Thank you very much for the quick reply and the code! I will start to play
around with the new code.
> Multivectors do not inherit from sympy symbols so solve does not work.
How hard would it be to implement this? Is it only a (possibly difficult,
time consuming) programming exercise or is there a fundamental reason why
this is hard to do for GA equations? How well do the solve routines deal
with noncommutative symbols, for instance?
Regarding division, I suppose it wouldn't be too hard to implement division
for blades and simply check if the squared norm is zero before dividing by
it.
Regarding speed improvements for orthogonal bases: is there any improvement
for 'almost orthogonal' bases, such as the CGA diag(1,1,1,1,-1) basis?
How would you rate the new version of the software in terms of stability
and correctness?
Best,
tsc
On Sunday, March 3, 2013 5:10:53 PM UTC+1, brombo wrote:
>
> 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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