[Proj] Dozier's TM method---my summary

[Gerald I. Evenden]

I think the basic idea presented by Dozier sounds feasible but his execution 
falls short of the goal.

Although the the method of computing elliptic integral of the second kind with 
complex argument may be OK it has troubles with large arguments when 
evaluating the Jacobian Zeta function.  The second, and I believe the more 
serious problem, is the Newton-Raphson method employed.  He has expanded the 
basic real function and applied it to a complex variable.  I am not sure that 
this is appropriate and searching through the net and all leads me to the 
conclusion that we are getting into deep water when dealing with the complex 
plane.  Also, it looks like we are also dealing with multiple roots, 
especially when longitude exceeds a certain value (suggested to be 
(pi/2)*(1-k))---a factor not addressed in Dozier's solution.

Note that the cusp (poorly displayed in the previously mention gif url) 
appears to be the beginning of the multiple root solution.  I say poorly 
displayed as the gradation of the equitorial parallel should *smoothly* begin 
a swing to the north OR (importantly) to the south.  Selecting the north or 
south root becomes a practical problem for a projection program and is a 
problem with any method dealing with the comprehensive TM projection.

Because I have no training and no experience in working with complex variable 
problems and have failed to find any practical material related to alternate 
methods to compute elliptic integrals with complex arguments and, more 
importantly, a Newton-Raphson routine for determining roots of complex 
functions, I have decide to suspend again any activity on the Dozier method.

One has to know when to throw in the towel.  ;-)
-- 
Jerry and the low-riders: Daisy Mae and Joshua
"Cogito cogito ergo cogito sum"
   Ambrose Bierce, The Devil's Dictionary