[Om Joshi <om+pyt...@omajoshi.com>]
> I spent a couple hours putting this together:
> https://github.com/omajoshi/complex_math
>
> It does lazy updating, staying in polar form until an addition, then staying
> in
> rectangular form until the next multiplication.
>
> I implemented Tim Peters' tests (from two emails back):
> https://github.com/omajoshi/complex_math/blob/main/test.py
>
> Here are my results vs his:
> https://github.com/omajoshi/complex_math/blob/main/test.txt
>
> The library may be useless, but as far as I can it is associative (at least
> on Tim's tests).
Ya, polar is much more natural for expressing complex multiplication.
The Cartesian form _can_ be made associative too, but I'm not aware
of any implementation that actually does so. Proof: there are, up to
isomorphism, two 4-element groups, both of which happen to be
commutative. Pick either one, and name each of the 4 elements with one
of the 4 complex zeroes. Then let the group's "multiplication" table
_define_ the result.
That's major hackery, though, and doesn't seem to follow from any
"natural" way of doing 754 arithmetic on the components.
Confounding it: some other results for signed zeroes are defined by
various semi-authoritative sources, like Annex G of the most recent C
standards. In particular, "the angle" ("phase", or "argument") of a
complex number is computed by carg(z), where special cases are
supposed to do the same as atan2(cimag(z), creal(z)),. Which is more
arbitrary gibberish for signed zeroes:
...
print(y, x, atan2(y, x))
...
0.0 0.0 0.0
0.0 -0.0 3.141592653589793
-0.0 0.0 -0.0
-0.0 -0.0 -3.141592653589793
Which Python's cmath.phase() satisfies on my box:
>>> cmath.phase(complex(0.0, 0.0))
0.0
>>> cmath.phase(complex(-0.0, 0.0))
3.141592653589793
>>> cmath.phase(complex(0.0, -0.0))
-0.0
>>> cmath.phase(complex(-0.0, -0.0))
-3.141592653589793
But I don't suggest anyone spend time on this. It's a bottomless pit,
and nobody cares :-)
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