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Notebook[{

Cell[CellGroupData[{
Cell["\<\
Illustration of Zeros of Polynomials Orthogonal on the Unit Circle \
with Respect to Weights Having Jump Discontinuities in Higher \
Derivatives\
\>", "Title"],

Cell[CellGroupData[{

Cell["Some Basic Definitions", "Section"],

Cell[TextData[{
  "The following command defines ",
  StyleBox["ell", "Input"],
  " equally spaced angles of discontinuity in the interval (-\[Pi],\[Pi]).  \
In order of increasing angle the angles are indexed by ",
  StyleBox["j", "Input"],
  " varying between ",
  StyleBox["1", "Input"],
  " and ",
  StyleBox["ell", "Input"],
  ".  The set of angles is symmetric about zero."
}], "Text"],

Cell[BoxData[
    \(\(thetajump[ell_Integer, j_Integer] = 
        Pi*\((2*j - 1)\)/ell - Pi;\)\)], "Input"],

Cell[TextData[{
  "The following command defines the weight function \[Phi](\[Theta]) as a \
function of the angle ",
  StyleBox["theta", "Input"],
  " in the interval ending with ",
  StyleBox["thetajump[ell,j]", "Input"],
  ".  The vector ",
  StyleBox["w", "Input"],
  " of length ",
  StyleBox["ell", "Input"],
  " contains the parameters for the weight function, and the integer ",
  StyleBox["k", "Input"],
  " determines the order of the derivative in which the weight will have jump \
discontinuities at the angles ",
  StyleBox["thetajump[ell,j]", "Input"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(phi[ell_Integer, w_, k_Integer, j_Integer, theta_] := 
        1 + Exp[w[\([j]\)]]*
            Abs[Sin[ell*\((theta - thetajump[ell, j])\)/2]]^k;\)\)], "Input"],

Cell[TextData[{
  "The following command defines the Fourier coefficient of the weight \
\[Phi](\[Theta]) with wavenumber ",
  StyleBox["q", "Input"],
  ".  Here ",
  StyleBox["q", "Input"],
  " is allowed to be real so that a limit can be taken to integer values to \
correctly handle resonances."
}], "Text"],

Cell[BoxData[
    \(\(FourierCoeff[ell_Integer, w_, k_Integer, q_]\  := \ 
        Sum[Integrate[
              phi[ell, w, k, j, theta]*Exp[\(-I\)*q*theta], {theta, 
                thetajump[ell, j] - 2*Pi/ell, thetajump[ell, j]}], {j, 1, 
              ell}]/\((2*Pi)\);\)\)], "Input"],

Cell[TextData[{
  "The following command defines the Toeplitz matrix of dimension ",
  StyleBox["n+1", "Input"],
  " corresponding to the weight function \[Phi](\[Theta]) specified by the \
number of discontinuities ",
  StyleBox["ell", "Input"],
  ", the weight parameter vector ",
  StyleBox["w", "Input"],
  ", and the order of the discontinuity ",
  StyleBox["k", "Input"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(ToeplitzMatrix[ell_Integer, w_, k_Integer, n_Integer] := 
        Module[{FC = FourierCoeff[ell, w, k, q], fcs}, 
          fcs = Table[Limit[FC, q \[Rule] m], {m, \(-n\), n}]; 
          Table[fcs[\([j1 - j2 + n + 1]\)], {j1, 0, n}, {j2, 0, 
              n}]];\)\)], "Input"],

Cell["\<\
The following command defines the field V(\[Theta])=-log(\[Phi](\
\[Theta])).\
\>", "Text"],

Cell[BoxData[
    \(\(v[ell_Integer, w_, k_Integer, j_Integer, 
          theta_] := \(-Log[phi[ell, w, k, j, theta]]\);\)\)], "Input"],

Cell["\<\
The following command defines the mean value of V(\[Theta]) to be \
the result of a numerical integration to be carried out when the procedure is \
called.\
\>", "Text"],

Cell[BoxData[
    \(\(v0[ell_Integer, w_, k_Integer] := 
        Sum[NIntegrate[
              v[ell, w, k, j, theta], {theta, thetajump[ell, j] - 2*Pi/ell, 
                thetajump[ell, j]}, WorkingPrecision \[Rule] 50], {j, 1, 
              ell}]/\((2*Pi)\);\)\)], "Input"],

Cell[TextData[{
  "The following command returns a vector of the limiting values of the ",
  StyleBox["k", "Input"],
  "th order derivative of V(\[Theta]) evaluated as the points of \
discontinuity are approached from below."
}], "Text"],

Cell[BoxData[
    \(\(vderiv[ell_Integer, w_, k_Integer] := 
        Table[Limit[D[v[ell, w, k, j, theta], {theta, k}], 
            theta \[Rule] thetajump[ell, j]], {j, 1, ell}];\)\)], "Input"],

Cell[TextData[{
  "The following command uses the previous command to compute the jumps in \
the ",
  StyleBox["k", "Input"],
  "th derivative of V(\[Theta]) at the points of discontinuity."
}], "Text"],

Cell[BoxData[
    \(\(delta[ell_Integer, w_, 
          k_Integer] := \((\((\(-1\))\)^k)\)*RotateLeft[vderiv[ell, w, k]] - 
          vderiv[ell, w, k];\)\)], "Input"],

Cell[TextData[{
  "The following command defines a vector of values of \[CapitalOmega](\
\[Theta]) (the Cauchy transform of V(\[Theta])) at the points ",
  StyleBox["thetajump[ell,j]", "Input"],
  " of discontinuity of the ",
  StyleBox["k", "Input"],
  "th derivative of V(\[Theta]).  The values are obtained by numerical \
integration when the procedure is called."
}], "Text"],

Cell[BoxData[
    \(\(omega[ell_Integer, w_, k_Integer] := \ 
        Table[2*Im[\(-Sum[
                    NIntegrate[
                      I*Exp[I*
                            theta]*\((v[ell, w, k, j2, theta] - 
                              v[ell, w, k, j2, thetajump[ell, j1]])\)/\((Exp[
                                I*theta] - 
                              Exp[I*thetajump[ell, j1]])\), {theta, 
                        thetajump[ell, j2] - 2*Pi/ell, 
                        thetajump[ell, j2]}], {j2, 1, ell}]\)/\((2*Pi*
                    I)\)], {j1, 1, ell}];\)\)], "Input"],

Cell[TextData[{
  "The following command defines a polynomial proportional to the orthogonal \
polynomial of degree ",
  StyleBox["n", "Input"],
  " with respect to the weight \[Phi](\[Theta]) specified by the data ",
  StyleBox["ell ", "Input"],
  "(the number of points of discontinuity), ",
  StyleBox["w ", "Input"],
  "(the vector of weight parameters), and ",
  StyleBox["k ", "Input"],
  "(the order of the derivative having discontinuities).  The returned \
polynomial is a function of the variable ",
  StyleBox["z", "Input"],
  ".  The coefficients are minors of a Toeplitz matrix, which are computed by \
first scaling the Toeplitz matrix elements by ",
  StyleBox["Exp[v0[ell,w,k]]", "Input"],
  ", which stabilizes the calculation with respect to numerical overflow or \
underflow according to the strong Szego limit theorem for Toeplitz \
determinants.  The calculation of the matrix elements is done to 50 digits of \
accuracy."
}], "Text"],

Cell[BoxData[
    \(\(scaledpi[ell_Integer, w_, k_Integer, n_Integer, z_] := 
        Module[{t = ToeplitzMatrix[ell, w, k, n]}, 
          Sum[\((\((\(-1\))\)^\((n - p)\))\)*\((z^p)\)*
              Det[N[Exp[v0[ell, w, k]]*
                    t[\([Table[m, {m, 1, n}], 
                        Join[Table[m, {m, 1, p}], 
                          Table[m, {m, p + 2, n + 1}]]]\)], 50]], {p, 0, 
              n}]];\)\)], "Input"],

Cell[TextData[{
  "The following command defines a constant multiple of the rational function \
whose zeros inside the unit circle are possible points of attraction for \
spurious zeros of the orthogonal polynomial of degree ",
  StyleBox["n", "Input"],
  "."
}], "Text"],

Cell[BoxData[
    \(\(scaledf[ell_Integer, w_, k_Integer, n_Integer, z_] := 
        Module[{deltavec = delta[ell, w, k], omegavec = omega[ell, w, k]}, 
          Sum[\[IndentingNewLine]deltavec[\([j]\)]*
              Exp[I*\((n + 1)\)*thetajump[ell, j]]*
              Exp[I*omegavec[\([j]\)]]/\((Exp[I*thetajump[ell, j]] - 
                    z)\), {j, 1, ell}]];\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Plot Generating Program", "Section"],

Cell[TextData[{
  "This code fragment takes as input the number ",
  StyleBox["ell", "Input"],
  " of discontinuities of the derivative of order ",
  StyleBox["k", "Input"],
  " of the weight, which is parameterized by a real vector ",
  StyleBox["w", "Input"],
  " of length ",
  StyleBox["ell", "Input"],
  ", and also the degree ",
  StyleBox["n", "Input"],
  " of the polynomial of interest.\n\nThis module computes the orthogonal \
polynomial of degree ",
  StyleBox["n", "Input"],
  " with respect to the weight \[Phi](\[Theta]) on the unit circle specified \
by the remaining parameters.  The roots of this polynomial are then found \
numerically, and a graphics object called ",
  StyleBox["pidots", "Input"],
  " is created to store them drawn as small black dots for subsequent \
display.  A graphics object called ",
  StyleBox["unitcircle", "Input"],
  " is created storing the unit circle (drawn in black) with exterior tick \
marks indicating the angles ",
  StyleBox["thetajump[ell,j]", "Input"],
  " of discontinuity of the ",
  StyleBox["k", "Input"],
  "th derivative of the weight.  A graphics object called ",
  StyleBox["limitcircle", "Input"],
  " contains the circle (drawn in green) to which the majority of the zeros \
are supposed to be attracted as the degree grows.  A graphics object called \
",
  StyleBox["zerofree", "Input"],
  " contains the circle (drawn in red) outside of which there should be no \
zeros of the polynomial (asymptotically as the degree grows).  The roots of \
the rational function whose roots are supposed to characterize the spurious \
zeros are computed and stored as lavender dots in a graphics object called ",
  StyleBox["fdots", "Input"],
  ".  Finally, the graphics objects are all rendered together to give a \
complete picture of the zeros of the polynomial and their relation to various \
asymptotic quantities of interest.  This image is the output of the routine."
}], "Text"],

Cell[BoxData[
    \(\(zeroplot[ell_, w_, k_, n_] := 
        Module[{polyroots = 
              z /. NSolve[scaledpi[ell, w, k, n, z] \[Equal] 0, z], 
            polyrootsre, polyrootsim, 
            spuriousroots = 
              z /. NSolve[scaledf[ell, w, k, n, z] \[Equal] 0, z], 
            spuriousrootsre, spuriousrootsim, pidots, unitcircle, tickmarks, 
            limitcircle, fdots}, \[IndentingNewLine]polyrootsre = 
            Re[polyroots]; \[IndentingNewLine]polyrootsim = 
            Im[polyroots]; \[IndentingNewLine]spuriousrootsre = 
            Re[spuriousroots]; \[IndentingNewLine]spuriousrootsim = 
            Im[spuriousroots]; \[IndentingNewLine]pidots = 
            ListPlot[Transpose[{polyrootsre, polyrootsim}], 
              AspectRatio \[Rule] 1, DisplayFunction \[Rule] Identity, 
              PlotStyle \[Rule] 
                PointSize[1/68]]; \[IndentingNewLine]unitcircle = 
            ParametricPlot[{Cos[theta], Sin[theta]}, {theta, 0, 2*Pi}, 
              DisplayFunction \[Rule] 
                Identity]; \[IndentingNewLine]tickmarks = 
            Table[ParametricPlot[{r*Cos[thetajump[ell, j]], 
                  r*Sin[thetajump[ell, j]]}, {r, 1, 1.05}, 
                DisplayFunction \[Rule] Identity], {j, 1, 
                ell}]; \[IndentingNewLine]limitcircle\  = \ 
            ParametricPlot[{\((1 - \((k + 1)\)*Log[n]/n)\)*
                  Cos[theta], \((1 - \((k + 1)\)*Log[n]/n)\)*
                  Sin[theta]}, {theta, 0, 2*Pi}, PlotStyle \[Rule] Hue[1/3], 
              DisplayFunction \[Rule] 
                Identity]; \[IndentingNewLine]zerofree = 
            ParametricPlot[{\((1 - k*Log[n]/n)\)*
                  Cos[theta], \((1 - k*Log[n]/n)\)*Sin[theta]}, {theta, 0, 
                2*Pi}, PlotStyle \[Rule] Hue[0], 
              DisplayFunction \[Rule] Identity]; \[IndentingNewLine]fdots = 
            ListPlot[Transpose[{spuriousrootsre, spuriousrootsim}], 
              PlotStyle \[Rule] {Hue[2/3, 1/3, 1], PointSize[0.04]}, 
              DisplayFunction \[Rule] Identity]; \ 
          Show[unitcircle, tickmarks, limitcircle, zerofree, fdots, pidots, 
            AspectRatio \[Rule] 1, 
            PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, 
            Axes \[Rule] False, Frame \[Rule] False, 
            DisplayFunction \[Rule] $DisplayFunction]];\)\)], "Input"],

Cell[TextData[{
  "Here is an example of its use.  Note that the previous definitions must be \
executed before ",
  StyleBox["zeroplot[]", "Input"],
  " is called."
}], "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(zeroplot[3, {1/2, \(-1\)/2, 0}, 2, 51]\)], "Input"],

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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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Mistroke
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Mfstroke
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Mistroke
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Mistroke
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Cell[CellGroupData[{

Cell["A Version Without The Zeros of the Polynomial", "Section"],

Cell[TextData[{
  "This is the same as ",
  StyleBox["zeroplot[]", "Input"],
  ", but it does not calculate the orthogonal polynomial or its zeros.  This \
routine is useful for locating the possible spurious zeros, as it is \
significantly faster than ",
  StyleBox["zeroplot[]", "Input"],
  " when ",
  StyleBox["n", "Input"],
  " is large."
}], "Text"],

Cell[BoxData[
    \(\(spuriousplot[ell_Integer, w_, k_Integer, 
          n_Integer] := \[IndentingNewLine]Module[{spuriousroots = 
              z /. NSolve[scaledf[ell, w, k, n, z]], spuriousrootsre, 
            spuriousrootsim, unitcircle, tickmarks, limitcircle, fdots}, \ 
          spuriousrootsre = Re[spuriousroots]; 
          spuriousrootsim = Im[spuriousroots]; \ 
          unitcircle = 
            ParametricPlot[{Cos[theta], Sin[theta]}, {theta, 0, 2*Pi}, 
              DisplayFunction \[Rule] 
                Identity]; \[IndentingNewLine]tickmarks = 
            Table[ParametricPlot[{r*Cos[thetajump[ell, j]], 
                  r*Sin[thetajump[ell, j]]}, {r, 1, 1.05}, 
                DisplayFunction \[Rule] Identity], {j, 1, 
                ell}]; \[IndentingNewLine]limitcircle\  = \ 
            ParametricPlot[{\((1 - \((k + 1)\)*Log[n]/n)\)*
                  Cos[theta], \((1 - \((k + 1)\)*Log[n]/n)\)*
                  Sin[theta]}, {theta, 0, 2*Pi}, PlotStyle \[Rule] Hue[1/3], 
              DisplayFunction \[Rule] 
                Identity]; \[IndentingNewLine]zerofree = 
            ParametricPlot[{\((1 - k*Log[n]/n)\)*
                  Cos[theta], \((1 - k*Log[n]/n)\)*Sin[theta]}, {theta, 0, 
                2*Pi}, PlotStyle \[Rule] Hue[0], 
              DisplayFunction \[Rule] Identity]; \[IndentingNewLine]fdots = 
            ListPlot[Transpose[{spuriousrootsre, spuriousrootsim}], 
              PlotStyle \[Rule] {Hue[2/3, 1/3, 1], PointSize[0.04]}, 
              DisplayFunction \[Rule] Identity]; \ 
          Show[unitcircle, tickmarks, limitcircle, zerofree, fdots, 
            AspectRatio \[Rule] 1, 
            PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, 
            Axes \[Rule] False, Frame \[Rule] False, 
            DisplayFunction \[Rule] $DisplayFunction]];\)\)], "Input"],

Cell[TextData[{
  "Here is an example of its use.  The earlier definitions, with the \
exception of ",
  StyleBox["FourierCoeff[]", "Input"],
  ", ",
  StyleBox["ToeplitzMatrix[]", "Input"],
  ", and ",
  StyleBox["scaledpi[]", "Input"],
  ", must be executed before ",
  StyleBox["spuriousplot[]", "Input"],
  " is called."
}], "Text"],

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Cell[BoxData[
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Mfstroke
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Mfstroke
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