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{PARA 0 "" 0 "" {TEXT -1 123 "As an example of Heun's method for syste
ms of first-order equations,\nconsider the following nonlinear predato
r-prey model.\n" }{XPPEDIT 18 0 "dF/dt = (-0.03 + 0.001*R)*F" "/*&%#dF
G\"\"\"%#dtG!\"\"*&,&$\"\"$!\"#F'*&$\"\"\"!\"$F%%\"RGF%F%F%%\"FGF%" }
{TEXT -1 14 "              " }{XPPEDIT 18 0 "F(0) = 20" "/-%\"FG6#\"\"
!\"#?" }{MPLTEXT 1 0 1 "\n" }{XPPEDIT 18 0 "dR/dt = (0.05 - 0.001*F)*R
" "/*&%#dRG\"\"\"%#dtG!\"\"*&,&$\"\"&!\"#F%*&$\"\"\"!\"$F%%\"FGF%F'F%%
\"RGF%" }{TEXT -1 90 "                 R(0) = 50\n\nFirst, define the \+
functions that give the value of the slope:\n" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 36 "f := (F,R,t) -> (-0.03 + 0.001*R)*F;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"FG%\"RG%\"tG6\"6$%)operatorG%&ar
rowGF**&,&$!\"$!\"#\"\"\"9%$F3F1F39$F3F*F*" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 36 "g := (F,R,t) -> (0.05 - 0.001*F)*R;\n" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"FG%\"RG%\"tG6\"6$%)operatorG%&arrowG
F**&,&$\"\"&!\"#\"\"\"9$$!\"\"!\"$F39%F3F*F*" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 78 "Then, define the step size and the initial values for e
ach variable and for t." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "h := 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "t[0] := 0;  F[0] :=
 20;  R[0] := 50;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"tG6#\"\"!F'
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"FG6#\"\"!\"#?" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"RG6#\"\"!\"#]" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 201 "Then for each variable, the new value is ``old value + s
tep size times slope.''\nThe slope is the average of the derivative at
 two different points.\nAfter one step, the values are calculated as f
ollows:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
320 "t[1] := t[0] + h:\n\nslope1F := f(F[0],R[0],t[0]):\nslope1R := g(
F[0],R[0],t[0]):\n\nslope2F := f(F[0]+h*slope1F,R[0]+h*slope1R,t[1]):
\nslope2R := g(F[0]+h*slope1F,R[0]+h*slope1R,t[1]):\n\nslopeF := ( slo
pe1F + slope2F )/2:\nslopeR := ( slope1R + slope2R )/2:\n\nF[1] := F[0
] + h*slopeF:\nR[1] := R[0] + h*slopeR:\n\n[F[1],R[1],t[1]];" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#7%$\"+++$>/#!\")$\"+++A^^F&\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 391 "Recall that Euler's method gave F
[1] := 20.4 and R[1] := 51.5.\nThe true values are approximately F[1] \+
:=  20.4194 and R[1] := 51.5121.  You can\nsee that Heun's method, whi
le requiring about twice as many calculations, returns a\nmuch more ac
curate answer.\n\nAs in the Euler example, let's create three sequence
s of points:  one in the t-F plane,\none in the t-R plane, and one in \+
the F-R plane." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We need to defi
ne the number of steps and the step size:" }{MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "N := 200; h := 1;" }}{PARA 0 "" 0 "" 
{TEXT -1 51 "and the first point in each of the three sequences:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "heun_ptsF := [t[0],F[0]];" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "heun_ptsR := [t[0],R[0]];" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 26 "heun_ptsFR := [F[0],R[0]];" }}{PARA 0 "" 0 "" 
{TEXT -1 37 "Now we can proceed with the for loop." }{MPLTEXT 1 0 0 "
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for n from 0 to N-1 do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "   t[n+1] :=  t[n] + h;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "   s
lope1F := f(F[n],R[n],t[n]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "   \+
slope1R := g(F[n],R[n],t[n]);" }}{PARA 0 "" 0 "" {TEXT -1 65 "        \+
            First slope is the slope from Euler's method." }{MPLTEXT 
1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "   slope2F := f(F[n]+h*s
lope1F,R[n]+h*slope1R,t[n+1]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " \+
  slope2R := g(F[n]+h*slope1F,R[n]+h*slope1R,t[n+1]);" }}{PARA 0 "" 0 
"" {TEXT -1 91 "                    Second slope is calculated at the \+
points resulting from Euler's method." }{MPLTEXT 1 0 0 "" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "   slopeF := ( slope1F + slope2F )/2;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "   slopeR := ( slope1R + slope2R )/
2;" }}{PARA 0 "" 0 "" {TEXT -1 74 "                   Final value for \+
slope is the average of the two slopes." }{MPLTEXT 1 0 0 "" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "   F[n+1] := F[n] + h*slopeF;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "   R[n+1] := R[n] + h*slopeR;" }}{PARA 0 "" 
0 "" {TEXT -1 60 "                    new value = old value + stepsize
 * slope" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "   n
ew_ptF := [ t[n+1], F[n+1] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  \+
 new_ptR := [ t[n+1], R[n+1] ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "
   new_ptFR := [ F[n+1], R[n+1] ];" }}{PARA 0 "" 0 "" {TEXT -1 78 "   \+
                 define the newest points in the three sequences of po
ints." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "   heun
_ptsF := heun_ptsF, new_ptF;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "   \+
heun_ptsR := heun_ptsR, new_ptR;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 
"   heun_ptsFR := heun_ptsFR, new_ptFR;" }}{PARA 0 "" 0 "" {TEXT -1 
90 "                    modify each of the three sequences by adding t
he new point at the end." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 
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>%\"hG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*heun_ptsFG7$\"\"!\"
#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*heun_ptsRG7$\"\"!\"#]" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+heun_ptsFRG7$\"#?\"#]" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([heun_ptsF],labels = [`t`,`F`]
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0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 307 "Note that the cu
rve in the phase plane appears to be closed, this indicates that there
 is likely a periodic\nsolution; after a certain amount of time the po
pulations of F and R return to their original states\nand the process \+
repeats itself.  Note, though, that in Euler's method, the curve in th
e phase plane" }}{PARA 0 "" 0 "" {TEXT -1 102 "did not close; a smalle
r step size would have to be chosen to verify that the solutions were
\nperiodic." }{MPLTEXT 1 0 1 " " }}}}{MARK "10" 0 }{VIEWOPTS 1 1 0 1 
1 1803 }