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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 112 "Unfortunately, the fsolve command in Map
le does not work very well with this problem:  it is too general a too
l." }}{PARA 0 "" 0 "" {TEXT -1 50 "This is a quick-and-dirty gradient \+
descent solver." }}{PARA 0 "" 0 "" {TEXT -1 146 "As these calculations
 are to be performed once to determine the parameters, the run time is
 not relevant so long as it is reasonable (under 10 s)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 787 "GradientDescent := proc( expr, xr,
 yr )\n    local x, y, x0, y0, h, i, delta, minval, newval, dx, dy, x1
, y1;\n    x := lhs( xr );\n    y := lhs( yr );\n    x0 := rhs( xr );
\n    y0 := rhs( yr );\n\n    h := 0.1;\n    minval := evalf( eval( ex
pr, [x = x0, y = y0] ) );\n\n    for i from 1 to 300 do\n        dx :=
 evalf( eval( diff( expr, x ), [x = x0, y = y0] ) );\n        dy := ev
alf( eval( diff( expr, y ), [x = x0, y = y0] ) );\n\n        delta := \+
sqrt( dx^2 + dy^2 );\n\n        x1 := x0 - dx*h/delta;\n        y1 := \+
y0 - dy*h/delta;\n\n        newval := evalf( eval( expr, [x = x1, y = \+
y1] ) );\n\n        if newval < minval then\n            x0 := x1;\n  \+
          y0 := y1;\n            minval := newval;\n        else\n    \+
        h := h/3.0;\n        end if;\n    end do;\n\n    \{x = x0, y =
 y0\};\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 245 "This is the
 function which finds the best fitting clamped cubic Bezier curve of a
 function on a range. \nThe two function calls are BestFittingClampedC
ubicBezierCurve( f, a..b ) or more commoly BestFittingClampedCubicBezi
erCurve( f(x), x = a..b )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
2313 "BestFittingClampedCubicBezierCurve := proc( expr::algebraic, var
rng::\{name = algebraic..algebraic, algebraic..algebraic\} )\n    opti
on remember;\n    local f, x, a, b, points, eqn1, eqn2, fa, fb, Dfa, D
fb, sgnDfa, sgnDfb, var1, val1, var2, val2, integral, pts;\n\n    if v
arrang::('algebraic'..'algebraic') then\n        (a, b) := op( varrng \+
);\n        f := expr;\n    else\n        (a, b) := op( rhs( varrng ) \+
);\n        f := unapply( expr, lhs( varrng ) );\n    end if;\n\n    p
oints := [[x1, y1], [x2, y2], [x3, y3], [x4, y4]];\n \n    eqn1 := (1 \+
- t)^3*x1 + 3*(1 - t)^2*t*x2 + 3*(1 - t)*t^2*x3 + t^3*x4;\n    eqn2 :=
 (1 - t)^3*y1 + 3*(1 - t)^2*t*y2 + 3*(1 - t)*t^2*y3 + t^3*y4;\n\n    f
a := limit( f(x), x = a, 'right' );\n    fb := limit( f(x), x = b, 'le
ft' );\n    Dfa := limit( diff( f(x), x ), x = a, 'right' );\n    Dfb \+
:= limit( diff( f(x), x ), x = b, 'left' );\n    sgnDfa := signum( abs
( Dfa ) - 1 );\n    sgnDfb := signum( abs( Dfb ) - 1 );\n\n    eqn1 :=
 eval( eqn1, [x1 = a, x4 = b] );\n    eqn2 := eval( eqn2, [y1 = fa, y4
 = fb] );\n\n    points := eval( points, [x1 = a, x4 = b, y1 = fa, y4 \+
= fb] );\n\n    if not sgnDfa::'numeric' then\n        error \"cannot \+
determine the magnitude of the derivative at %1\", a;\n    elif sgnDfa
 <= 0 then\n        eqn2 := eval( eqn2, y2 = fa + Dfa*(x2 - a) );\n   \+
     points := eval( points, [y2 = fa + Dfa*(x2 - a)] );\n        var1
 := x2;\n        val1 := (a + b)/2;\n    else\n        eqn1 := eval( e
qn1, x2 = (y2 - fa)/Dfa + a );\n        points := eval( points, [x2 = \+
(y2 - fa)/Dfa + a] );\n        var1 := y2;\n        val1 := (f(a) + f(
b))/2;\n    end if;\n\n    if not sgnDfb::'numeric' then\n        erro
r \"cannot determine the magnitude of the derivative at %1\", b;\n    \+
elif sgnDfb <= 0 then\n        eqn2 := eval( eqn2, y3 = fb + Dfb*(x3 -
 b) );\n        points := eval( points, [y3 = fb + Dfb*(x3 - b)] );\n \+
       var2 := x3;\n        val2 := (a + b)/2;\n    else\n        eqn1
 := eval( eqn1, x3 = (y3 - fb)/Dfb + b );\n        points := eval( poi
nts, [x3 = (y3 - fb)/Dfb + b] );\n        var2 := y3;\n        val2 :=
 (f(a) + f(b))/2;\n    end if;\n\n    integral := Int( (eqn2 - f(eqn1)
)^2*diff( eqn1, t ), t = 0..1 );\n    pts := GradientDescent( integral
, var1 = val1, var2 = val2 );\n\n    eval( points, pts ), sqrt( evalf(
 eval( integral, pts ) ) ), eval( eqn1, pts ), eval( eqn2, pts );\nend
 proc:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "expr := x*exp(-x)
:\na := 0:\nb := 2:" }}}{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "result := ClampedBestFittingCubicBezier( expr, x = a.
.b ): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 179 "The result is evaluate
d to five decimal digits of precision.  The first list are the points,
 the second is the root mean squared error, and the last two are the p
arametric curves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(
 %, 7 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6&7&7$$\"\"!F&F%7$$\"(fG![!
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*&F4F=FJF=F=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "This plots both t
he expression and the best fitting Bezier curve." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 134 "plots[display]( \n   plot( expr, x = a..b, n
umpoints = 1000 ),\n   plot( [result[-2..-1], t = 0..1], color = blue,
 numpoints = 1000 )\n);" }}{PARA 13 "" 0 "" {TEXT -1 0 "" }}{PARA 13 "
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