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Cell[CellGroupData[{
Cell["The Runge -Kutta Method", "Subsection"],

Cell[BoxData[{
    FormBox[
    \(As\ we' ve\ noted\ before, \ 
      Euler' s\ method\ is\ \ a\ fairly\ intuitive\ but\ not\), TextForm], 
    FormBox[
    \(terribly\ efficient\ means\ for\ obtaining\ numerical\ approximations\ 
        for\), TextForm], 
    FormBox[
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        FormBox[\(the\ solution\ of\ an\ initial\ value\ problem\),
          "TextForm"], "\n"}], TextForm], 
    FormBox[
      RowBox[{
        FormBox[\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
          y' \((x)\)\  = \ f \((x, y)\), \ \ \ y \((x\_0)\)\  = \ y\_0\),
          "TextForm"], "\n"}], TextForm], 
    FormBox[
    \(The\ Runge - 
        Kutta\ method\ is\ a\ much\ more\ efficient\ 
          \((and\ much\ less\ \nintuitive)\)\ numerical\ method . \ \ The\ 
          following\ routine\ generates\ an\), TextForm], 
    FormBox[
    \(approximate\ solution\ for\ the\ above\ initial\ value\ problem\ using
        \), TextForm], 
    FormBox[
    \(Runge - 
        Kutta . \ \ \ The\ syntax\ is\ the\ same\ as\ it\ was\ for\ Euler' s\ 
          method\), TextForm], 
    FormBox[
    \(in\ the\ demonstration\ from\ Section\ \(8.1 . \)\), TextForm]}], "Text"],

Cell[BoxData[
    \(RKMethod[f_, {x0_, y0_}, h_, n_]\  := \n\t\n\ \ \ \ \ 
      Module[\n\t\t\ \ \ \ \ \ \ {RKStep}, \n\t\t\n\t\t\ \ \ \ \ \ \ \ 
        RKStep[{x_, y_}]\  := \ \n\t\t\t\ \ \ \ \ \ 
          Module[{k1, k2, k3, k4}, \n
            \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k1\  = \ f[x, y]; 
            \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
            k2\  = \ f[x + h/2, \ y + \ k1*h/2]; \n
            \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
            k3\  = \ f[x + h/2, y + \ k2*h/2]; \n\t\t\t\t\t\t\t\t\t\t\ \ 
            k4\ \  = \ f[x + h, y + k3*h]; \n
            \t\t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
            N[{x + h, y + h*\((k1\  + \ 2*k2\  + \ 2*k3\  + \ k4)\)/6}, 12]\n
            \t\t\t\ \ \ \ \ ]; \n\t\t\n\t\t\ \ \ \ \ \ \ \ 
        Interpolation[\n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
          NestList[RKStep, {x0, y0}, n], \n\t\t\t\ \ \ \ \ \ \ \ \ \ \ \ 
          InterpolationOrder\  -> \ 1\n\t\t\t\ \ \ \ \ ]\n\t\ \ \ \ \ ]\)], 
  "Input"],

Cell[BoxData[{
    \(TextForm\`As\ before, \ 
    we' ll\ try\ out\ our\ method\ on\ the\ initial\ value\ problem\), 
    \(TextForm\`\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
    y' \((x)\)\  = \ y \((x)\), \ \ \ \ \ \ \ \ y \((0)\)\  = \ 1\), 
    \(TextForm
    \`and\ compare\ the\ results\ with\ the\ exact\ solution\ y \((x)\)\  = \ 
      \(e\^x . \)\ \ \ \ \)}], "Text"],

Cell[BoxData[
    \(f[x_, y_]\  = \ y\)], "Input"],

Cell["\<\
With a ridiculously large step size h=1, we obtain the following 
approximation of e.\
\>", "Text"],

Cell[BoxData[
    \(approx\  = \ RKMethod[f, {0, 1}, 1, 1]\)], "Input"],

Cell[BoxData[
    \(approx[1]\)], "Input"],

Cell["\<\
That's two digits of accuracy already!  If we cut the step size by \
a
factor of ten, we obtain\
\>", "Text"],

Cell[BoxData[
    \(approx\  = \ RKMethod[f, {0, 1},  .1, 10]\)], "Input"],

Cell[BoxData[
    \(approx[1]\)], "Input"],

Cell[BoxData[{
    \(TextForm
    \`which\ is\ accurate\ to\ \((at\ least)\)\ 6\ \(places!\)\ \ To\ obtain\ 
      higher\ accuracy, \n
    we' d\ have\ to\ also\ specify\ more\ accuracy\ for\ the\ "N"\ command\ 
      used\), 
    \(TextForm
    \`in\ defining\ RKMethod . \ \ \ It\ turns\ out\ that\ local\ truncation\ 
      error\ is\), 
    \(TextForm
    \`proportional\ to\ \(h\^5\) for\ this\ method\ and\ global\ truncation\ 
      error\ is\), 
    \(TextForm\`proportional\ to\ h\^4 . \ \ \ That\ is, \ 
    for\ every\ factor\ of\ ten\ that\ we\ shrink\), 
    \(TextForm\`h, \ we\ gain\ a\ factor\ of\ 10, 
    000\ \ \((i . e . \ 4\ more\ decimal\ places)\)\), 
    \(TextForm\`in\ accuracy . \ \ \ Not\ \(bad . \)\)}], "Text"],

Cell["\<\
In closing, let's graphically examine the difference 
between the exact and approximate solutions using h=.25 
and x ranging from 0 to 4.\
\>", "Text"],

Cell[BoxData[
    \(approx\  = \ RKMethod[f, {0, 1},  .25, 16]\)], "Input"],

Cell[BoxData[
    \(Plot[{approx[x], \ Exp[x]}, {x, 0, 4}]\)], "Input"],

Cell["...virtually indistinguishable.", "Text"],

Cell[BoxData[
    \(Plot[approx[x] - Exp[x], {x, 0, 4}]\)], "Input"],

Cell["Why do you suppose the difference oscillates so much?", "Text"]
}, Open  ]]
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