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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 120 "As an example of Heun's method for secon
d-order equations,\nconsider the following damped, nondriven mass-spri
ng  model.\n" }{XPPEDIT 18 0 "d^2x/dt^2 + 2*(dx/dt) + 5*x = 0" "/,(*(%
\"dG\"\"#%\"xG\"\"\"*$%#dtG\"\"#!\"\"F(*&\"\"#F(*&%#dxGF(F*F,F(F(*&\"
\"&F(F'F(F(\"\"!" }{TEXT -1 9 "         " }{XPPEDIT 18 0 "x(0) = 1; dx
/dt = 0" "C$/-%\"xG6#\"\"!\"\"\"/*&%#dxG\"\"\"%#dtG!\"\"F'" }{MPLTEXT 
1 0 1 "\n" }{TEXT -1 192 "\nBy considering velocity to be a second ind
ependent variable, we\ncan implement Heun's scheme for systems of seco
nd-order equations.\nThe initial-value problem above is equivalent to \+
the system\n" }{XPPEDIT 18 0 "dx/dt = v" "/*&%#dxG\"\"\"%#dtG!\"\"%\"v
G" }{TEXT -1 28 " ,                          " }{XPPEDIT 18 0 "x(0) = \+
1" "/-%\"xG6#\"\"!\"\"\"" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "dv/dt = -2v
 - 5x" "/*&%#dvG\"\"\"%#dtG!\"\",&*&\"\"#F%%\"vGF%F'*&\"\"&F%%\"xGF%F'
" }{TEXT -1 11 " ,         " }{XPPEDIT 18 0 "v(0) = 0" "/-%\"vG6#\"\"!
F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := (x,v,t) -> v;" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG:6%%\"xG%\"vG%\"tG6\"6$%)operat
orG%&arrowGF*9%F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "g :=
 (x,v,t) -> -2*v-5*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG:6%%\"x
G%\"vG%\"tG6\"6$%)operatorG%&arrowGF*,&9%!\"#9$!\"&F*F*" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 56 "We need to choose the number of steps and
 the step size:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
18 "N := 50; h := 0.1;" }}{PARA 0 "" 0 "" {TEXT -1 51 "and the first p
oint in each of the three sequences:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "heun_ptsx := [t[0],x[0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
heun_ptsv := [t[0],v[0]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "heun_p
tsxv := [x[0],v[0]];" }}{PARA 0 "" 0 "" {TEXT -1 37 "Now we can procee
d 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 "   slope1x := f(x[n],v[n],t[n]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "   slope1v := g(x[n],v[n],t[n]);" }
}{PARA 0 "" 0 "" {TEXT -1 65 "                    First slope is the s
lope from Euler's method." }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "   slope2x := f(x[n]+h*slope1x,v[n]+h*slope1x,t[n+1])
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "   slope2v := g(x[n]+h*slope1x
,v[n]+h*slope1v,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 "   s
lopex := ( slope1x + slope2x )/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "   slopev := ( slope1v + slope2v )/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 "   x
[n+1] := x[n] + h*slopex;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "   v[n
+1] := v[n] + h*slopev;" }}{PARA 0 "" 0 "" {TEXT -1 60 "              \+
      new value = old value + stepsize * slope" }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "   new_ptx := [ t[n+1], x[n+1] ];" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "   new_ptv := [ t[n+1], v[n+1] ];
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "   new_ptxv := [ x[n+1], v[n+1]
 ];" }}{PARA 0 "" 0 "" {TEXT -1 78 "                    define the new
est points in the three sequences of points." }{MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "   heun_ptsx := heun_ptsx, new_ptx;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "   heun_ptsv := heun_ptsv, new_
ptv;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "   heun_ptsxv := heun_ptsxv
, new_ptxv;" }}{PARA 0 "" 0 "" {TEXT -1 90 "                    modify
 each of the three sequences by adding the 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 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#
]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%*heun_ptsxG7$\"\"!\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*heun_ptsvG7$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%+heun_ptsxvG7$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
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"heunplot := plot([heun_ptsx],style=POINT):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "exactplot := plot(exp(-t)*(cos(2*t)+sin(2*t)/2),t=
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "display(\{heunplot,exactplot
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 271 "There is a little improvement ove
r the results from Euler's method (note in particular the improved\nac
curacy of the times at which x(t) crosses zero) but there is still som
e discrepancy.\n\nIn general, the error in the Euler method is roughly
 proportional to the step size " }{XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT 
-1 0 "" }{TEXT -1 58 ";\nthe error in the Heun method is roughly propo
rtional to " }{XPPEDIT 18 0 "h^2" "*$%\"hG\"\"#" }{TEXT -1 201 ".\n(Th
e constants of proportionality depend on the original differential equ
ation, but you can get\nan idea of what the constant is by looking at \+
the change in the results for a few different\nstep sizes." }}}}{MARK 
"14 0 5" 190 }{VIEWOPTS 1 1 0 1 1 1803 }