[with patches, needs review] New module complex_mpc using lib mpc for complex multiprecision arithmetic

replace the previous patch (rebase against 3.2)

There seems to have been some bitrot due to recent changes in Sage. I have uploaded a very slightly modified version of Philippe's patch, which applies cleanly against 3.2. A number of doctests fail for various reasons. I will try to look into this soon.

coercion problems resolved, more functions

The coercion problems have been solved, all doctests now succeed in new version. The whole set of mpc functions involving complex numbers is now in the interface.

apply after complex_mpc.p0.patch

Philippe,

Great work! I'll do my best to review this for 3.2.2. Right now I notice that the patch doesn't work in 3.2.1 because of #4580. I'm attaching a tiny patch that fixes that (should be applied after complex_mpc.p0.patch.

There are a few small issues that I ran into, but since this is going to be (for now) an optional part of Sage and nothing depends on it, I think we can fix these later (I'm keeping a list of them so they don't get lost).

However, there is a policy of 100% doctest for all new code. The coverage is now:

[ghitza@artin sage]$ sage -coverage rings/complex_mpc.pyx
----------------------------------------------------------------------
rings/complex_mpc.pyx
SCORE rings/complex_mpc.pyx: 68% (46 of 67)

Missing documentation:
	 * _repr_ (self):
	 * _latex_(self):
	 * _an_element_(self):
	 * random_element(self, bound):
	 * __hash__(self):
	 * prec(self):
	 * _set(self, z, y=None, base=10):
	 * _repr_(self):
	 * _latex_(self):
	 * ModuleElement _add_(self, ModuleElement right):
	 * ModuleElement _sub_(self, ModuleElement right):
	 * RingElement _mul_(self, RingElement right):
	 * RingElement _div_(self, RingElement right):
	 * ModuleElement _neg_(self):
	 * __create__MPComplexField_version0 (prec, rnd):
	 * __create_MPComplexNumber_version0 (parent, s, base=10):


Missing doctests:
	 * _rnd(self):
	 * _rnd_re(self):
	 * _rnd_im(self):
	 * _real_field(self):
	 * _imag_field(self):


Possibly wrong (function name doesn't occur in doctests):
	 * _element_constructor_(self, z):
	 * _coerce_map_from_(self, S):
	 * bint is_exact(self) except -2: return False def is_finite(self):
	 * Element _call_(self, z):
	 * Element _call_(self, x):

----------------------------------------------------------------------

So the missing docstrings and doctests will have to be added before this patch can be merged. I would happily do this but the earliest I can get to it is Thursday or so (and that's being optimistic).

To end on a positive note: this looks good. It will take a bit more work, but if we can show that (a) the performance is improved, or doesn't suffer too much, and (b) the switch can be made seamlessly, then it will be easy to convince people that we should switch the core of our complex numbers functionality over to MPC.

apply after complex_mpc.p0.patch and trac4446_fix.patch

I added a patch trac4446_doctests.patch, which does a number of things:

• adds doctests for all functions except three internal use only functions
• changes _repr_ of complex numbers so that it agrees with the way complex numbers are currently printed in Sage
• makes MPComplexField inherit from ParentWithGens?, being generated over its real field by the square root of -1 (just as it is now); this required adding a few functions. So now one can do

sage: from sage.rings.complex_mpc import MPComplexField
sage: MPC.<j> = MPComplexField()
sage: j^2
-1.00000000000000 + 0.000000000000000*I

Note that only the last three patches should be applied, in order: complex_mpc.p0.patch, trac4446_fix.patch, and trac4446_doctests.patch

One more patch trac4446_norm.patch with:

* Alex Ghitza listed in authors list and copyright notice

* complex_mpc uses a copy of mpfr rounding mode list instead of its private one

* the abs and norm documentation has been improved.