Editing Appendix/Ramblings/NumericallyDeterminedEigenvectors (section)

==Varying the Oscillation Frequency==
===Approach===
First, we fix <math>~q</math>, <math>~\gamma_e</math>, and <math>~\gamma_c</math>; in the example, here ([[#Properties_of_21Analytic_Solution|as above]]) we choose:  <math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>.  For this example, we will also retain the constraint, <math>~g^2 = \mathcal{B}</math>, in which case,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~0.3902664 \, .</math>
  </td>
</tr>
</table>
</div>
Next, we pick various values of the (square of the) dimensionless oscillation frequency, <math>~\sigma_c^2</math>, and, in order to ensure that the dimensional frequency in the envelope matches the dimensional frequency of the core, from each value we set,

<div align="center">
<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\mathfrak{F}_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\sigma_c^2 + 8}{\gamma_c} - 6 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{env} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{\gamma_e} \biggl[ \sigma_c^2 \biggl(\frac{\rho_e}{\rho_c} \biggr)^{-1} + 8\biggr]- 6 \, .</math>
  </td>
</tr>
</table>
</div>

For the finite-difference algorithm, we divide the core &#8212; radial coordinate range, <math>~0 \le \xi \le 1</math> &#8212; into N<sub>core</sub> zones, and the envelope &#8212; radial coordinate range, <math>~1\le \xi \le 1/q</math> &#8212; into N<sub>env</sub> zones.  This means that the spacing between successive radial zones in the core and envelope is, respectively,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~\Delta_c \equiv \frac{1}{\mathrm{N}_\mathrm{core}}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\Delta_e \equiv \frac{q^{-1} - 1}{\mathrm{N}_\mathrm{env}} \, .</math>
  </td>
</tr>
</table>
</div>

Starting at the center of the configuration <math>~(\xi = 0)</math>, where we arbitrarily set the value of the eigenfunction to <math>~x_0 = 1</math>, the value of the eigenfunction at the first grid point away from the center <math>~(\xi = \Delta_c)</math> is,
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<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x_{k=1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ 1 - \frac{\Delta_c^2 \mathfrak{F}_\mathrm{core}}{2g^2} \biggr]x_0  \, .
</math>
  </td>
</tr>
</table>
</div>
Thereafter &#8212; moving out toward and just beyond the interface location <math>~(\xi = 1)</math>, the radial coordinate of each successive grid point is <math>~\xi_k = k\Delta_c</math>, and the numerically determined value of the eigenfunction at each successive grid point <math>~(k = 1 \rightarrow \mathrm{N}_\mathrm{core})</math> is,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~
x_{k+1}
</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{[4\xi_k (g^2 - \xi_k^2)  - 2\Delta_c^2 \xi_k \mathfrak{F}_\mathrm{core} ]x_k
+ [ \Delta_c( 4g^2 - 6\xi_k^2 )  - 2\xi_k (g^2 - \xi_k^2)]  x_{k-1}  }{[2\xi_k (g^2 - \xi_k^2)  +   \Delta_c( 4g^2 - 6\xi_k^2 )  ] } \, .
</math>
  </td>
</tr>
</table>
</div>
Then, at the interface, which is associated with <math>~k = \mathrm{N}_\mathrm{core}</math>, we define the reference slope as,
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<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~x_q'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{x_{k+1} - x_{k-1}}{2\Delta_c} \, .</math>
  </td>
</tr>
</table>
</div>

Next, we move outward into the envelope, using the integer index, <math>~n = 1 \rightarrow \mathrm{N}_\mathrm{env}</math>, to label successive radial grid locations <math>~(\xi_n = 1 + n\Delta_e)</math>. Letting the value of the eigenfunction at the interface be represented by <math>~x_q</math>, at the first grid location outside the interface <math>~(\xi = 1 + \Delta_e)</math>, the value of the eigenfunction is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math align="right">
~x_{n=1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 2\Delta_e [2( 1  - q^3  ) - \Delta_e ( 3  - 6q^3  )  ] x_q'
+ [4( 1  - q^3  ) - 2\Delta_e^2 ( q^3  \mathfrak{F}_\mathrm{env}  -\alpha_e  ) ] x_q }{ 4( 1  - q^3  )  } \, .
</math>
  </td>
</tr>
</table>
</div>

Thereafter, moving outward through the envelope to the surface, the value of the eigenfunction at each successive grid location is,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
x_{n+1}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{[ \Delta_e ( 3  - 6q^3 \xi_n^3 ) \xi_n - 2\xi_n^2( 1  - q^3 \xi_n^3 ) ] x_{n-1} + [4\xi_n^2( 1  - q^3 \xi_n^3 ) 
- 2\Delta_e^2 ( q^3  \mathfrak{F}_\mathrm{env} \xi_n^3 -\alpha_e  ) ] x_{n} }{  [2\xi_n^2( 1  - q^3 \xi_n^3 )  + \Delta_e ( 3  - 6q^3 \xi_n^3 ) \xi_n ] } \, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red"><b>TEST:</b></font>  &nbsp; We tested this finite-difference algorithm on a grid of resolution, <math>~\mathrm{N}_\mathrm{core} = \mathrm{N}_\mathrm{core} = 50</math>, by first setting <math>~\sigma_c^2 = 28.91158</math>.  The resulting, numerically constructed eigenfunction matched to high accuracy the analytically generated eigenfunction shown, above, as [[#Figure1|Figure 1]]; see also, the middle image in the top panel of [[#Figure2|Figure 2]].  Representative values of the numerically determined eigenfunction, <math>~x(\xi)</math> at various discrete grid locations are provided in Table 1, along with the numerically determined value of the slope at the interface, <math>~x_q'</math>.  At each grid location, the associated value of the dimensionless radius, <math>~r/R</math>, may be obtained by simply multiplying each tabulated value of <math>~\xi</math> by the parameter, <math>~q</math>.

<table border="1" cellpadding="5" align="center">
<tr>
  <th align="center" colspan="7">
<font size="+1">Table 1:</font><br />
Example Numerical Determination of Eigenfunction
  </th>
</tr>
<tr>
  <td align="center" colspan="7">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(q,\gamma_e,\gamma_c)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(0.684012, 1.194030, 1.845579)</math>
  </td>
</tr>
<tr><td align="center" colspan="3">and</td></tr>
<tr>
  <td align="center" colspan="3">
<math>~\sigma_c^2 = 28.91158</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <th align="center" colspan="3">Core</th>
  <td align="center" rowspan="10">&nbsp;</td>
  <th align="center" colspan="3">Envelope</th>
</tr>
<tr>
  <td align="center" colspan="3">
<table border="0" cellpadding="5" align="center">

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  <td align="right">
<math>~g^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1.323609</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~14</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.02</math>
  </td>
</tr>
</table>
  </td>
  <td align="center" colspan="3">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\alpha_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-0.35</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\mathfrak{F}_\mathrm{env}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~62.74338</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Delta_e</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0.00923926</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center"><math>~k</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~x</math></td>
  <td align="center"><math>~n</math></td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~x</math></td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">0.00</td>
  <td align="center">1.000000</td>
  <td align="center">0</td>
  <td align="center">1.00</td>
  <td align="center">-0.057649</td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">0.02</td>
  <td align="center">0.997885</td>
  <td align="center">1</td>
  <td align="center">1.0092393</td>
  <td align="center">-0.076955</td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">0.04</td>
  <td align="center">0.997182</td>
  <td align="center">2</td>
  <td align="center">1.0184785</td>
  <td align="center">-0.095792</td>
</tr>
<tr>
  <td align="center"><math>~\vdots</math></td>
  <td align="center"><math>~\vdots</math></td>
  <td align="center"><math>~\vdots</math></td>
  <td align="center"><math>~\vdots</math></td>
  <td align="center"><math>~\vdots</math></td>
  <td align="center"><math>~\vdots</math></td>
</tr>
<tr>
  <td align="center">49</td>
  <td align="center">0.98</td>
  <td align="center">-0.015811</td>
  <td align="center">49</td>
  <td align="center">1.452724</td>
  <td align="center">0.466484</td>
</tr>
<tr>
  <td align="center">50</td>
  <td align="center">1.00</td>
  <td align="center">-0.057649</td>
  <td align="center">50</td>
  <td align="center">1.461963</td>
  <td align="center">0.535957</td>
</tr>
<tr>
  <td align="center" colspan="3"><math>~x_q' = -2.113043</math></td>
  <td align="center" colspan="3">&nbsp;</td>
</tr>
</table>

===Results===

====Motivated by Analytic21====
Continuing with our analysis of the equilibrium model that is defined by the parameters,  <math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>, we have used the above-described numerical algorithm, to construct 26 different eigenfunctions that simultaneously satisfy the LAWE of the core and the LAWE of the envelope for 26 different values of <math>~\sigma_c^2</math> in the range, <math>~300 \ge \sigma_c^2 \ge 0</math>.   The curve traced by a sequence of small circular markers (red = core; green = envelope) in the bottom panel of Figure 2 displays each of these numerically constructed eigenfunctions in succession &#8212; in order of ''decreasing'' values of <math>~\sigma_c^2</math> &#8212; in the form of a looped animation sequence.  Also displayed in each frame of the animation, for reference, is the relevant value of <math>~\sigma_c^2</math>, as well as an unchanging, smooth, thin red/green curve that traces the ''analytically'' derived eigenfunction shown in Figure 1, for which <math>~\sigma_c^2 = 28.91158</math>.  

<div align="center" id="Figure2">
<table border="1" align="center" cellpadding="5">
<tr><th align="center">
<font size="+1">Figure 2:</font><br />
<br />
<math>~(q,\gamma_e,\gamma_c) = ( 0.6840119, 1.1940299, 1.845579)</math>
</th></tr>
<tr><td align="center">
[[File:ImageTrio.png|500px|center|Three movie frames]]
</td></tr>
<tr><td align="center">
[[File:EigenfunctionMovie1.gif|500px|center|Eigenfunction movie]]
</td></tr>
</table>
</div>

Three frames from the animation sequence have been displayed side-by-side in the top panel of Figure 2.  This image montage is presented, in part, to illustrate the degree to which our numerically generated eigenfunction matches the analytically generated eigenfunction in the ''specific'' case  <math>~(\sigma_c^2 = 28.9)</math> for which we have been able to obtain an analytic solution to the combined/matched, core/envelope LAWEs.

====Motivated by Analytic22====

We have also numerically constructed an eigenfunction that matches our [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration22|accompanying analytic Illustration22]].  In Figure 3, the numerically derived solution has been plotted on top of the analytically derived solution.
<div align="center" id="Figure3">
<table border="1" align="center" cellpadding="5">
<tr><th align="center">
<font size="+1">Figure 3:</font><br />
<br />
<math>~(q,\gamma_e,\gamma_c) = ( 0.886575, 1.798817, 1.021798)</math>
</th></tr>
<tr><td align="center">
[[File:NumericalModel22.png|350px|center|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration22]]
</td></tr>
</table>
</div>

====Motivated by Analytic31====

We have also numerically constructed an eigenfunction that matches our [[Appendix/Ramblings/AdditionalAnalyticallySpecifiedEigenvectors00Bipolytropes#Illustration31|accompanying analytic Illustration31]].  In Figure 4, the numerically derived solution has been plotted on top of the analytically derived solution.
<div align="center" id="Figure4">
<table border="1" align="center" cellpadding="5">
<tr><th align="center">
<font size="+1">Figure 4:</font><br />
<br />
<math>~(q,\gamma_e,\gamma_c) = ( 0.4059596, 1.180462, 1.008887)</math>
</th></tr>
<tr><td align="center">
[[File:NumericanModel31.png|350px|center|Numerically generated eigenfunction plotted on top of the analytically derived, Illustration31]]
</td></tr>
</table>
</div>