " " }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " x := L[i][1];" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 34 " if \+ (x < xm) then xm := x fi;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 7 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " xm;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 21 "xmaxoflist := proc(L)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " local i,x, xm;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " xm:= L[1][1];" }{MPLTEXT 1 205 0 "" } }{PARA 205 "> " 0 "" {MPLTEXT 1 205 30 " for i from 2 to nops(L) do " }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " \+ x := L[i][1];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 34 " if (x > xm) then xm := x fi;" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 7 " od;" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " xm;" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 21 "yminoflist := proc(L)" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " local i,y, ym;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " ym:= L[1][2];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 30 " for i from 2 to nops(L) do" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " y := L[i][1];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 34 " if (y < ym) then ym := y fi;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 7 " od;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " ym;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" } {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 21 "ymaxoflist := proc(L) " }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 16 " loca l i,y, ym;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " ym:= L[1][2];" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 30 " for i from 2 to nops(L) do" }{MPLTEXT 1 205 0 " " }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 20 " y := L[i][1];" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 34 " if \+ (y > ym) then ym := y fi;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 7 " od;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 5 " ym;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 35 "id := x -> x; # identity function #" }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I#idG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arro wGF%F%9$F%F%F%" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 33 "c obweb := proc( f, n, a, xrange )" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 75 " local thread1, thread2, thread3, e psilon, x1, x2, y1, y2, yrange," }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 37 " xmin,xmax,ymin,ymax;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 17 " \+ " }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 27 " yrange := xrange;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 25 " epsilon := 0.1;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 45 " thread1 := maket hread( f, 2*n, a );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 49 " x1 := op(1,xrange); x2 := op(2,xrange);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 49 " \+ y1 := op(1,yrange); y2 := op(2,yrange);" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 38 " xmin := xminoflist(thread1);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 38 " xmax := \+ xmaxoflist(thread1);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 38 " ymin := yminoflist(thread1);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 38 " ymax := \+ ymaxoflist(thread1);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 10 " " }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 44 " if (x1 > xmin) then x1 := xmin fi;" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 44 " \+ if (x2 < xmax) then x2 := xmax fi;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 44 " if (y1 > ymin) then y1 := ymin \+ fi;" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 44 " \+ if (y2 < ymax) then y2 := ymax fi;" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 10 " " }{MPLTEXT 1 205 0 " " }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 37 " x1 := x1 - epsilo n*(x2-x1);" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 37 " x2 := x2 + epsilon*(x2-x1);" }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 48 " thread2 := discretize(f, 20, x1, x2 );" } {MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 48 " \+ thread3 := discretize(id, 1, x1, x2 );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 10 " " }{MPLTEXT 1 205 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 46 " plot( \{ thread1, thr ead2, thread3 \}," }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 65 " x = x1..x2, y=y1..y2, style = LINE, thickness=2 );" }{MPLTEXT 1 205 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 4 "end:" }{MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 20 "double := x -> 2*x; " }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I'doubleG6\" f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,$9$\"\"#F%F%F%" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 13 "evalf(ln(2));" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#$\"+1=ZJp!#5" }{TEXT 206 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 51 "Example 1 We want \+ to find the solutions to 2^x=x^10" }}{PARA 204 "" 0 "" {TEXT 204 81 "M ethod 1. We define f1(x)=(10/ln(2))*(ln(x)) and apply the fixed point \+ algorithm." }}{PARA 204 "" 0 "" {TEXT 204 18 "Define a function " } {TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 31 "f1 := x -> (10 /ln(2))*(ln(x)); " }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I#f1G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,$*&-I#lnG6$I*p rotectedGF1I(_syslibGF%6#\"\"#!\"\"-F/6#9$\"\"\"\"#5F%F%F%" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 25 "iterprint( f1, 10, 50 ); " }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"!\"#]" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+!>cQk&!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"#$\"+B$4'=e!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"$$\"+yCgie!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"%$\"+K$pM(e!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"&$\"+?59we!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"'$\"+9rzwe!\")" }{TEXT 206 0 "" }} {PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"($\"+)=ep(e!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\")$\"+Hx*p(e!\")" }{TEXT 206 0 " " }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"*$\"+Ou+xe!\")" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"#5$\"+>)4q(e!\")" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 99 "applies double a total of four times t o compute and print the sequence 1, 2, 4, 8, 16. The command " }{TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 24 "iterplot( f1, 10, 50 ); " }{MPLTEXT 1 205 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6$7-7$$\"\"!F)$\"#]F)7$$\"\"\"F)$\"3>CZx*=cQk &!#;7$$\"\"#F)$\"3PN??A$4'=eF17$$\"\"$F)$\"3Iu\\ixCgieF17$$\"\"%F)$\"3 _C6)3LpM(eF17$$\"\"&F)$\"3A%=b'=59weF17$$\"\"'F)$\"3/-[t8rzweF17$$\"\" (F)$\"3/V/_(=ep(eF17$$\"\")F)$\"35;a)*Gx*p(eF17$$\"\"*F)$\"31T " 0 "" {MPLTEXT 1 207 0 "" }} }{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 209 69 "Method 2. We use g1(x)=2^(0.1*x) and apply the fix ed point algorithm." }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 17 " g1:=x->2^(0.1*x);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I#g1G6\"f*6#I\" xGF%F%6$I)operatorGF%I&arrowGF%F%)\"\"#*&$\"\"\"!\"\"F09$F0F%F%F%" } {TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 26 "iterpr int( g1, 10, -0.5 );" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"!$!\"&!\" \"" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+*Gj$f '*!#5" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"#$\"+peC p5!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"$$\"+\" 3Ip2\"!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"%$ \"+QQ]x5!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"& $\"+\"pYv2\"!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$ \"\"'$\"+#*)\\v2\"!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"($\"+J,bx5!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\")$\"+\\,bx5!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"*$\"+],bx5!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"#5$\"+],bx5!\"*" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 21 "iterplot(g1,10,-0.5);" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6$7-7$$\"\"!F)$!3 ++++++++]!#=7$$\"\"\"F)$\"3I+++*Gj$f'*F,7$$\"\"#F)$\"3'*******oeCp5!#< 7$$\"\"$F)$\"3)******43Ip2\"F77$$\"\"%F)$\"3%******z$Q]x5F77$$\"\"&F)$ \"3++++\"pYv2\"F77$$\"\"'F)$\"3'******>*)\\v2\"F77$$\"\"(F)$\"3'****** 48]v2\"F77$$\"\")F)$\"3!*******[,bx5F77$$\"\"*F)$\"3)*******\\,bx5F77$ $\"#5F)FY-%&COLORG6&%$RGBG$Fgn!\"\"$F)F]oF^o-%+AXESLABELSG6$Q!6\"Fbo-% &STYLEG6#%&POINTG-%%VIEWG6$;F^oFfn;$!*.5bJ&!\"*$\"+`6546F_p" 1 5 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}{TEXT 208 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 209 123 "Method 3. We apply the Newton's method to obtain the th ird root. Notice the Newton's method is x(n+1)=x(n)-h(x(n))/h'(x(n))" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 209 21 "First, we define h(x)" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 15 "h:=x->2^x-x^10;" }} {PARA 206 "" 1 "" {XPPMATH 20 "6#>I\"hG6\"f*6#I\"xGF%F%6$I)operatorGF% I&arrowGF%F%,&)\"\"#9$\"\"\"*$F/\"#5!\"\"F%F%F%" }{TEXT 206 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 13 "diff(h(x),x);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#,&*&)\"\"#I\"xG6\"\"\"\"-I#lnG6$I*protected GF-I(_syslibGF(6#F&F)F)*$F'\"\"*!#5" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 10 "h1:=x->x-(" }{MPLTEXT 1 207 29 "2^x-x ^10)/(2^x*ln(2)-10*x^9);" }{MPLTEXT 1 207 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I#h1G6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&9$\" \"\"*&,&)\"\"#F-F.*$F-\"#5!\"\"F.,&*&F1F.-I#lnG6$I*protectedGF;I(_sysl ibGF%6#F2F.F.*$F-\"\"*!#5F5F5F%F%F%" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 20 "iterprint(h1,10,-2);" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"!!\"#" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"\"$!+&fb+!=!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"#$!+1G@?;!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"$$!+_;me9!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"%$!+eW898!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 " " {XPPMATH 20 "6$\"\"&$!+*>mk=\"!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"'$!+v%fz2\"!\"*" }{TEXT 206 0 "" }}{PARA 206 " " 1 "" {XPPMATH 20 "6$\"\"($!+QJUc**!#5" }{TEXT 206 0 "" }}{PARA 206 " " 1 "" {XPPMATH 20 "6$\"\")$!+h%Q*)\\*!#5" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"*$!+f,8y$*!#5" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"#5$!+$>96P*!#5" }{TEXT 206 0 "" }}} {EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 207 19 "iterplot(h1,10,-2);" }} {PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6&-%'CURVESG6$7-7 $$\"\"!F)$!\"#F)7$$\"\"\"F)$!3K4)e_fb+!=!#<7$$\"\"#F)$!3#fWdk!G@?;F17$ $\"\"$F)$!3m*4/Hlh'e9F17$$\"\"%F)$!3Ms;peW898F17$$\"\"&F)$!3/_yc*>mk= \"F17$$\"\"'F)$!3M=nlv%fz2\"F17$$\"\"(F)$!3K)G,@9Bk&**!#=7$$\"\")F)$!3 w$[SBYQ*)\\*FP7$$\"\"*F)$!3Wc@Qf,8y$*FP7$$\"#5F)$!3=Rny#>96P*FP-%&COLO RG6&%$RGBG$Fgn!\"\"$F)F_oF`o-%+AXESLABELSG6$Q!6\"Fdo-%&STYLEG6#%&POINT G-%%VIEWG6$;F`oFfn;$!2kU9;xd7-#!#;$!1uCkwk`e\"*Fap" 1 5 2 0 10 1 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" }}{TEXT 208 0 "" }}}{EXCHG {PARA 208 "" 0 "" {TEXT 209 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 9 "Example 2" }}{PARA 204 "" 0 " " {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 136 "The function \"cob web\" is useful for demonstrating graphicaly how a map is iterated. It is also useful for for finding fixed points. Try " }{TEXT 204 0 "" }} {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 28 "cobweb( f1, 20, 50, 0..60); " }{MPLTEXT 1 205 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%'CURVESG6$7$7$$!\"'\"\"!F( 7$$\"3W************fm!#;F,-%&COLORG6&%$RGBG$\"#5!\"\"$F*F5F6-F$6$757$$ \"++++g7!\"*$\"+QtBMLF=7$$\"++++!*[F=$\"+lW$)*G#!\")7$$\"++++?&)F=$\"+ JM&34$FE7$$\"++++:7FE$\"+4W)Gg$FE7$$\"++++y:FE$\"++`-!)RFE7$$\"++++T>F E$\"+8#G(yUFE7$$\"++++/BFE$\"+7)og_%FE7$$\"++++nEFE$\"+?f9PZFE7$$\"+++ +IIFE$\"+!*eC@\\FE7$$\"++++$R$FE$\"+E&*[%3&FE7$$\"++++cPFE$\"+e^7J_FE7 $$\"++++>TFE$\"+@AAk`FE7$$\"++++#[%FE$\"+W22'[&FE7$$\"++++X[FE$\"+gZU) f&FE7$$\"++++3_FE$\"+Wvl-dFE7$$\"++++rbFE$\"+6W')*z&FE7$$\"++++MfFE$\" +AI$4*eFE7$$\"++++(H'FE$\"+jFfwfFE7$$\"++++gmFE$\"+t-XdgFE-F06&F2F6F3F 6-F$6$7K7$$\"#]F*$F*F*7$F[s$\"3M+++!>cQk&F.7$F_sF_s7$F_s$\"3u*****HK4' =eF.7$FcsFcs7$Fcs$\"3;+++yCgieF.7$FgsFgs7$Fgs$\"3#)*****>LpM(eF.7$F[tF [t7$F[t$\"3/+++?59weF.7$F_tF_t7$F_t$\"3-+++9rzweF.7$FctFct7$Fct$\"3q** ***z=ep(eF.7$FgtFgt7$Fgt$\"3++++Hx*p(eF.7$F[uF[u7$F[u$\"3C+++Ou+xeF.7$ F_uF_u7$F_u$\"3))******=)4q(eF.7$FcuFcu7$Fcu$\"3)******RS5q(eF.7$FguFg u7$Fgu$\"3!)*****za5q(eF.7$F[vF[v7$F[v$\"3y*****>e5q(eF.7$F_vF_v7$F_v$ \"3#)*****4f5q(eF.7$FcvFcv7$Fcv$\"31+++%f5q(eF.7$FgvFgvFivFivFivFivFiv FivFivFivFivFiv-F06&F2F3F3F6-%+AXESLABELSG6$Q\"x6\"Q\"yF`w-%&STYLEG6#% %LINEG-%%VIEWG6$;$!#gF5$\"$m'F5;F6$\"#gF*-%*THICKNESSG6#\"\"#" 1 6 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" }} {TEXT 208 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 28 "cobweb ( g1, 10, -0.5, 0..2);" }{MPLTEXT 1 207 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%'CURVESG6$7$7$$!3++++++++v!#=F( 7$$\"3#************\\F#!#U@F7\"F.7$$\"3/+++++D@=F.$\"3,+++#y`X8\"F.7$$\"3#***********\\s>F.$ \"3/+++W3^Y6F.7$$\"3\")*********\\P7#F.$\"3-+++@Rfe6F.7$F,$\"3'******4 M/3<\"F.-F06&F2F6F3F6-F$6$777$$!3++++++++]F*$F7F77$Fas$\"3I+++*Gj$f'*F *7$FesFes7$Fes$\"3'*******oeCp5F.7$FisFis7$Fis$\"3)******43Ip2\"F.7$F] tF]t7$F]t$\"3%******z$Q]x5F.7$FatFat7$Fat$\"3++++\"pYv2\"F.7$FetFet7$F et$\"3'******>*)\\v2\"F.7$FitFit7$Fit$\"3'******48]v2\"F.7$F]uF]u7$F]u $\"3!*******[,bx5F.7$FauFau7$Fau$\"3)*******\\,bx5F.7$FeuFeuFguFgu-F06 &F2F3F3F6-%+AXESLABELSG6$Q\"x6\"Q\"yF^v-%&STYLEG6#%%LINEG-%%VIEWG6$;$! #v!\"#$\"%vA!\"$;$!\"&F5$\"#?F5-%*THICKNESSG6#\"\"#" 1 6 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" "Curve 3" }}{TEXT 208 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 27 "cobweb( h1, 10, -2, \+ -2..2);" }{MPLTEXT 1 207 0 "" }}{PARA 207 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6)-%'CURVESG6$7$7$$!3!**************R#!#F*7$$!3#************f\">F *$!3>$H1\"ei[CV\"F*$!3#GSw;I2/H\"F*7$$!3%**************=\"F*$!3;foN6?&33\"F*7$ $!3a************z%*!#=$!3%3I-:lIiP*FY7$$!3i************fqFY$!3;jfYOW^# Q\"F*7$$!3B++++++SYFY$!3O*=K%Hzsx=F*7$$!3.++++++?AFY$!3[f;fdFmk;F*7$$ \"3/+++++++?!#>$!3!Q(*))3/&pA9F*7$$\"35++++++?EFY$!3A\"[QR=%z!=\"F*7$$ \"3/++++++S]FY$!3y%oQj\">k!p*FY7$$\"3'*************fuFY$!3G*R#QMf2()GF *7$$\"3!*************z)*FY$\"3Z(396FYC8\"F*7$$\"3)*************H7F*$\" 3'>P>I0g69\"F*7$$\"3)************>Z\"F*$\"3MyrXUI_K8F*7$$\"3(********* ***Rr\"F*$\"3rW=,r$o[a\"F*7$$\"3'************f&>F*$\"3![6(R`2?h#F*$\"3/SHE'*[_y>F*7$F,$\"3OwR/$4Zh>#F*-F/6&F1F5F2F5 -F$6$777$$!\"#F6$F6F67$Fas$!3!******\\fb+!=F*7$FesFes7$Fes$!31+++1G@?; F*7$FisFis7$Fis$!3)******>lh'e9F*7$F]tF]t7$F]t$!31+++eW898F*7$FatFat7$ Fat$!3/+++*>mk=\"F*7$FetFet7$Fet$!37+++v%fz2\"F*7$FitFit7$Fit$!37+++QJ Uc**FY7$F]uF]u7$F]u$!3[+++h%Q*)\\*FY7$FauFau7$Fau$!3!*******e,8y$*FY7$ FeuFeu7$Feu$!3q*****H>96P*FY7$FiuFiu-F/6&F1F2F2F5-%+AXESLABELSG6$Q\"x6 \"Q\"yFbv-%&STYLEG6#%%LINEG-%%VIEWG6$;$!2/++++++S#!#;$\"$W#Fbs;$!#?F4$ \"#?F4-%*THICKNESSG6#\"\"#" 1 6 2 0 10 2 2 6 1 4 2 1.0 45.0 45.0 1 0 " Curve 1" "Curve 2" "Curve 3" }}{TEXT 208 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 141 "Notice that the cobweb is used for fixed point algo rithm and h1 is used for the Newton's method (see above) so the graph \+ does not look right." }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 " " 0 "" {TEXT 204 9 "Example 3" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }} {PARA 204 "" 0 "" {TEXT 204 99 "The iter function can be used to imple ment Newton's method for finding roots. Let us use it to find" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 66 "the square root of two, whi ch is a root of the quadratic function " }{TEXT 204 0 "" }}{PARA 204 " " 0 "" {TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 24 " f \+ := x -> x*x - 2; " }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I\"fG6\"f*6#I\"xGF%F%6$I)operatorGF%I&arrowGF%F%,&*&9$\"\"\"F.F /F/!\"#F/F%F%F%" }{TEXT 206 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }} {PARA 204 "" 0 "" {TEXT 204 40 "Newton tells us to iterate the functio n " }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 205 "> \+ " 0 "" {MPLTEXT 1 205 30 " g :=x-> x - (f(x)/(2*x));" }{MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6#>I\"gG6\"f*6#I\"xGF%F%6$I) operatorGF%I&arrowGF%F%,&9$\"\"\"*&-I\"fGF%6#F-F.F-!\"\"#F3\"\"#F%F%F% " }{TEXT 206 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 101 "with a reasonable guess for the starting va lue. Try the example below, which iterates g 10 times with" }{TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 20 "1 as initial value. " } {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 205 "> " 0 " " {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }} {PARA 205 "> " 0 "" {MPLTEXT 1 205 28 " iterprint( g, 10, 1 ); " } {MPLTEXT 1 205 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"!\"\"\"" } {TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"\"$\"+++++:!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"#$\"+nmm;9!\"* " }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"$$\"+'o:UT\"! \"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"%$\"+iN@99 !\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"&$\"+iN@9 9!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"'$\"+iN@ 99!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"($\"+iN @99!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\")$\"+i N@99!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"\"*$\"+ iN@99!\"*" }{TEXT 206 0 "" }}{PARA 206 "" 1 "" {XPPMATH 20 "6$\"#5$\"+ iN@99!\"*" }{TEXT 206 0 "" }}}{EXCHG {PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 204 "" 0 "" {TEXT 204 0 "" }}{PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{EXCHG {PARA 205 "> " 0 "" {MPLTEXT 1 205 0 "" }}}{PARA 209 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }