(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 5.0'

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Notebook[{

Cell[CellGroupData[{
Cell["Babbage", "Title",
  FontSize->18],

Cell["\<\
A brief explanation of the method of differences, used in Charles Babbage's \
Difference Engine #2\
\>", "Subtitle",
  FontSize->16],

Cell["\<\
Original Mathematica code by Ed Theren, rewritten by Scott Centoni.\
\>", "Subtitle",
  FontSize->16],

Cell["\<\
The Difference Engines were designs for large machines to automatically print \
out tables of mathematical functions by extrapolation from a small number of \
points, using the method of differences.\
\>", "Text"],

Cell[CellGroupData[{

Cell["initialization", "Section"],

Cell[BoxData[
    \(CleanSlate[]\)], "Input"],

Cell[CellGroupData[{

Cell["imports", "Subsection"],

Cell[BoxData[
    \(Needs["\<Graphics`Graphics`\>"]\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["prologs", "Subsection"],

Cell["Automatically display matrices in MatrixForm.", "Text"],

Cell[BoxData[
    \(\($PrePrint = If[MatrixQ[#], MatrixForm[#], #] &;\)\)], "Input"],

Cell["Improve the appearance of text in graphs.", "Text"],

Cell[BoxData[
    \(\(\(\($TextStyle = {FontFamily \[Rule] "\<Palatino\>", 
          FontSize \[Rule] 12}\) \)\(;\)\(\ \)\(\($FormatType = 
        TraditionalForm\) \)\(;\)\(\ \)\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["misc", "Subsection"],

Cell["\<\
Print non-matrix 2D tables in a format that keeps the items centered on each \
line.\
\>", "Text"],

Cell[BoxData[
    \(CheckerInterleave[x_] := 
      Most@Flatten[\({#, "\<\>"} &\) /@ x, 1]\)], "Input"],

Cell[BoxData[
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                n - Length[x]}]\[IndentingNewLine]}, \[IndentingNewLine]Join[
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          padding\ ]\[IndentingNewLine]]\)], "Input"],

Cell[BoxData[
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            x, \[IndentingNewLine]TableSpacing \[Rule] {4, 
              1}\[IndentingNewLine]]\[IndentingNewLine]]\)], "Input"],

Cell["Pascal's triangle", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Table[Binomial[n, k], {n, 0, 8}, {k, 0, n}]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

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Cell[CellGroupData[{

Cell["method of differences", "Section"],

Cell["\<\
Perhaps the best way of thinking of the Difference Engine is as a machine for \
extrapolating tables of numbers. For instance, if we have a sequence of \
numbers 0,1,8,27,64, we recognize this as a list of the first few cubes. But \
if we did not make use of this information, how could we predict the next \
number? One way to approach this is to look at the successive differences \
from one item to the next:\
\>", "Text"],

Cell[BoxData[
    \(Differences[x_] := Rest[x] - Most[x]\)], "Input"],

Cell[BoxData[
    \(\(seq = {0, 1, 8, 27, 64, 125, 216};\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Differences@%\)], "Input"],

Cell[BoxData[
    \({1, 7, 19, 37, 61, 91}\)], "Output"]
}, Open  ]],

Cell["\<\
Hmm, not quite obvious what the pattern is, so let's repeat the process\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Differences@%\)], "Input"],

Cell[BoxData[
    \({6, 12, 18, 24, 30}\)], "Output"]
}, Open  ]],

Cell["\<\
Aha...well, we can predict the next item in this sequence, but let's do it \
again anyway\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Differences@%\)], "Input"],

Cell[BoxData[
    \({6, 6, 6, 6}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Differences@%\)], "Input"],

Cell[BoxData[
    \({0, 0, 0}\)], "Output"]
}, Open  ]],

Cell["\<\
We can see the pattern more clearly by showing all the differences \
together.\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(dat = NestList[Differences, \ seq, \ 5]\)], "Input"],

Cell[BoxData[
    \({{0, 1, 8, 27, 64, 125, 216}, {1, 7, 19, 37, 61, 91}, {6, 12, 18, 24, 
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}, Open  ]],

Cell[CellGroupData[{

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Cell[BoxData[
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Cell["\<\
Of course, all the higher-order differences are exactly zero, so that taking \
the first item in each list of differences forms a new sequence, 0, 1, 6, 6, \
0, ...\
\>", "Text"],

Cell["\<\
Now, with a bit of thought, it should be evident that this process can be \
reversed: if we have the sequence of coefficients, we can generate the \
sequence of values. That is precisely what the Difference Engine does: One \
inputs the difference coefficients, and cranks the machine to repeatedly add \
them up to obtain the values.  Matrix multiplication is a useful way to \
express this process (though it was not developed until after the Difference \
Engine was designed).\
\>", "Text"],

Cell[BoxData[
    \(crm[x_] := \(crm[x] = 
        Table[If[0 \[LessEqual] j - i < 2, 1, 0], {i, x}, {j, 
            x}]\)\)], "Input"],

Cell[BoxData[
    \(cri[x_] := \(cri[x] = 
        Table[If[i \[LessEqual] j, \((\(-1\))\)\^\(i + j\), 0], {i, x}, {j, 
            x}]\)\)], "Input"],

Cell["\<\
This is the matrix representing cranking through a Difference Engine with 6 \
wheels:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(crm[6]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"1", "1", "0", "0", "0", "0"},
            {"0", "1", "1", "0", "0", "0"},
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            {"0", "0", "0", "1", "1", "0"},
            {"0", "0", "0", "0", "1", "1"},
            {"0", "0", "0", "0", "0", "1"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
This is the matrix that represents cranking that machine backward:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cri[6]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
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            {"0", "0", "0", "0", "0", "1"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
}, Open  ]],

Cell["\<\
If we do first one, then the other, then we get the identity matrix\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(crm[6] . cri[6]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
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      Function[ BoxForm`e$, 
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Cell[BoxData[
    \(Crank[x_] := crm[Length[x]]  x\)], "Input"],

Cell[BoxData[
    \(ReverseCrank[x_] := cri[Length[x]] . x\)], "Input"],

Cell["This basically cranks the machine once:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Crank[{0, 1, 6, 6}]\)], "Input"],

Cell[BoxData[
    \({1, 7, 12, 6}\)], "Output"]
}, Open  ]],

Cell["That is, 0+1=1, 1+6=7, 6+6=12.", "Text"],

Cell["Let's crank the machine again:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Crank@%\)], "Input"],

Cell[BoxData[
    \({8, 19, 18, 6}\)], "Output"]
}, Open  ]],

Cell["So 1+7=8, 7+12=19, 12+6=18.", "Text"],

Cell["Again:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Crank@%\)], "Input"],

Cell[BoxData[
    \({27, 37, 24, 6}\)], "Output"]
}, Open  ]],

Cell["Or, better yet, several steps shown in sequence:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(NestList[Crank, \ {0, 1, 6, 6, 0}, \ 7]\)], "Input"],

Cell[BoxData[
    TagBox[
      RowBox[{"(", "\[NoBreak]", GridBox[{
            {"0", "1", "6", "6", "0"},
            {"1", "7", "12", "6", "0"},
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            {"27", "37", "24", "6", "0"},
            {"64", "61", "30", "6", "0"},
            {"125", "91", "36", "6", "0"},
            {"216", "127", "42", "6", "0"},
            {"343", "169", "48", "6", "0"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
        MatrixForm[ BoxForm`e$]]]], "Output"]
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Cell["\<\
And we see that the first column has reproduced our table of function values, \
plus one more--and we could keep on cranking the machine as long as we \
wanted. In most cases, we're only concerned with the values taken on \
sequentially by the register shown in the first column.\
\>", "Text"],

Cell[BoxData[
    \(CrankTimes[regs_, n_] := First /@ NestList[Crank, regs, n]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CrankTimes[{0, 1, 6, 6, 0}, 20]\)], "Input"],

Cell[BoxData[
    \({0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 
      2744, 3375, 4096, 4913, 5832, 6859, 8000}\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(CrankThrough[regs_] := CrankTimes[regs, Length[regs] - 1]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CrankThrough[{0, 1, 6, 6, 0}]\)], "Input"],

Cell[BoxData[
    \({0, 1, 8, 27, 64}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["using the machine to crank out its own coefficients", "Section"],

Cell["\<\
There is a trick to use the machine to generate its own coefficients from a \
table of regularly-spaced values. If you reverse the sign of every other item \
in the sequence before and after cranking through the list, it just churns \
out the coefficients.\
\>", "Text"],

Cell[BoxData[
    \(Alternate[x_] := x \((\(-1\))\)\^Range[0, Length[x] - 1]\)], "Input"],

Cell[BoxData[
    \(DeCrankThrough[regs_] := 
      Alternate@\(CrankThrough@\(Alternate@regs\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(DeCrankThrough[{0, 1, 8, 27, 64}]\)], "Input"],

Cell[BoxData[
    \({0, 1, 6, 6, 0}\)], "Output"]
}, Open  ]],

Cell[TextData[{
  "If we input ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " numbers as function values, we can output ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " numbers as coefficients without making any assumptions about higher-order \
terms.  If we output more than ",
  Cell[BoxData[
      \(TraditionalForm\`n\)]],
  " numbers, we assume that the higher-order differences are all zero."
}], "Text"],

Cell["Let's look at what happens in the general case:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Array[a, \ 5]\)], "Input"],

Cell[BoxData[
    \({a[1], a[2], a[3], a[4], a[5]}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(DeCrankThrough[Array[a, \ 5]] // TableForm\)], "Input"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {\(a[1]\)},
          {\(\(-a[1]\) + a[2]\)},
          {\(a[1] - 2\ a[2] + a[3]\)},
          {\(\(-a[1]\) + 3\ a[2] - 3\ a[3] + a[4]\)},
          {\(a[1] - 4\ a[2] + 6\ a[3] - 4\ a[4] + a[5]\)}
          },
        RowSpacings->1,
        ColumnSpacings->3,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
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        a[ 1], 
        Plus[ 
          Times[ -1, 
            a[ 1]], 
          a[ 2]], 
        Plus[ 
          a[ 1], 
          Times[ -2, 
            a[ 2]], 
          a[ 3]], 
        Plus[ 
          Times[ -1, 
            a[ 1]], 
          Times[ 3, 
            a[ 2]], 
          Times[ -3, 
            a[ 3]], 
          a[ 4]], 
        Plus[ 
          a[ 1], 
          Times[ -4, 
            a[ 2]], 
          Times[ 6, 
            a[ 3]], 
          Times[ -4, 
            a[ 4]], 
          a[ 5]]}]]], "Output"]
}, Open  ]],

Cell["\<\
This is very similar to the definition of a derivative, but note that the \
derivatives of different orders are being approximated at different points. \
We do recover the original points by cranking through the machine in the \
normal way:\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CrankThrough@%\)], "Input"],

Cell[BoxData[
    \({a[1], a[2], a[3], a[4], a[5]}\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["scaling the step size", "Section"],

Cell["\<\
The technique of using the Difference Engine to generate its own coefficients \
from a sequence of equally-spaced values is of limited practical benefit.  \
It's more useful if we can adjust the coefficients afterward to produce a \
more finely-spaced sequence.\
\>", "Text"],

Cell[TextData[{
  "To begin with, we consider the coefficients that correspond to the simple \
integer powers, ",
  Cell[BoxData[
      \(TraditionalForm\`1, x, x\^2, x\^3,  ... \)]]
}], "Text"],

Cell[BoxData[
    \(\(polycoeff[0] = {1};\)\)], "Input"],

Cell[BoxData[
    \(polycoeff[n_] := DeCrankThrough[Table[k\^n, {k, 0, n}]]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(CheckerForm[Table[polycoeff[k], {k, 0, 8}]]\)], "Input"],

Cell[BoxData[
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}, Open  ]],

Cell["\<\
We'll try the cube function again, but this time sample it with a step of two \
instead of one.\
\>", "Text"],

Cell[BoxData[
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Cell[CellGroupData[{

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Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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Cell["\<\
We see that instead of the coefficient list {0,1,6,6,0,...} we got before, we \
get larger values. But we can't just multiply the coefficients by a constant \
to change the step size.\
\>", "Text"],

Cell["\<\
It turns out that we can multiply the coefficient list by a matrix that will \
scale the step. To scale a 7th-order coefficient list by one half, we \
multiply it on the left by\
\>", "Text"],

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\(7\/256\), \(-\(21\/1024\)\), \(33\/2048\)},
                {"0", 
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Cell["\<\
This recovers the coefficient list we saw before, and the same technique \
works with any function.\
\>", "Text"],

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Cell[TextData[{
  "The general scaling matrix (of the appropriate order) is a function of the \
scaling factor ",
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  "."
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Cell[CellGroupData[{

Cell["polynomial approximation of non-polynomial functions", "Section"],

Cell["\<\
Now, the real point of the Difference Engine is not to tabulate polynomial \
functions for their own sake, but because they can be good approximations of \
non-polynomial functions, like logarithmic and trigonometric functions.  \
There are several ways to come up with the coefficients to supply the \
Difference Engine, and they do not all have the same accuracy.\
\>", "Text"],

Cell[BoxData[
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Cell[BoxData[
    \(MaxAbs[x_] := Max@\(Abs@x\)\)], "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

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Note that Fit works only with machine precision. The more general function \
FindFit allows one to specify precision, though.\
\>", "Text"],

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We see that the rms error and maximum error are quite poor for the Taylor \
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coefficients or least squared error minimization.\
\>", "Text"],

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