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Notebook[{

Cell[CellGroupData[{
Cell["Variational Schemes for Bounded Domains", "Chapter",
  CounterAssignments->{{"Chapter", 5}},
  CellTags->"CHA variational"],

Cell[TextData[{
  "To evaluate a particular cell, simply hit \"Shift+Enter\" with the cursor \
positioned in that cell.  If ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " asks to evaluate initialization cell, click \"Yes\".  See the ",
  StyleBox["Mathematica ",
    FontSlant->"Italic"],
  "Help browser or wolfram.com for more information on using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "."
}], "Text"],

Cell["\<\
This notebook and those for the other seven chapters use a common stylesheet \
that has been imported into the notebook.  This style sheet supports \
chapters, section, subsections, numbered equations, numbered figures and \
citations.  A separate copy of the stylesheet can be download from the same \
page where this notebook was downloaded.  \
\>", "Text"],

Cell[TextData[{
  StyleBox["COPYRIGHT ISSUES:",
    FontWeight->"Bold"],
  "  The authors reserve all copyrights associated with this work.  Any of \
the material appearing in these notebooks (such as the polyhedral meshes in \
chapter 7) can be used and modified without restriction as long as the use is \
non\[Hyphen]commercial.  We simply ask that you acknowledge the authors when \
using material from these notebooks. For those readers interested in \
commercial use of the material in these notebooks, please contact \
jwarren@cs.rice.edu."
}], "Text"],

Cell[CellGroupData[{

Cell["Helper functions from previous chapters", "Subsubsection"],

Cell[BoxData[
    \(plotCoeff[p_, opts___] := 
      Show[Graphics[{Line[p], PointSize[0.02], RGBColor[1, 0, 0], 
            Map[Point, p]}], 
        Join[{opts}, {Axes \[Rule] True, 
            Ticks \[Rule] {Automatic, Automatic}}]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(makeCoeff[genFun_, {{minX_, maxX_}, {minY_, maxY_}}, k_, x_, y_] := 
      With[{expFun = Expand[genFun]}, \n\t
        Table[With[{rowY = 
                Table[Coefficient[expFun, y, 
                    j], {j, \(2\^k\) minY, \(2\^k\) maxY}]}, \n\t\t\t\tTable[
              Coefficient[rowY, x, 
                i], {i, \(2\^k\) minX, \(2\^k\) maxX}]]]]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(plotCoeff3D[p_, plotArgs___] := 
      Show[Graphics3D[\[IndentingNewLine]Table[
            Polygon[{p\[LeftDoubleBracket]i, j\[RightDoubleBracket], 
                p\[LeftDoubleBracket]i + 1, j\[RightDoubleBracket], 
                p\[LeftDoubleBracket]i + 1, j + 1\[RightDoubleBracket], 
                p\[LeftDoubleBracket]i, 
                  j + 1\[RightDoubleBracket]}], {i, \(Dimensions[
                    p]\)\[LeftDoubleBracket]1\[RightDoubleBracket] - 
                1}, {j, \(Dimensions[
                    p]\)\[LeftDoubleBracket]2\[RightDoubleBracket] - 1}]], 
        Join[{plotArgs}, {Axes \[Rule] True}]]\)], "Input",
  InitializationCell->True]
}, Closed]],

Cell[CellGroupData[{

Cell["Inner products for stationary subdivision schemes", "Section"],

Cell[CellGroupData[{

Cell["Exact derivatives", "Subsection"],

Cell[CellGroupData[{

Cell["Compute exact interpolation mask for uniform schemes", "Subsubsection"],

Cell[TextData[{
  "Turn off warning messages from ",
  Cell[BoxData[
      \(Solve\)]]
}], "Text"],

Cell[BoxData[
    \(\(Off[Solve::svars];\)\)], "Input",
  InitializationCell->True],

Cell[TextData[{
  "Computation of left eigenvector of the form ",
  Cell[BoxData[
      \(n\ S \[Equal] n\)]],
  " for uniform subdivision matrix ",
  Cell[BoxData[
      \(S\)]],
  " with mask ",
  Cell[BoxData[
      \(s[x]\)]],
  ". The entries of this vector defined the exact interpolation mask for a \
uniform subdivision scheme.  Convert into equivalent generating function \
expression of the form ",
  Cell[BoxData[
      \(n[x] s[x] \[Equal] n[x\^2] + x*r[x\^2]\)]],
  " where ",
  Cell[BoxData[
      \(r[x]\)]],
  " is an arbitrary residual term.  Normalize via ",
  Cell[BoxData[
      \(n[1] \[Equal] 1\)]],
  ". "
}], "Text"],

Cell[BoxData[
    \(evalMask[s_] := 
      Module[{leng = Exponent[s, x], n, \[ScriptN], 
          eqs}, \[IndentingNewLine]n = 
          Function[x, 
            Sum[\[ScriptN][i] x\^\(i - 1\), {i, 
                leng - 1}]]; \[IndentingNewLine]eqs = 
          CoefficientList[s*n[x] - \ x\ n[x\^2], {x}]; \[IndentingNewLine]n[
            x] /. \(Solve[{eqs\[LeftDoubleBracket]
                    Range[2, Length[eqs], 2]\[RightDoubleBracket] \[Equal] 0, 
                n[1] \[Equal] 
                  1}]\)\[LeftDoubleBracket]1\[RightDoubleBracket]]\)], "Input",\

  InitializationCell->True,
  CellTags->"EQN compute interpolation mask"],

Cell[BoxData[{
    \(\(cubic = \(1\/8\) \((1 + x)\)\^4;\)\), "\[IndentingNewLine]", 
    \(\(fourPt = \(-1\)\/16 + \(9\/16\) x\^2 + 
          x\^3 + \(9\/16\) x\^4 - \(1\/16\) x\^6;\)\)}], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[{
    \(evalMask[cubic]\), "\[IndentingNewLine]", 
    \(evalMask[fourPt]\)}], "Input"],

Cell[BoxData[
    \(1\/6 + \(2\ x\)\/3 + x\^2\/6\)], "Output"],

Cell[BoxData[
    \(x\^2\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Compute exact derivative masks for uniform schemes", "Subsubsection"],

Cell[BoxData[
    \(derivMask[s_, k_] := 
      Expand[\(\((1 - x)\)\^k\) 
          evalMask[Simplify[\(\(2\^k\) s\)\/\((1 + x)\)\^k]]]\)], "Input",
  InitializationCell->True],

Cell["Compute exact masks for cubic B\[Hyphen]splines", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(derivMask[cubic, 0]\), "\[IndentingNewLine]", 
    \(derivMask[cubic, 1]\), "\[IndentingNewLine]", 
    \(derivMask[cubic, 2]\)}], "Input"],

Cell[BoxData[
    \(1\/6 + \(2\ x\)\/3 + x\^2\/6\)], "Output"],

Cell[BoxData[
    \(1\/2 - x\^2\/2\)], "Output"],

Cell[BoxData[
    \(1 - 2\ x + x\^2\)], "Output"]
}, Open  ]],

Cell["Compute exact derivative masks for four point rule", "Text"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(derivMask[fourPt, 0]\), "\[IndentingNewLine]", 
    \(derivMask[fourPt, 1]\)}], "Input"],

Cell[BoxData[
    \(x\^2\)], "Output"],

Cell[BoxData[
    \(\(-\(1\/12\)\) + \(2\ x\)\/3 - \(2\ x\^3\)\/3 + x\^4\/12\)], "Output"]
}, Open  ]]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Exact inner products", "Subsection",
  CellTags->"exact inner prod"],

Cell[CellGroupData[{

Cell["Computing the inner product mask for uniform schemes", "Subsubsection"],

Cell[TextData[{
  "The function ",
  Cell[BoxData[
      \(ipMask\)]],
  " takes two subdivision masks ",
  Cell[BoxData[
      \(s[x]\)]],
  " and ",
  Cell[BoxData[
      \(t[x]\)]],
  " and computes the inner product mask for the ",
  Cell[BoxData[
      \(i\)]],
  "th and ",
  Cell[BoxData[
      \(j\)]],
  "th derivatives of the associated scaling functions."
}], "Text"],

Cell[BoxData[
    \(ipMask[s_, t_, {i_, j_}] := 
      With[{rT = \((t /. {x \[Rule] 1\/x})\) 
              x\^Exponent[t, x]}, \[IndentingNewLine]Expand[\(\((1 - 
                    x)\)\^i\) \(\((x - 1)\)\^j\) 
            evalMask[
              Simplify[\(1\/2\) \(\(\(2\^i\) 
                        s\)\/\((1 + x)\)\^i\) \(\(2\^j\) rT\)\/\((1 + \
x)\)\^j]]]]\)], "Input",
  InitializationCell->True],

Cell[CellGroupData[{

Cell[BoxData[
    \(ipMask[cubic, cubic, {0, 0}]\)], "Input"],

Cell[BoxData[
    \(1\/5040 + 
      x\/42 + \(397\ x\^2\)\/1680 + \(151\ x\^3\)\/315 + \(397\ x\^4\)\/1680 \
+ x\^5\/42 + x\^6\/5040\)], "Output"]
}, Open  ]]
}, Closed]],

Cell[CellGroupData[{

Cell["Evaluation of inner products for uniform schemes", "Subsubsection"],

Cell[TextData[{
  Cell[BoxData[
      \(evalInner\)]],
  " takes a mask ",
  Cell[BoxData[
      \(e\)]],
  " (as a list of coefficients) and two list of coefficients ",
  Cell[BoxData[
      \(p\)]],
  " and ",
  Cell[BoxData[
      \(q\)]],
  " and returns ",
  Cell[BoxData[
      \(\(p\^T\) E\ q\)]],
  " where ",
  Cell[BoxData[
      \(E\)]],
  " is the circulant matrix defined by mask ",
  Cell[BoxData[
      \(e\)]]
}], "Text"],

Cell[BoxData[
    \(evalInner[e_, p_, q_] := 
      With[{ec = CoefficientList[e, x]}, \[IndentingNewLine]q . 
          ListConvolve[ec, p, \(Length[ec] + 1\)\/2]]\)], "Input",
  InitializationCell->True]
}, Closed]]
}, Closed]],

Cell[CellGroupData[{

Cell["Example: Exact enclosed area for parametric curves", "Subsection"],

Cell[CellGroupData[{

Cell["Exact enclosed area for the four\[Hyphen]point curves", "Subsubsection"],

Cell[TextData[{
  "Given a closed parametric curves ",
  Cell[BoxData[
      \({\[ScriptP][\[ScriptX]], \[ScriptQ][\[ScriptX]]}\)]],
  ", the length of this curve correponds to the integral"
}], "Text"],

Cell[BoxData[
    RowBox[{\(1\/2\), 
      RowBox[{\(\[Integral]\_\[CapitalOmega]\), 
        RowBox[{
          RowBox[{"Det", "[", 
            RowBox[{"(", GridBox[{
                  {\(\[ScriptP][\[ScriptX]]\), \(\(\[ScriptP]\^\[Prime]\)[\
\[ScriptX]]\)},
                  {\(\[ScriptQ][\[ScriptX]]\), \(\(\[ScriptQ]\^\[Prime]\)[\
\[ScriptX]]\)}
                  }], ")"}], 
            "]"}], \(\(\[DifferentialD]\[ScriptX]\)\(\ \)\(.\)\)}]}]}]], \
"Equation"],

Cell[TextData[{
  "For example, the area of circle of radius ",
  Cell[BoxData[
      \(r\)]],
  " is"
}], "Text"],

Cell[BoxData[
    \(circle[\[Theta]_] := r {Cos[\[Theta]], Sin[\[Theta]]}\)], "Input"],

Cell[BoxData[
    RowBox[{\(area[\[ScriptP]_, {min_, max_}]\), ":=", 
      RowBox[{\(1\/2\), 
        RowBox[{\(\[Integral]\_min\%max\), 
          RowBox[{
            RowBox[{"Det", "[", 
              RowBox[{"{", 
                RowBox[{\(\[ScriptP][\[ScriptX]]\), ",", 
                  RowBox[{
                    SuperscriptBox["\[ScriptP]", "\[Prime]",
                      MultilineFunction->None], "[", "\[ScriptX]", "]"}]}], 
                "}"}], "]"}], \(\[DifferentialD]\[ScriptX]\)}]}]}]}]], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(area[circle, {0, 2  \[Pi]}]\)], "Input"],

Cell[BoxData[
    \(\[Pi]\ r\^2\)], "Output"]
}, Open  ]],

Cell["Compute enclosed area mask for cubic B\[Hyphen]spline", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(cubicArea = ipMask[cubic, cubic, {0, 1}]\)], "Input"],

Cell[BoxData[
    \(\(-\(1\/720\)\) - \(7\ x\)\/90 - \(49\ x\^2\)\/144 + \(49\ x\^4\)\/144 \
+ \(7\ x\^5\)\/90 + x\^6\/720\)], "Output"]
}, Open  ]],

Cell["\<\
Define exact unit square for four point scheme via five fold control points \
and compute its enclosed area\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(evalInner[
      cubicArea, \[IndentingNewLine]{0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 
        1, 1, 0, 0, 0, 0, 0}, \[IndentingNewLine]{0, 0, 0, 0, 0, 0, 0, 0, 0, 
        0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]\)], "Input"],

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

Cell["\<\
Compute approximate area of unit circle, note that approximation leads to \
substantial error\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(evalInner[
      cubicArea, \[IndentingNewLine]Table[Cos[\(2  \[Pi]\ i\)\/80], {i, 80}] // 
        N, \[IndentingNewLine]Table[Sin[\(2  \[Pi]\ i\)\/80], {i, 80}] // 
        N]\)], "Input"],

Cell[BoxData[
    \(3.135139317714538`\)], "Output"]
}, Open  ]],

Cell["Compute enclosed area mask for four point rule", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(fourPtArea = ipMask[fourPt, fourPt, {0, 1}]\)], "Input",
  CellTags->"fourPtArea"],

Cell[BoxData[
    \(1\/665280 + \(4\ x\)\/10395 - \(481\ x\^2\)\/73920 + \(731\ \
x\^3\)\/6930 - \(3659\ x\^4\)\/5280 + \(3659\ x\^6\)\/5280 - \(731\ \
x\^7\)\/6930 + \(481\ x\^8\)\/73920 - \(4\ x\^9\)\/10395 - 
      x\^10\/665280\)], "Output"]
}, Open  ]],

Cell["\<\
Define exact unit square for four point scheme via five fold control points, \
check normalization\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(evalInner[
      fourPtArea, \[IndentingNewLine]{0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 
        1, 1, 0, 0, 0, 0, 0}, \[IndentingNewLine]{0, 0, 0, 0, 0, 0, 0, 0, 0, 
        0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}]\)], "Input"],

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

Cell["\<\
Note much better accuracy of approximation of interpolating scheme for circle\
\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(ListPlot[
      Table[{n, 
          evalInner[
            fourPtArea, \[IndentingNewLine]Table[
                Cos[\(2  \[Pi]\ i\)\/n], {i, n}] // 
              N, \[IndentingNewLine]Table[Sin[\(2  \[Pi]\ i\)\/n], {i, n}] // 
              N]}, {n, 8, 20}], PlotJoined \[Rule] True, 
      PlotRange \[Rule] All]\)], "Input",
  CellTags->"FIG circle area"],

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Cell[CellGroupData[{

Cell["Solve for inner product matrix", "Subsubsection"],

Cell[TextData[{
  "Construct inner product matrix ",
  Cell[BoxData[
      \(E\)]],
  " from finite element basis with subdivision matrix ",
  Cell[BoxData[
      \(S\&^\)]],
  " for the finite element basis functions.  In particular, we are interested \
in the inner product matrix ",
  Cell[BoxData[
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  " such that"
}], "Text"],

Cell[BoxData[
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\[ScriptN]\^\((2)\)\)[\[ScriptX] - i] \(\[ScriptN]\^\((2)\)\)[\[ScriptX] - 
              j] \[DifferentialD]\[ScriptX]\)], "NumberedEquation"],

Cell[TextData[{
  "Since inner product uses second derivatices, use ",
  Cell[BoxData[
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  " quadratic B\[Hyphen]splines with local subdivision matrix ",
  Cell[BoxData[
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}], "Text"],

Cell[BoxData[
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                {"0", "3", "1"},
                {"0", "1", "3"},
                {"0", "0", "3"},
                {"0", "0", "1"}
                }], ")"}]}]}], ";"}]], "Input"],

Cell[TextData[{
  "Define rules for creating unknown entries in inner product matrix.  The \
function ",
  Cell[BoxData[
      \(idx\)]],
  " numbers entries of unknown matrix ",
  Cell[BoxData[
      \(unkE\)]],
  ". Construct energy matrix with unknowns, allow for two rows of special \
ruls (i.e. ",
  Cell[BoxData[
      \(i\)]],
  " and ",
  Cell[BoxData[
      \(j\)]],
  " start ranging from ",
  Cell[BoxData[
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Cell[CellGroupData[{

Cell[BoxData[
    \(\((unkE = 
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\[IndentingNewLine]idx[\(-2\), \(-1\)] = \(-2\); \
\[IndentingNewLine]idx[\(-1\), \(-2\)] = \(-3\); \
\[IndentingNewLine]idx[\(-2\), \(-2\)] = \(-4\); \[IndentingNewLine]idx[i_, 
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                3}]])\) // MatrixForm\)], "Input"],

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  Cell[BoxData[
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              e[2] \[Equal] 1}]\)\[LeftDoubleBracket]1\[RightDoubleBracket] // 
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            {"0", "0", "0", "1", \(-4\), "6"}
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Cell["Implementation of finite inner product matrix", "Subsubsection"],

Cell[TextData[{
  "Define a function ",
  Cell[BoxData[
      \(EE[k, n]\)]],
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  Cell[BoxData[
      \(\([0, n]\)\)]],
  " for the resolution ",
  Cell[BoxData[
      \(\(1\/2\^k\) \[DoubleStruckCapitalZ]\)]],
  "."
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Cell[BoxData[
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        n_] := \[IndentingNewLine]With[{diff = 
            4\^k*With[{row = 
                    Join[{1, \(-2\), 1}, 
                      Table[0, {\(2\^k\) n - 2}]]}, \n\t\t\t\tTable[
                  RotateRight[row, i], {i, 
                    0, \(2\^k\) n - 2}]]}, \(\((1\/2)\)\^k\) 
          Transpose[diff] . diff]\)], "Input",
  InitializationCell->True],

Cell[BoxData[
    \(General::"spell1" \(\(:\)\(\ \)\) 
      "Possible spelling error: new symbol name \"\!\(row\)\" is similar to \
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Cell[BoxData[
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            {"0", "0", "0", "0", "1", \(-2\), "1"}
            }], "\[NoBreak]", ")"}],
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        MatrixForm[ BoxForm`e$]]]], "Output"]
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}, Open  ]],

Cell[CellGroupData[{

Cell["A multi\[Hyphen]scale relation for natural cubic splines", "Subsection"],

Cell["See end of next section for plot of figure 6.4", "Text"]
}, Open  ]],

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Cell["Subdivision rules for natural cubic splines", "Subsection",
  CellTags->"SUBSEC compute S"],

Cell[CellGroupData[{

Cell["Solve for subdivision matrix", "Subsubsection"],

Cell[TextData[{
  "Proceed to use variational approach for construct natural cubic splines \
via ",
  Cell[BoxData[
      \(E . S \[Equal] \(1\/8\) U . E\)]],
  " where ",
  Cell[BoxData[
      \(U\)]],
  " is upsampling matrix.  "
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\((truncE = \(EE[0, 6]\)\[LeftDoubleBracket]{1, 2, 3, 4, 
              5}\[RightDoubleBracket])\) // MatrixForm\)], "Input"],

Cell[BoxData[
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            {"1", \(-2\), "1", "0", "0", "0", "0"},
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            {"1", \(-4\), "6", \(-4\), "1", "0", "0"},
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            {"0", "0", "1", \(-4\), "6", \(-4\), "1"}
            }], "\[NoBreak]", ")"}],
      Function[ BoxForm`e$, 
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Cell[TextData[{
  "Define function that upsamples a list ",
  Cell[BoxData[
      \(l\)]],
  "."
}], "Text"],

Cell[BoxData[
    \(upsample[l_] := 
      Table[If[OddQ[i], 
          l\[LeftDoubleBracket]\(i + 1\)\/2\[RightDoubleBracket], 
          0*l\[LeftDoubleBracket]1\[RightDoubleBracket]], {i, 
          2  Length[l] - 1}]\)], "Input",
  InitializationCell->True,
  CellTags->"EQN upsampling"],

Cell[TextData[{
  "Define upsampled version of ",
  Cell[BoxData[
      \(truncE\)]],
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Cell[CellGroupData[{

Cell[BoxData[
    \(\((uE = 
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            }], "\[NoBreak]", ")"}],
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Cell[TextData[{
  "Define indexing rules for unknown entries in ",
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  ".  Note that I used different rules than for ",
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Cell[BoxData[
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\[IndentingNewLine]idx[\(-3\), \(-1\)] = \(-3\); \
\[IndentingNewLine]idx[\(-3\), \(-3\)] = \(-4\); \
\[IndentingNewLine]idx[\(-2\), \(-3\)] = \(-2\); \[IndentingNewLine]idx[i_, 
                j_] := Abs[j - i]; \[IndentingNewLine]Table[
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  "Solve for subdivision matrix ",
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Cell[BoxData[
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Cell[TextData[{
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Cell["Plot figure 6.4", "Text"],

Cell[CellGroupData[{

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\[IndentingNewLine]PlotRange \[Rule] {\(-15\), 
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