Don't know too much about MatLab, so I can't say. I've always thought it was more a Engineer's number cruncher than a symbolic math package, so I never looked into it.
As for the speed thing. The only package I am using directly is Miscellaneous`WorldPlot`, which I suspect was written more as a demonstration of cartography than an attempt to do much that is useful. Certainly for simple calculations like 999999! Mathematica takes only a minute or two, so its not necessarily a sluggard.
As for not having found the best functions, decide for yourself. This is the standard Mollweide Projection and its definition in Mathematica's WorldPlot is: Mollweide = Module[{th}, (N[{Sqrt[8]/Pi #2 Degree/60 Cos[th], Sqrt[2] Sin[th]}/.FindRoot[2 th + Sin[2 th] == Pi Sin[#1/60 Degree],{th,0}]] )&];
And here is the definition of Eckert IV as well as the code I wrote for it: EckertIV = Module[{Theta}, (N[ { (2/Sqrt[Pi (4+Pi)]) (Degree/60) #2 (1 + Cos[Theta])) , 2 Sqrt[Pi/(4+Pi) Sin[Theta] } /. FindRoot[ Theta + Sin[Theta] Cos[Theta] + 2 Sin[Theta] == (2 +Pi/2) Sin[#1/60 Degree] , {Theta, 0} ]]) &];
They're almost identical (although I had to clean mine up a bit, since I used greek symbols that didn't translate well into html). They DO do lots of redundant root finding though. A lookup table for Theta would greatly speed things up.
MatLab vs Mathematica in an Invariant Homolocage!
As for the speed thing. The only package I am using directly is Miscellaneous`WorldPlot`, which I suspect was written more as a demonstration of cartography than an attempt to do much that is useful. Certainly for simple calculations like 999999! Mathematica takes only a minute or two, so its not necessarily a sluggard.
As for not having found the best functions, decide for yourself. This is the standard Mollweide Projection and its definition in Mathematica's WorldPlot is:
Mollweide =
Module[{th},
(N[{Sqrt[8]/Pi #2 Degree/60 Cos[th],
Sqrt[2] Sin[th]}/.FindRoot[2 th + Sin[2 th] ==
Pi Sin[#1/60 Degree],{th,0}]] )&];
And here is the definition of Eckert IV as well as the code I wrote for it:
EckertIV =
Module[{Theta}, (N[
{ (2/Sqrt[Pi (4+Pi)]) (Degree/60) #2 (1 + Cos[Theta]))
, 2 Sqrt[Pi/(4+Pi) Sin[Theta]
} /.
FindRoot[ Theta + Sin[Theta] Cos[Theta] + 2 Sin[Theta] == (2 +Pi/2) Sin[#1/60 Degree]
, {Theta, 0}
]]) &];
They're almost identical (although I had to clean mine up a bit, since I used greek symbols that didn't translate well into html). They DO do lots of redundant root finding though. A lookup table for Theta would greatly speed things up.