[Kstars-devel] Atmospheric Refraction trouble

[Akarsh Simha]

On Tue, Jul 24, 2012 at 09:14:14PM -0700, Daniel-Marian Baboiu wrote:
> On Mon, 2012-07-23 at 16:09 +0400, Aleksey Khudyakov wrote:
> > On 21.07.2012 05:31, Akarsh Simha wrote:
> > >> Maybe it works only because of the existence of the below-altCrit
> > >> bug. The bug makes sure we use a function with the opposite slope
> > >> later, so maybe it comes back and converges after many more cycles
> > >> than it should actually take.
> > >
> > It uses plain old iteration method for solving equations in form x=f(x)
> > 
> > x_{n+1} = f( x_{n} )
> > 
> > We could write equation in form
> > 
> > h0 = h - correction(h0)
> > 
> > h0 - unrefracted height
> > h  - with refraction
> > 
> > This method doesn't always converge in generic case but it should always 
> > converge in our case.
> 
> It is called "fixed point iteration", and there is a mathematical reason
> why (and when) it converges. In short, equation
> x=f(x)
> can be solved by iterations (fixed point iterations) if |f'(x)|<1; it is
> the same method used to solve Kepler's equation for elliptical orbits.
> And because refraction correction changes very little with altitude,
> convergence is very quick.
> 
> For atmospheric refraction, there are various formulas, for refract as
> well as for unrefract. But the refraction formulas are not invertible,
> and the existing formulas are not exact inverse to one another; applying
> repeatedly refract/unrefract analytical formulas may diverge. 

Thanks for the explanations! I understand this now. I've come across
this while studying the logistic map -- but those systems usually have
multiple fixed points, because they are not invertible maps. In our
case, it should certainly converge (I checked on a plot that the slope
everywhere is less than 1).

The reason my iteration did not seemed to work was because of a
"silly" mistake. Once I found and fixed it, it's alright now.

Regards
Akarsh