Editing Appendix/Ramblings/NumericallyDeterminedEigenvectors (section)

==Additional Possibilities==

* We should be able to ''numerically'' identify a wide range of quantized radial modes of oscillation by specifying a physically reasonable surface slope then, for each successive quantum node count, tuning the choice of <math>~\sigma_c^2</math> until the desired slope has been encountered.

* It seems unlikely that any of our analytically derived eigenfunctions will happen to satisfy the specified boundary condition.  The analytic functions have nevertheless proven to be useful in the sense that they provide a terrific check for the computational algorithm that, then, can be used to identify physically meaningful solutions numerically.

* Tie this new (numerical) technique into the [[SSC/Stability/BiPolytrope00CompareApproaches#Comparing_Stability_Analyses_of_Zero-Zero_Bipolytropes|associated discussion]] of the relationship between stability determinations via LAWEs and global free-energy considerations.

* The Figure 2 animation displays numerically determined solutions, <math>x(r/R)</math>, to the combined/matching LAWEs only for ''non-negative'' values of <math>\sigma_c^2</math>; these should be relevant only to ''stable'' radial oscillations.  We should also see what can be learned from solutions associated with ''negative'' values of <math>~\sigma_c^2</math>, which may be relevant to unstable radial modes of oscillation.

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