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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "This Maple Procedure impl
ements the Improved Euler Approximation to a differential equation of \+
the type " }{XPPEDIT 18 0 "dy/dx = f(x,y);" "6#/*&%#dyG\"\"\"%#dxG!\"
\"-%\"fG6$%\"xG%\"yG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 531 "ImprovedEuler:= proc(f,xstart, ystart, xend, n)\nloc
al h, i, k, y, x, k1, k2;\nh := evalf((xend - xstart)/n);\ny:=ystart: \+
  x:=xstart:\nprintf(\"%s  %s          %s %f \\n\",\"Step\",\"x\",\"y \+
with h =\",h);\nfor i from 1 to n+1 do\n    printf(\"%d  %f    %f \\n
\",i-1,x,y); # display current values\n    k1 := f(x,y): \011\011\011 \+
    # the left-hand slope\n\011\011  k2 := f(x+h,y+h*k1): \011# the ri
ght-hand slope\n\011    k := (k1 + k2)/2:\011\011  # the average slope
\011\n\011    y := y + h*k:  \011\011    # Euler step to update y\n
\011    x := x + h:     \011\011   # update x\nod:\nprint(f);\nend:" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "The Syntax is as follows: Improv
edEuler(function, initial x, initial y, final x, number of steps);" }}
{PARA 0 "" 0 "" {TEXT -1 53 "Below we compute the Improved Euler Appro
ximation to " }{XPPEDIT 18 0 "dy/dx = x+y;" "6#/*&%#dyG\"\"\"%#dxG!\"
\",&%\"xGF&%\"yGF&" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y(0) = 1;" "6#/-%
\"yG6#\"\"!\"\"\"" }{TEXT -1 16 ", with 10 steps." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y) -> x + y;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&9$\"\"
\"9%F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ImprovedEul
er(f,0,1,1,10);" }}{PARA 6 "" 1 "" {TEXT -1 36 "Step  x          y wit
h h = .100000 " }}{PARA 6 "" 1 "" {TEXT -1 24 "0  0.000000    1.000000
 " }}{PARA 6 "" 1 "" {TEXT -1 23 "1  .100000    1.110000 " }}{PARA 6 "
" 1 "" {TEXT -1 23 "2  .200000    1.242050 " }}{PARA 6 "" 1 "" {TEXT 
-1 23 "3  .300000    1.398465 " }}{PARA 6 "" 1 "" {TEXT -1 23 "4  .400
000    1.581804 " }}{PARA 6 "" 1 "" {TEXT -1 23 "5  .500000    1.79489
4 " }}{PARA 6 "" 1 "" {TEXT -1 23 "6  .600000    2.040857 " }}{PARA 6 
"" 1 "" {TEXT -1 23 "7  .700000    2.323147 " }}{PARA 6 "" 1 "" {TEXT 
-1 23 "8  .800000    2.645578 " }}{PARA 6 "" 1 "" {TEXT -1 23 "9  .900
000    3.012364 " }}{PARA 6 "" 1 "" {TEXT -1 25 "10  1.000000    3.428
162 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6$%\"xG%\"yG6\"6$%)operatorG
%&arrowGF',&9$\"\"\"9%F-F'F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}}{MARK "1 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }