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All the impl ementations of numerical methods take five arguments: a function of tw o variables " }{XPPEDIT 18 0 "g(x,y)" "6#-%\"gG6$%\"xG%\"yG" }{TEXT -1 14 ", the initial " }{TEXT 258 1 "x" }{TEXT -1 14 ", the initial " }{TEXT 259 1 "y" }{TEXT -1 46 ", the step size, and the number of iter ations." }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 5 "Euler" }}{PARA 0 "" 0 " " {TEXT 0 5 "Euler" }{TEXT -1 42 " is an implementation of the Euler M ethod." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "Euler := proc(g,x 0,y0,h,N) local x,y,eta,k; x:=x0; y[0]:=y0; for k from 0 to N-1 do eta :=y[k]; y[k+1]:=evalf(eta+h*g(x,eta)); x:=evalf(x+h) od; RETURN([y[i] \+ $ i=0..N]) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&EulerGR6'%\"gG%# x0G%#y0G%\"hG%\"NG6&%\"xG%\"yG%$etaG%\"kG6\"F1C&>8$9%>&8%6#\"\"!9&?(8' F:\"\"\",&9(F>!\"\"F>%%trueGC%>8&&F86#F=>&F86#,&F=F>F>F>-%&evalfG6#,&F EF>*&9'F>-9$6$F4FEF>F>>F4-FM6#,&F4F>FQF>-%'RETURNG6#7#-%\"$G6$&F86#%\" iG/F\\o;F:F@F1F16\"" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 14 "Error Es timate" }}{PARA 0 "" 0 "" {TEXT 0 2 "er" }{TEXT -1 188 " estimates the global discretization error. Its five arguments are: an array contai ning the approximation of the solution, an expression in x representin g the exact solution, the initial " }{TEXT 260 1 "x" }{TEXT -1 48 ", t he step size, and the number of iterations. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "er := proc(y,z,x0,h,N) local e; e:=max(evalf(ab s(y[k+1]-evalf(subs(x=x0+k*h,z))))$k=1..N); RETURN(evalf(e)) end;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#erGR6'%\"yG%\"zG%#x0G%\"hG%\"NG6#% \"eG6\"F.C$>8$-%$maxG6#-%\"$G6$-%&evalfG6#-%$absG6#,&&9$6#,&%\"kG\"\" \"FDFDFD-F96#-%%subsG6$/%\"xG,&9&FD*&FCFD9'FDFD9%!\"\"/FC;FD9(-%'RETUR NG6#-F96#F1F.F.6\"" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 3 "RK4" }} {PARA 0 "" 0 "" {TEXT 0 3 "rk4" }{TEXT -1 29 " is an implementation of RK4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 320 "rk4 := proc(g,x0,y 0,h,N) local x,y,k,l,xi,eta,w1,w2,w3,w4; l:=h/2; x:=x0; y[0]:=y0; for \+ k from 0 to N-1 do xi:=x+l; eta:=y[k]; w1:=evalf(g(x,eta)); w2:=evalf( g(xi,eta+l*w1)); w3:=evalf(g(xi,eta+l*w2)); w4:=evalf(g(x+h,eta+h*w3)) ; y[k+1]:=evalf(eta+1/6*h*(w1+2*w2+2*w3+w4)); x:=evalf(x+h) od; RETURN ([y[i] $ i=0..N]) end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$rk4GR6'% \"gG%#x0G%#y0G%\"hG%\"NG6,%\"xG%\"yG%\"kG%\"lG%#xiG%$etaG%#w1G%#w2G%#w 3G%#w4G6\"F7C'>8',$9'#\"\"\"\"\"#>8$9%>&8%6#\"\"!9&?(8&FGF>,&9(F>!\"\" F>%%trueGC*>8(,&FAF>F:F>>8)&FE6#FJ>8*-%&evalfG6#-9$6$FAFT>8+-FZ6#-Fgn6 $FQ,&FTF>*&F:F>FXF>F>>8,-FZ6#-Fgn6$FQ,&FTF>*&F:F>FjnF>F>>8--FZ6#-Fgn6$ ,&FAF>F,&FTF>*&FFboF>F>>&FE6#,&FJF>F>F>-FZ6#,&FTF>*&F,*FXF>Fj nF?FboF?FjoF>F>#F>\"\"'>FA-FZ6#F_p-%'RETURNG6#7#-%\"$G6$&FE6#%\"iG/Fiq ;FGFLF7F76\"" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 8 "Plotting" }} {PARA 0 "" 0 "" {TEXT 0 8 "plotlist" }{TEXT -1 150 " displays an appro ximate solution obtained by one of the implementation above. The four arguments are: an array containing the solution, the initial " } {TEXT 261 1 "x" }{TEXT -1 46 ", the step size, and the number of itera tions." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "plotlist := proc( y,x0,h,N) local p,P; p:=[[x0+h*i, y[i+1]]$i=0..N]; P:=PLOT(CURVES(p)): RETURN (P) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)plotlistGR6&%\" yG%#x0G%\"hG%\"NG6$%\"pG%\"PG6\"F.C%>8$7#-%\"$G6$7$,&9%\"\"\"*&9&F9%\" iGF9F9&9$6#,&F8%-%%PLOTG6#-%'CURVESG6#F1-%'RETURNG6 #FFF.F.6\"" }}}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "We apply these procedures to the problems in " }{TEXT 266 19 "Maple Assignment #3" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 4 "1. " }{TEXT 262 8 "y' = x+y" }{TEXT -1 3 " ; " }{TEXT 263 2 " y" }{TEXT -1 9 "(1) = -3." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := (x,y) -> x+y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF ),&9$\"\"\"9%F/F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "d eq := diff(y(x),x)=f(x,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de qG/-%%diffG6$-%\"yG6#%\"xGF,,&F,\"\"\"F)F." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 175 "### WARNING: `dsolve` has been extensively rewritt en, many new result forms can occur and options are slightly different , see help page for details\ndsolve(\{deq,y(1)=-3\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(F'!\"\"F)\"\"\"*&-%$expGF&F* -F-6#F*F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Z := rhs(%): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "P:=plot(Z,x=1..5):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Y:=Euler(f,1,-3,.2,20):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "W:=rk4(f,1,-3,.2,20):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "er(Y,Z,1,.2,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9]0E;!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "er(W,Z,1,.2,20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"'slC!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P1:=plotlist( Y,1,.2,20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Q1:=plotlist (W,1,.2,20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "Y:=Euler(f, 1,-3,.1,40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "W:=rk4(f,1, -3,.1,40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "er(Y,Z,1,.1,4 0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*H%*)Q$*!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "er(W,Z,1,.1,40);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"&Ln\"!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P2:=plotlist(Y,1,.1,40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Q2:=plotlist(W,1,.1,40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y:=Euler(f,1,-3,.05,80):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "W:=rk4(f,1,-3,.05,80):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "er(Y,Z,1,.05,80);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"*\"*3n.&!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "er(W,Z ,1,.05,80);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%q5!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P3:=plotlist(Y,1,.05,80):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Q3:=plotlist(W,1,.05,80):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{P,P1,P2,P3\});" }}{PARA 13 "" 1 "" {GLPLOT2D 292 292 292 {PLOTDATA 2 "6(-%'CURVESG6#77 7$$\"\"\"\"\"!!\"$7$$\"#7!\"\"$!#MF/7$$\"#9F/$!$%Q!\"#7$$\"#;F/$!%GVF+ 7$$\"#=F/$!&O([!\"%7$$\"#?F/$!'K)[&!\"&7$$\"#AF/$!(%)f='!\"'7$$\"#CF/$ !)3=$)p!\"(7$$\"#EF/$!*'p\")**y!\")7$$\"#GF/$!+_.yf*)!\"*7$$\"#IF/$!+U O<>5FZ7$$\"#KF/$!+q$3I;\"FZ7$$\"#MF/$!+W+hJ8FZ7$$\"#OF/$!+`?$*H:FZ7$$ \"#QF/$!+k%=Rw\"FZ7$$\"#SF/$!+d@qS?FZ7$$\"#UF/$!+)eU)oBFZ7$$\"#WF/$!+1 6heFFZ7$$\"#YF/$!+FLLAKFZ7$$\"#[F/$!+#***zuPFZ7$$\"#]F/$!+!**fPV%FZ-F$ 6#7KF'7$$\"#6F/$!#KF/7$F-$!$T$F77$$\"#8F/$!%JOF+7$F3$!&T'QFB7$$\"#:F/$ !'^5TFH7$F9$!(h:P%FN7$$\"#$!*\"))eV\\FZ7$$\"#>F/$!+\"p ZzD&Fjn7$FD$!+gCu$f&Fjn7$$\"#@F/$!+1n6`fFjn7$FJ$!+x$G%QjFjn7$$\"#BF/$! 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}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "Here \+ the Euler approximations are remarkably close to each other, and as be fore, the RK4 approximations are indistinguishable." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "6. Approximation of " }{TEXT 273 1 "e" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := (x,y) -> y; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operator G%&arrowGF)9%F)F)6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Y : = Euler(f,0,1,.01,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " Y[101]; evalf(exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MQ\"[q#! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The Euler approximatio n is good to two digits. We have " }{XPPEDIT 18 0 "y[100] < exp(1)" " 6#2&%\"yG6#\"$+\"-%$expG6#\"\"\"" }{TEXT -1 35 ". The reason is that \+ the solution " }{XPPEDIT 18 0 "y=exp(x)" "6#/%\"yG-%$expG6#%\"xG" } {TEXT -1 156 " is convex, and in this case the Euler method will alway s yield an approximation which lies below the exact solution. This fo llows from the Taylor theorem:" }}{PARA 258 "" 0 "" {TEXT -1 2 "y(" } {TEXT 274 3 "x+h" }{TEXT -1 6 ") = y(" }{TEXT 275 1 "x" }{TEXT -1 7 ") + y'(" }{TEXT 276 1 "x" }{TEXT -1 2 ") " }{TEXT 277 1 "h" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"\"\"\"\"\"#!\"\"" }{TEXT -1 5 " y''(" }{XPPEDIT 18 0 "xi" "6#%#xiG" }{TEXT -1 2 ") " }{XPPEDIT 18 0 "h^2" "6#*$%\"hG\"\"#" }{TEXT -1 5 " > y(" }{TEXT 278 1 "x" }{TEXT -1 7 ") + y'(" }{TEXT 279 1 "x" }{TEXT -1 2 ") " }{TEXT 280 1 "h" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "We perform the same computatio ns with RK4:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Y:=rk4(f,0,1 ,.01,100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Y[101]; evalf (exp(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J=G=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+G=G=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The approximation is now good to 8 digits. Note that now " }{XPPEDIT 18 0 "y[100]>exp(1)" "6#2-%$expG6# \"\"\"&%\"yG6#\"$+\"" }{TEXT -1 1 "." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "7. Approximation of " }{XPPEDIT 18 0 "Pi" "6#%#PiG" } {TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "f := (x,y) \+ -> 4/(1+x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6 \"6$%)operatorG%&arrowGF),$*$,&*$9$\"\"#\"\"\"F3F3!\"\"\"\"%F)F)6\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Y := Euler(f,0,0,.01,100): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Y[101]; evalf(Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+')fd^J!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 81 "Again the approximation is good to two si gnificant digits. However, now we have " }{XPPEDIT 18 0 "y[100]>Pi" " 6#2%#PiG&%\"yG6#\"$+\"" }{TEXT -1 21 ", since the solution " } {XPPEDIT 18 0 "y = 4*arctan(x)" "6#/%\"yG*&\"\"%\"\"\"-%'arctanG6#%\"x GF'" }{TEXT -1 12 " is concave." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 42 "We perform the same computations with RK4 :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Y := rk4(f,0,0,.01,100) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Y[101]; evalf(Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+^EfTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+aEfTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 52 "The approximation is now good to 9 digits . We have " }{XPPEDIT 18 0 "y[100]