Editing Appendix/Ramblings/NumericallyDeterminedEigenvectors (section)

===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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<math>~\frac{\rho_e}{\rho_c} = \frac{2q^3}{1+2q^3}</math>
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<math>~\approx</math>
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<math>~0.3902664 \, .</math>
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</table>
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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,

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

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<math>~\mathfrak{F}_\mathrm{env} </math>
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<math>~=</math>
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<math>~\frac{1}{\gamma_e} \biggl[ \sigma_c^2 \biggl(\frac{\rho_e}{\rho_c} \biggr)^{-1} + 8\biggr]- 6 \, .</math>
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</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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<math>~\Delta_c \equiv \frac{1}{\mathrm{N}_\mathrm{core}}</math>
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&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; 
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<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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<math>~
x_{k=1}
</math>
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<math>~=</math>
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<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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<math>~
x_{k+1}
</math>
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<math>~\approx</math>
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<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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<math>~x_q'</math>
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<math>~=</math>
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<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,

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<math align="right">
~x_{n=1}
</math>
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<math>~=</math>
  </td>
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<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>
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</table>
</div>

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

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<math>~
x_{n+1}
</math>
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<math>~=</math>
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<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>
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</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>
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<font size="+1">Table 1:</font><br />
Example Numerical Determination of Eigenfunction
  </th>
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<math>~(q,\gamma_e,\gamma_c)</math>
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  <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>
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  <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">
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<math>~g^2</math>
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<math>~=</math>
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<math>~1.323609</math>
  </td>
</tr>

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

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  <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">

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

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  <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>