Editing Appendix/ComputerAlgorithms/EFE (section)

==jacobi7 (25 June 2016)==

Chandrasekhar provides two relations that define the equilibrium
properties of Jacobi ellipsoids.  One defines the relationship between
the pair of axis ratios (b/a,c/a); the other provides an expression
for the corresponding rotation frequency (squared), &Omega;<sup>2</sup>.  These are detailed in our [[ThreeDimensionalConfigurations/JacobiEllipsoids#Equilibrium_Conditions_for_Jacobi_Ellipsoids|accompanying discussion of Jacobi Ellipsoids]].
Our desire, here, is to develop a tool &#8212; probably employing a Newton-Raphson
technique &#8212; that finds the root(s) of the first of these expressions, namely,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~f_J</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{b}{a}\biggr)^2 \biggl[ \frac{2(1-A_1)-A_3}{1 - (b/a)^2} \biggr]-\biggl(\frac{c}{a}\biggr)^2  A_3 =0 \, .</math>
  </td>
</tr>
</table>
Such an iteration technique will require evaluation of the function
as well as evaluation of its first derivative.

So far, we have developed and debugged a subroutine (fJ) that evaluates
the analytic expression for fJ; it also evaluates the first derivative
of, fJprime, with respect to (b/a) while holding (c/a) fixed.

DOUBLE PRECISION ROUTINES:
File fJ.for contains the subroutine (fJ) that evaluates the function
and its derivative. In addition, jacobi7.for is a main program that
calls this subroutine for 25 different axis-ratio pairs that define
the Jacobi sequence (according to EFE); Subroutine fJ is included
as part of this jacobi7.for file.

<ul>
<li>gfortran -c jacobi7.for -ffree-form</li>
<li>gfortran -o exec jacobi7.o doubleELib.o</li>
<li>./exec > output</li>
</ul>

For supplementary information, read our accompanying discussion titled, 
"[[ThreeDimensionalConfigurations/JacobiEllipsoids#Roots_of_the_Governing_Relation|Roots of the Governing Relation]]."