I've just found the sympy GA module, and I must say it looks really neat!
I'd like to use it for automatic differentiation and equation solving in
high-dimensional conformal geometric algebra. While experimenting with the
module, I've run into a few problems though. I'm not familiar with sympy
internals, so before I dive into the source of this module to fix things,
I'd like to see if anyone can tell me if these are really bugs or if I'm
misunderstanding something.
All examples below are run in isympy, just pulled from github. I
initialized with:
from sympy.galgebra.GA import *
e1,e2,e3= MV.setup('e1 e2 e3', '1 0 0, 0 1 0, 0 0 1')
1. Division:
When I perform e1 / e1, I get an error: "TypeError: unsupported operand
type(s) for /: 'MV' and 'MV'". Is this not implemented? For non-null
vectors (i.e. x >> x != 0), the inverse is x / (x >> x). The inverse of a
product of invertible vectors is just the reverse product of the inverses.
For an invertible blade A, this reduces to A / (A >> A).
2. Solving equations:
I tried to solve some basic equations, e.g.
>> solve( (x - e1) * e2, x)
_1*e1e2/e2
This is correct, but strangely it involves division which doesn't appear
to work when typed into the terminal. Other examples that act strange:
>> solve( (x - e1) >> e2, x)
[]
>> solve( (x - e1) | e2, x)
[0]
Even though | and >> should both implement contraction.
3. Efficiency:
I tried MV.setup on some high-dimensional algebras, and noticed that it
takes very long to initialize beyond n=8. The multivector basis is 2^n, so
a slowdown is to be expected, but 2^8 = 256, which seems a bit low. Would
it be possible to speed up the implementation, e.g. using
bitarray<https://pypi.python.org/pypi/bitarray> to
represent the presence or absence of a multivector basis element, and
compute products using a combination of scalar multiplication on numpy
arrays and bitwise operations on bitarrays?
4. Notation for outer product and powers:
I find the notation '**' for the wedge product confusing. Consider
typing this into a terminal:
>> e1*e1
e1**2
>> e1**2
2*e1
Why is the result of e1*e1 (e1 squared) written in the usual python
exponentiation notation e1**2, while at the same time we cannot use that
notation to perform exponentiation? I think it would be best to use ** for
exponentiation, since that is the python standard. We can use ^ for the
wedge, which is visually most natural anyway.
5. Contraction with scalar doesn't work.
When I compute e1 >> 2, or any other contraction with a scalar, I get
None. The correct results is 0 when a (multi)vector is contracted onto a
scalar. When a scalar is contracted onto a multivector, the result should
be the same as scalar multiplication.
If I can make it all work I might be interested in implementing a module
for conformal GA, and one for linking sympy to GAViewer. The latter would
make it possible to visualize results from symbolic computations in a 3D
viewer.
Thanks in advance for any help!
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