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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 47 "The Improved Euler (or Heun) method for s
olving" }{MPLTEXT 1 0 1 " " }{XPPEDIT 18 0 "diff(y,t) = f(t,y);" "6#/-
%%diffG6$%\"yG%\"tG-%\"fG6$F(F'" }{TEXT -1 31 " with an initial condit
ion of  " }{XPPEDIT 18 0 "y(t0) = y0;" "6#/-%\"yG6#%#t0G%#y0G" }{TEXT 
-1 33 ".  We assume a fixed spacing (by " }{XPPEDIT 18 0 "h;" "6#%\"hG
" }{TEXT -1 12 ") mesh with " }{XPPEDIT 18 0 "n+1;" "6#,&%\"nG\"\"\"\"
\"\"F%" }{TEXT -1 8 " points." }{MPLTEXT 1 0 305 "\n\nHeunMethod := pr
oc( f, t0, y0, h, n)\n                    local a,ft,hh,i,t,y,yn;\n  h
h := evalf(h/2.0);\n  t := evalf(t0);\n  y := evalf(y0);\n  a := [[t,y
]];\n  for i from 1 to n do\n    ft := f(t,y);\n    yn := y + h * ft;
\n    t := t + h;\n    y := y + hh * (ft + f(t,yn));\n    a := [op(a),
[t,y]];\n  od;\nend;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%+HeunMethodGR6'%\"fG%#t0G%#y0G%\"hG%\"nG6)
%\"aG%#ftG%#hhG%\"iG%\"tG%\"yG%#ynG6\"F4C'>8&-%&evalfG6#,$9'$\"+++++]!
#5>8(-F96#9%>8)-F96#9&>8$7#7$FAFF?(8'\"\"\"FP9(%%trueGC'>8%-9$6$FAFF>8
*,&FFFP*&F<FPFUFPFP>FA,&FAFPF<FP>FF,&FFFP*&F7FP,&FUFP-FW6$FAFZFPFPFP>F
K7$-%#opG6#FKFMF4F4F4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f \+
:= (t,y)->t/y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"tG%\"yG6
\"6$%)operatorG%&arrowGF)*&9$\"\"\"9%!\"\"F)F)F)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "hm := HeunMethod( f,0,1,0.1,10 );" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%#hmG7-7$\"\"!$\"\"\"F'7$$F)!\"\"$\"++++05!
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\"\"&F,$\"+W-3=6F/7$$\"\"'F,$\"+)zRi;\"F/7$$\"\"(F,$\"+5lq?7F/7$$\"\")
F,$\"+-hn!G\"F/7$$\"\"*F,$\"+3JTX8F/7$$\"#5F,$\"+]JE99F/" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "plot(hm);" }}{PARA 13 "" 1 "" 
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F17$$\"1+++++++!)F.$\"1+++-hn!G\"F17$$\"1+++++++!*F.$\"1+++3JTX8F17$F)
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