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MathGroup Archive 1998

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Re: Symbolic Orthogonalisation of Function spaces


  • To: mathgroup@smc.vnet.net
  • Subject: [mg10709] Re: Symbolic Orthogonalisation of Function spaces
  • From: Paul Abbott <paul@physics.uwa.edu.au>
  • Date: Fri, 30 Jan 1998 04:24:44 -0500
  • Organization: University of Western Australia
  • References: <6ahqdq$po6@smc.vnet.net>

Tommy Nordgren wrote:

> Do anyone out there know about a package for efficient symbolic
> Gram-Schmidt orthogonalisation. The package bundled with Mathematica
> 2.2 performs this very inefficiently, among other things because the
> generated functions is not reduced to simples fotm. My need is to
> generate the set of polynomials that are orthogonal with respect to
> integration over the volume of the unit sphere, but I will eventually
> need to do this for other 3-d spaces.

Note that SphericalHarmonics are built-in (orthonormal on surface of the
unit sphere) and that by a suitable choice of coordinates, many other
spaces can be separated and then you can use the wide range of built-in
orthogonal polynomials.  E.g., to generate the set of polynomials that
are orthogonal with respect to integration over the volume of the unit
sphere, spherical coordinates are preferable.

I have appended below a Mathematica 3.0 Notebook which generates the set
of polynomials that are orthogonal with respect to integration over the
(volume of the) unit cube.  The basic code would run in 2.2 but,
because Mathematica 3.0 has an AngleBracket and DoubleBracketingBar,
the notation is much cleaner in 3.0.

Cheers,
	Paul 

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul@physics.uwa.edu.au  AUSTRALIA                            
http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________

Notebook[{

Cell[CellGroupData[{
Cell["Gram-Schmidt Orthogonalization", "Section"],

Cell[TextData[{
  "See the",
  StyleBox[" Mathematica ",
    FontSlant->"Italic"],
  "Journal ",
  StyleBox["1",
    FontWeight->"Bold"],
  "(4): 45, ",
  StyleBox["2",
    FontWeight->"Bold"],
  "(4): 35."
}], "Text"],

Cell[TextData[{
  "Starting with the set ",
  Cell[BoxData[
      \(TraditionalForm\`{u\_1, u\_2, \[Ellipsis], u\_n}\)]],
  ", along with the ",
  StyleBox["projection operator",
    FontSlant->"Italic"]
}], "Text",
  CellTags->"projection operator"],

Cell[BoxData[
    \(TraditionalForm
    \`Projection(f_, g_) := 
      Simplify[f - \[LeftAngleBracket]f, g\[RightAngleBracket]\ g]\)], 
  "Input",
  AspectRatioFixed->True],

Cell[TextData[{
  "where ",
  Cell[BoxData[
      \(TraditionalForm\`\[LeftAngleBracket]f,
g\[RightAngleBracket]\)]],
  " is any suitable ",
  StyleBox["inner product",
    FontSlant->"Italic"],
  ", and ",
  StyleBox["norm",
    FontSlant->"Italic"]
}], "Text",
  CellTags->{"inner product", "norm"}],

Cell[BoxData[
    \(TraditionalForm
    \`\[LeftDoubleBracketingBar]f_\[RightDoubleBracketingBar] := 
      \@\[LeftAngleBracket]f, f\[RightAngleBracket]\)], "Input"],

Cell[TextData[{
  "Gram-Schmidt orthogonalization produces the ",
  StyleBox["orthonormal set",
    FontSlant->"Italic"],
  " ",
  Cell[BoxData[
      \(TraditionalForm\`{v\_1, v\_2, \[Ellipsis], v\_n}\)]],
  " by the iterative construction: ",
  Cell[BoxData[
      \(TraditionalForm\`v\_1 = u\_1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm
      \`v\_1 = v\_1
           
\/\[LeftDoubleBracketingBar]v\_1\[RightDoubleBracketingBar]\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm
      \`v\_2 = u\_2 - 
          \[LeftAngleBracket]u\_2, v\_1\[RightAngleBracket] v\_1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`\(\ 
      v\_2 = v\_2
            \/\[LeftDoubleBracketingBar]v\_2
              \[RightDoubleBracketingBar]\)\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm
      \`v\_3 = u\_3 - 
          \[LeftAngleBracket]u\_3, v\_1\[RightAngleBracket] v\_1 - 
          \[LeftAngleBracket]u\_3, v\_2\[RightAngleBracket] v\_2\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm
      \`\(v\_3 = 
        v\_3\/\[LeftDoubleBracketingBar]v\_3\[RightDoubleBracketingBar]\

      \)\)]],
  ", and so on.  Gram-Schmidt orthogonalization can be used to construct
an \
orthonormal set of vectors, polynomials, or functions." }], "Text",
  CellTags->{"Gram-Schmidt", "orthonormal set", "iterative
construction"}],

Cell["\<\
The above sequence of operations can be recognized as the pair-wise \
\"folding\" of the projection operator, followed by normalization. This
\
functional representation, leads to a neat and powerful implementation
of \
Gram-Schmidt orthogonalization using functional programming:\ \>",
"Text",
  CellTags->{"orthogonalization", "projection", "functional
programming"}],

Cell[BoxData[
    \(TraditionalForm
    \`\(GramSchmidt := 
      Fold[Append[#, 
             
\((#\/\[LeftDoubleBracketingBar]#\[RightDoubleBracketingBar]&)
                  \)[Fold[Projection, #2, #1]]]&, {}, #]&; \)\)],
"Input"],

Cell[TextData[{
  "The heart of ",
  Cell[BoxData[
      FormBox[
        StyleBox["GramSchmidt",
          "Input"], TraditionalForm]]],
  " is the iterative projection of each vector \[LongDash] implemented
using \
the functional operation ",
  Cell[BoxData[
      FormBox[
        StyleBox["Fold",
          "Input"], TraditionalForm]]],
  ".  The ",
  Cell[BoxData[
      FormBox[
        StyleBox["LinearAlgebra`Orthogonalization`",
          "Input"], TraditionalForm]]],
  " package contains a similar routine for Gram-Schmidt
orthogonalization."
}], "Text",
  CellTags->{"Fold", "LinearAlgebra`Orthogonalization`"}],

Cell["To turn the monomials", "Text",
  CellTags->{"Polynomials", "Interval", "Weight function", "Inner
product"}],

Cell[BoxData[
    \(TraditionalForm
    \`\(u = {1, x, y, z, x\^2, y\^2, z\^2, x\ y, x\ z, y\ z, x\ y\ z};
\)\)], 
  "Input"],

Cell["\<\
 into an orthonormal set over the unit cube with unit weight \ function,
we define the inner product:\ \>", "Text"],

Cell[BoxData[
    \(TraditionalForm
    \`\(\[LeftAngleBracket]f_, g_\[RightAngleBracket] := 
      Simplify[\[Integral]\_0\%1
            \(\[Integral]\_0\%1
              \(\[Integral]\_0\%1 Expand[f\ g] \[DifferentialD]x 
                \[DifferentialD]y \[DifferentialD]z\)\)]; \)\)],
"Input"],

Cell["Gram-Schmidt orthonormalisation yields the polynomials", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`\((v = GramSchmidt[u])\) // Timing\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{10.0900000000000322`\ Second, {1, 2\ \@3\ \((x - 1\/2)\), 
        2\ \@3\ \((y - 1\/2)\), 2\ \@3\ \((z - 1\/2)\), 
        6\ \@5\ \((x\^2 - x + 1\/6)\), 6\ \@5\ \((y\^2 - y + 1\/6)\), 
        6\ \@5\ \((z\^2 - z + 1\/6)\), 3\ \((2\ x - 1)\)\ \((2\ y -
1)\), 
        3\ \((2\ x - 1)\)\ \((2\ z - 1)\), 3\ \((2\ y - 1)\)\ \((2\ z -
1)\), 
        3\ \@3\ \((2\ x - 1)\)\ \((2\ y - 1)\)\ \((2\ z - 1)\)}}\)], 
  "Output"]
}, Open  ]],

Cell[TextData[{
  "We can speed this up by computing the closed-form for an ",
  StyleBox["arbtitrary",
    FontSlant->"Italic"],
  " monomial term"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\[Integral]\_0\%1
            \(\[Integral]\_0\%1
              \(\[Integral]\_0\%1\( x\^m\ y\^n\ z\^p\) \[DifferentialD]x

                \[DifferentialD]y \[DifferentialD]z\)\) /. {m \[Rule] m
- i, 
          n \[Rule] n - j, p \[Rule] p - k} // Simplify\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`\(-\(1\/\(\((i - m - 1)\)\ \((j - n - 1)\)\ \((k - p -
1)\)\)\)\)\)], 
  "Output"]
}, Open  ]],

Cell["and then use pattern-matching to compute the inner-product:",
"Text"],

Cell[BoxData[
    FormBox[
      RowBox[{\(\[LeftAngleBracket]f_, g_\[RightAngleBracket]\), ":=", 
        RowBox[{\(Expand[f\ g\ \(x\^i\) \(y\^j\) z\^k]\), "/.", 
          RowBox[{\(\(x\^m_\) \(y\^n_\) z\^p_\), "\[Rule]", 
            
            FormBox[\(-
                \(1\/\(\((i - m - 1)\)\ \((j - n - 1)\)\ 
                      \((k - p - 1)\)\)\)\),
              "TraditionalForm"]}]}]}], TraditionalForm]], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`\((v = GramSchmidt[u])\) // Timing\)], "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`{3.06999999999999317`\ Second, {1, 2\ \@3\ \((x - 1\/2)\), 
        2\ \@3\ \((y - 1\/2)\), 2\ \@3\ \((z - 1\/2)\), 
        6\ \@5\ \((x\^2 - x + 1\/6)\), 6\ \@5\ \((y\^2 - y + 1\/6)\), 
        6\ \@5\ \((z\^2 - z + 1\/6)\), 3\ \((2\ x - 1)\)\ \((2\ y -
1)\), 
        3\ \((2\ x - 1)\)\ \((2\ z - 1)\), 3\ \((2\ y - 1)\)\ \((2\ z -
1)\), 
        3\ \@3\ \((2\ x - 1)\)\ \((2\ y - 1)\)\ \((2\ z - 1)\)}}\)], 
  "Output"]
}, Open  ]],

Cell[TextData[{
  "The set ",
  Cell[BoxData[
      \(TraditionalForm\`v\)]],
  " are orthonormal:"
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`Outer[\[LeftAngleBracket]##\[RightAngleBracket]&, v, v, 1]\)], 
  "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {"1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"},
            {"0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0"},
            {"0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0"},
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            {"0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "1"}
            },
          ColumnAlignments->{Decimal}], ")"}], TraditionalForm]],
"Output"]
}, Open  ]]
}, Open  ]]
}
]



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