{VERSION 5 0 "SGI MIPS UNIX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "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 "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 2 6 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Diagnostic" 7 9 1 {CSTYLE "" -1 -1 "" 0 1 64 128 64 1 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE " " 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 3 "" 0 "" {TEXT -1 56 "Maple Summer Workshop '04: Programming in Maple Tutorial" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Newton's iteration: Michael Monagan, June 2004." }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "To solve " }{XPPEDIT 18 0 "x = cos( x);" "6#/%\"xG-%$cosG6#F$" }{TEXT -1 5 " for " }{TEXT 256 1 "x" } {TEXT -1 12 " we can use " }{TEXT 257 6 "fsolve" }{TEXT -1 32 " as fol lows where we have given " }{TEXT 258 6 "fsolve" }{TEXT -1 53 " 0.5 to use as an initial approximation for the root." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "f := x -> x-cos(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$c osG6#F-!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "fsolv e( f(x)=0, x=0.5 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Newton's method uses the iterat ion" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "x[k+1] = x[k ]-f(x[k])/`f'`(x[k]);" "6#/&%\"xG6#,&%\"kG\"\"\"F)F),&&F%6#F(F)*&-%\"f G6#&F%6#F(F)-%#f'G6#&F%6#F(!\"\"F8" }{TEXT -1 2 " ." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "If we want to p rogram Newton's method, Newton's method needs a function f(" }{TEXT 259 1 "x" }{TEXT -1 154 ") (of one parameter) and also the derivative \+ of the function as input. One way to do this is to use D as follows ( we could input f ' by hand of course)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "fp := D(f); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fpGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,&\"\"\"F--%$sinG6#9$F-F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Here is a minimal version of Newton's method. It does a fixed \+ number of steps." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "Newton := proc(f::procedure,fp::procedure,x0::numeric,n::posint)\nlocal x;\n x := evalf(x0);\n to n do\n x := x - evalf(f(x))/evalf(f p(x));\n od;\n x;\nend:\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "trace(Newton);\nNewton(f,fp,0.5,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'NewtonG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Digi tsG\"#5" }}{PARA 9 "" 1 "" {TEXT -1 38 "\{--> enter Newton, args = f, \+ fp, .5, 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"\"&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+qTA_v!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"xG$\"+hmT\"R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+R8&3R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+K8 &3R(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+K8&3R(!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}{PARA 9 "" 1 "" {TEXT -1 49 "<-- exit Newton (now at top level) = .7390851332\}" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K8&3R(!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 40;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'DigitsG\"#S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Newton(f ,fp,0.5,5);" }}{PARA 9 "" 1 "" {TEXT -1 38 "\{--> enter Newton, args = f, fp, .5, 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"\"&!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"Iu\\.XeT$))Rqr;UOc5%\"xG$\"IX?Nn3Dh#4oX\\Cz)\\hmT\"R(! #S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"IZ%z5_p$*f\")pxK!o!3#R8 &3R(!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"I:'*Q(\\&zZ:f_wT1; :K8&3R(!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"Ih,/M(Qn(37`lT1 ;:K8&3R(!#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ih,/M(Qn(37`lT1;:K8& 3R(!#S" }}{PARA 9 "" 1 "" {TEXT -1 79 "<-- exit Newton (now at top lev el) = .7390851332151606416553120876738734040161\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Ih,/M(Qn(37`lT1;:K8&3R(!#S" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 153 "This version stops either if we hit the root or if we \+ have a small relative error. This time we print out the accuracy (esti mated) of the approximations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 467 "Newton := proc(f::procedure,fp::procedure,x0::numeric,M::posint )\nlocal x,y,n,er,fx,gx;\n x := evalf(x0);\n for n to M do\n \+ fx := evalf(f(x));\n if fx=0.0 then return x end if;\n \+ gx := evalf(fp(x));\n if gx=0.0 then error \"In Newton: deri vative is 0\" fi;\n y := x - fx/gx;\n er := (x-y)/x;\n \+ printf(\"Step %d, error is %20e\\n\",n,er);\n if abs(er) \+ < 10^(2-Digits) then return y; fi;\n x := y;\n od;\n x;\n end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Newton(f,fp,0.5,10) ;" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 1, error is -5.104448e-0 1" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 2, error is 2.129274e-0 2" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 3, error is 7.648362e-0 5" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 4, error is 9.547563e-1 0" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 5, error is 1.487610e-1 9" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 6, error is 3.653165e-3 9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"IM,/M(Qn(37`lT1;:K8&3R(!#S" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "Another version, this time takin g the maxium number of steps M as an optional 4th argument. Also, thi s time we check that the values of the function are floating point con stants (real or complex)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%'NewtonG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 739 "Newton := proc(f::proc edure,fp::procedure,x0::numeric,maxits::posint)\nlocal M,x,y,n,er,fx,g x;\n if nargs=3 then M := 10 else M := maxits fi;\n x := evalf(x 0);\n for n to M do\n fx := evalf(f(x));\n if not typ e(fx,complex(float)) then \n error \"Function does not evalu ate to a number\" fi;\n if fx=0.0 then return x end if;\n \+ gx := evalf(fp(x));\n if not type(gx,complex(float)) then \n \+ error \"Derivative does not evaluate to a number\" fi;\n \+ if gx=0.0 then error \"In Newton: derivative is 0\" fi;\n y := x - fx/gx;\n er := (x-y)/x;\n printf(\"Step %d, erro r is %20e\\n\",n,er);\n if abs(er) < 10^(2-Digits) then return \+ y; fi;\n x := y;\n od;\n x;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Now running this at Digits=100 we can verify that th e convergence is quadratic." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Digits := 100;\nNewton(f,fp,0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"$+\"" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 1, error is \+ -5.104448e-01" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 2, error is \+ 2.129274e-02" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 3, error is \+ 7.648362e-05" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 4, error is \+ 9.547563e-10" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 5, error is \+ 1.487610e-19" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 6, error is \+ 3.611458e-39" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 7, error is \+ 2.128478e-78" }}{PARA 6 "" 1 "" {TEXT -1 37 "Step 8, error is \+ -1.353024e-100" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"_q>`m<$p5w$* f%fZN)[l%Gtdj!ol\\Yd2!*e " 0 "" {MPLTEXT 1 0 65 "L := proc(x) local y; y := x ; to n do y := a*y*(1-y) od; y-x end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGf*6#%\"xG6#%\"yG6\"F*C%>8$9$?(F*\"\"\"F0%\"nG%%trueG>F-*(%\"a GF0F-F0,&F0F0F-!\"\"F0,&F-F0F.F7F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\" \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(%\"aG\"\"\"%\"xGF&,&F&F&F'!\"\"F&F&F'F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n := 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "L(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&**)%\"aG\"\"#\"\"\"%\"xG F(,&F(F(F)!\"\"F(,&F(F(*(F&F(F)F(F*F(F+F(F(F)F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The degree of L in x grows exponentially with n ." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for n from 1 to 5 do n=deg ree(L(x),x) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"$\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"%\" #;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"&\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "n := 2; 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