[Proj] TM exact

[OvV_HN]

Threadery:

In the thread:
Re: [Proj] Finally: geodesic-1.0 is available
Charles Karney wrote on 09 Jan 2009 that he is improving his exact TM.
Oscar van Vlijmen replied with a short remark about the lat=90d problem in 
the inverse.
Daan Strebe replied on 09 Jan:

> I find the regions from the cusps to a short way along the outer boundary,
> just inside the boundary, to be most computationally problematic. However,
> I am not using Thompson/Lee's development through Jacobian elliptic 
> functions,
> so possibly your difficult terrain differs. I do not use Jacobian elliptic 
> functions
> at all; only an elliptic integral of the second kind. I am satisfied with 
> my results
> throughout the rest of the map, so I am keen to compare notes in case it 
> makes
> sense to bring both methods to bear, depending on the region.

Some time ago I translated the Dozier method and found some improvements:
* Corrected an error in the publication.
* Used much better complex elliptics.
* Used a more robust iteration scheme for implicit complex functions.
(by H. Bach, http://www.netlib.org/toms/365)
* I've done next to nothing with respect to iteration starting values, the 
situation beyond delta-longitude 90d, and some difficult areas very near the 
boundaries.

I published some test values around the 90*(1-eccentricity) point and Daan 
Strebe confirmed the results. See the proj posting from 29 Jun 2006.

Shortly after Charles Karney published his Lee method, I investigated the 
exact TM again and I found that Karneys method is probably better than mine.
Separating complex functions into real and imaginary parts proves to be 
computationally a very good idea. In some difficult regions my downhill 
iteration scheme in the complex plane needs hundreds of passes, whereas Lees 
method is ready in a dozen or so.
The starting value problem remains however and you need to check if the 
denominator (usually the derivative) gets zero.
BTW, I'm also very pleased with the meridian convergence and point scale 
factor solutions Charles Karney presented.

Oscar van Vlijmen