On 03/04/2013 03:41 PM, tsc wrote:
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?
I think there would be problems in solve due to the different types
of multivector multiplication. The documentation in GA.bip goes into
detail an to how the multivectors and operations are implemented.
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?
diag(1,1,1,1,-1) is an orthogonal basis (any metric tensor that is
diagonal gives an orthogonal basis).
How would you rate the new version of the software in terms of
stability and correctness?
The new version is simpler and uses sympy to do more of the heavy
lifting. I think it is much more reliable. Also I have been working in
collaboration with Alan Macdonald to provide the software for his new
book on geometric calculus so that we have gone through many more
examples. Two people can break code much faster than one. Look at the
examples in the example directory, specifically test.py and
test_latex.py. Note that the Latex doc gives detailed instructions on
how to install the module and get it working with latex.
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 <http://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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