[Kstars-devel] Atmospheric Refraction trouble

[Daniel-Marian Baboiu]

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.