(************** Content-type: application/mathematica **************
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Notebook[{

Cell[CellGroupData[{
Cell["\<\
SimpleFit[ ] routine  (a modified version of the standard \
Mathematica Fit[] routine)\
\>", "Subsection",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell["\<\
David Withoff at Wolfram Associates (WRI) has provided me with  a \
modified version of the Mathematica Fit[ ] routine for  doing a least-squares \
fit of a set of basis functions to a  list of data values. This modified \
version, which he named  SimpleFit[], is so useful that I would like to pass \
it along to to others; and so I'm posting it here with his permission.  A \
short illustration of the application of SimpleFit to a \
Hermite-gaussian-squared mode expansion is  given below.

Withoff's alternative to the standard Mathematic Fit[] command is defined as:\
\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(SimpleFit[data_, basis_, vars_] := 
      Block[{purebasis, designmatrix, designinverse, response, 
          fitcoefficients}, 
        purebasis = Function[Evaluate[vars], Evaluate[basis]]; 
        designmatrix = Apply[purebasis, data, 1]; 
        designinverse = PseudoInverse[designmatrix]; response = Last /@ data; 
        fitcoefficients = designinverse . response; 
        fitcoefficients]\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell["\<\

Let's apply this to a test function in which the data points which we want to \
match to be on a broadened unit-area gaussian function with gaussian spot \
size w0=2 between x=0 and x = 3\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(w0 = 2; g[x_] := Sqrt[2. /\((Pi\ w0^2)\)]\ Exp[\(-2\)\ x^2/w0^2]; 
    hgData = Table[{x, N[g[x]]}, {x, 0, 3, 0.3}];\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \({{0, 0.3989422804014327`}, {0.3`, 0.38138781546052414`}, {0.6`, 
        0.33322460289179967`}, {0.8999999999999999`, 
        0.26608524989875487`}, {1.2`, 0.19418605498321295`}, {1.5`, 
        0.12951759566589174`}, {1.7999999999999998`, 
        0.07895015830089418`}, {2.1`, 0.04398359598042719`}, {2.4`, 
        0.0223945302948429`}, {2.6999999999999997`, 
        0.010420934814422605`}, {3.`, 0.0044318484119380075`}}\)], "Output"]
}, Open  ]],

Cell["\<\
We want to match this data to a basis set of Hermite-gaussian \
squared functions h[n,x] in the form

     c[0] h[0,x] + c[1] h[1,x] + c[2] h[2,x]

where the h[n,x] are normalized Hermite-gaussian (HG) squared intensity \
profiles with spot size w = Sqrt[2].  (These correspond to the intensity \
profiles of lowest and higher  order modes in a stable-cavity laser device.)  \
Note that the  HG functions themselves are orthogonal, but the HG-squared  \
functions h[n,x] are not.  These functions as defined as follows:\
\>", "Text",\

  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(dum = Sqrt[1. /N[Pi]];\)\), "\n", 
    \(h[n_, 
        x_] = \((dum/\((2^n\ \(n!\))\))\)\ HermiteH[n, x]^2\ Exp[\(-x^2\)]; 
    h[n, x]\)}], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(\(0.5641895835477563`\ 2\^\(-n\)\ \[ExponentialE]\^\(-x\^2\)\ \
HermiteH[n, x]\^2\)\/\(n!\)\)], "Output"]
}, Open  ]],

Cell["\<\
One way to do this expansion would be to use the Fit[] command from \
Mathematica as follows.  We'll use three basis functions in this example, and \
call the result hgStandardFit:\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(hgBasis = Table[h[n, x], {n, 0, 2}];\)\n\), "\\[IndentingNewLine]", 
    \(hgStandardFit[x_] = Fit[hgData, hgBasis, x]\)}], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(0.37654699770794764`\ \[ExponentialE]\^\(-x\^2\) + 
      0.2579758947125718`\ \[ExponentialE]\^\(-x\^2\)\ x\^2 + 
      0.006311757206902178`\ \[ExponentialE]\^\(-x\^2\)\ \((\(-2\) + 4\ x\^2)\
\)\^2\)], "Output"]
}, Open  ]],

Cell["\<\
The problem with this result is that the expansion is expressed not \
in the normalized functions we want, but rather in terms of the underlying \
algebraic form of the Hermite functions.  As a result the numerical \
coefficients are not the c[n] values that we want.

Applying Withoff's SimpleFit[] command to the same data and basis set,  \
produces a list hgCoefficients which do contain the desired expansion \
coefficients:\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(hgCoefficients = SimpleFit[hgData, hgBasis, x]\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \({0.6674121761343624`, 0.22862518401204707`, 
      0.08949838693883516`}\)], "Output"]
}, Open  ]],

Cell["\<\
These are the expansion coefficients c[n]  that we want.  At least \
in simple cases like this one, applying a normalized basis set to a \
normalized (i.e., unity area) input function as above  should yield a \"total \
power\" given by the sum of the c[n] coefficients which is close to unity \
(though this will not always  happen, especially if some of the c[n] turn out \
to be  negative):\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(pwr = Apply[Plus, hgCoefficients]\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(0.9855357470852447`\)], "Output"]
}, Open  ]],

Cell["\<\

This is not bad, and gets closer to unity if you use more functions in the \
basis set.

The fitting function (call it hgSimpleFit) is then given by the dot product \
of the hgCoefficients and the hgBasis, and has the expanded form:\
\>", "Text",\

  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(hgSimpleFit[x_] = Expand[hgCoefficients . hgBasis]\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(0.4017940265355564`\ \[ExponentialE]\^\(-x\^2\) + 
      0.15698777940213726`\ \[ExponentialE]\^\(-x\^2\)\ x\^2 + 
      0.10098811531043472`\ \[ExponentialE]\^\(-x\^2\)\ x\^4\)], "Output"]
}, Open  ]],

Cell["\<\
To check that this hgSimpleFit result is OK, compare it with the \
expanded hgStandardFit result:\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Expand[hgStandardFit[x]]\)], "Input",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[BoxData[
    \(0.40179402653555635`\ \[ExponentialE]\^\(-x\^2\) + 
      0.15698777940213698`\ \[ExponentialE]\^\(-x\^2\)\ x\^2 + 
      0.10098811531043485`\ \[ExponentialE]\^\(-x\^2\)\ x\^4\)], "Output"]
}, Open  ]],

Cell["\<\
The expanded versions of hgStandardFit and hgSimpleFit agree, as \
they should.\
\>", "Text",
  PageWidth->WindowWidth,
  InitializationCell->True,
  ShowSpecialCharacters->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Plot[{g[x], hgSimpleFit[x]}, {x, 0, 2}];\)\)], "Input"],

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