On 03/03/2013 08:07 AM, tsc wrote:
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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I forgot to mention that the attached file in the previous response is
named GA.bip but is really GA.zip. I was renamed so that certain mail
programs would not reject the attachment.
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