[Axiom-developer] [#47 complexForm(log(%i) - log(-%i))] meaning of y > 0; symbolic floating point

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http://page.axiom-developer.org/zope/mathaction/47ComplexFormLogILogI/diff
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Dear Martin:

Thanks, and yes, I was wrong again (shame on me): the range of the single 
argument 'atan' does not cover $[0, 2\pi)$. But the only way to return results 
in case variables are involved in the arguments of 'atan(x,y)' is to use 
provisos (I thought I was able to avoid this with the sign convention). Even if 
we were able to make 'EXPR INT' into an 'OrderedRing', the tests such as 'if y 
> 0 then' will have a different meaning than when 'y' takes a value in 'INT' 
whatever the ordering on 'EXPR INT' may be. Such tests must be done using 
values of 'y', not the expression represented by 'y' (thus, the code for 
'myatan2' used these tests incorrectly for general expressions 'y'). So it 
seems we need 'OrderedRing' not on 'EXPR INT' but on 'INT', which is okay, 
except that in 'COMPLEX R' we have no way to know about what 'R' is (see also 
last paragraph in this reply).

One possible way is to use a three argument 'atan': in addition to the 'x,y' 
coordinates, also include the quadrant. This would allow a user to supply the 
crucial information and also make it automatic when the quadrant can be 
deduced. We can allow the quadrant information to be specified as "unknown" or 
"failed" and then return the expression unevaluated in that case.

The amount of simplification that should be carried out should also be part of 
the discussions. For example, in a two-argument version, 
'atan(2*(x^2+1),(x^2+1))' should be evaluated to 'atan(1/2)' but any such 
simplification either is difficult (not possible?) or may lose information on 
the quadrant if not carefully done. For example, 'atan(2*(x^2+1),-(x^2+1))' and 
'atan(-2*(x^2+1), (x^2+1))' should give different answers. But how can we test 
signs on expressions? (Note that this is *different* from how to make 'EXPR 
INT' into an 'OrderedRing' because we are talking really on the level of the 
base domain: signs of expressions as values, not as expressions.) It would seem 
that we need to get into the full-blown cylindrical algebraic decomposition 
(CAD) stuff, even just for the polynomial expressions. And 'INT' is not quite 
the base domain to perform these computations.

Stepping back a bit, let's see why 'EXPR INT' is involved in the original post 
to evaluate 'argument(-%i)'. Functions (and especially, when implemented as 
operators) like 'argument' and 'atan' perhaps should be defined only under a 
domain which is closed with respect to them (so 'argument: Complex R -> R' 
would be considered closed, but 'argument: Complex Integer -> Expression 
Integer' is not.) In 'Complex Integer' these functions are meaningful only 
because the set of Gaussian integers is mathematically a subset of the field of 
complex numbers. The Axiom construction is complicated by the fact that $\pi$ 
may be symbolic and hence lives in 'Expression R'. And once expanded to 
'Expression R', the problem with what to do with variable inputs further 
complicates the implementation. Thus in evaluating 'argument((-1+%i)::Complex 
Integer)', Axiom needs to invoke 'argument: Complex EXPR INT->EXPR INT'. Would 
it be appropriate to invoke something like 'argument: Complex Float -> Flo!
at' instead? (This would avoid the issue of variables in input arguments.)

This discussion naturally leads one to question why specific constants such as 
$\pi$ live in separate domains in Axiom (contrary to a purely mathematical 
model). *Must* these constants be numerically represented under 'Float'? Can 
one create something called 'SymbolicFloat' where roughly, 'SymbolicFloat' is a 
mix of floating point numbers, symbolic algebraic numbers, and symbolic 
transcendental numbers, and yet retains the properties of 'RealNumberSystem' 
and 'OrderedRing' (so whenever an inequality in this domain is to be tested, 
all symbolic constants will be temporarily converted to floating point with an 
accuracy sufficient to resolve the decision, but the symbolic constants still 
remain symbolic in subsequent computations)? 

William
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