Actually, I think you can just convert the symbols to multiplications,
but set them all as commutative=False so that they don't get
rearranged. Then you can apply factor() (which I believe basically
does the above algorithm for noncommutatives), and to convert back to
dot products, convert each multiplication pairwise, like a*b*c*d ->
<a, b>*<c, d> (also accounting for powers, like e1**2 == <e1, e1>).
I'm not 100% sure this won't produce a wrong answer, so it's worth
double checking it somehow (perhaps numerically).
This won't catch simplifications that require rearranging the inner
products, like <a, b> = <b, a> (or <a, b> = conjugate(<b, a>) as the
case may be).
Aaron Meurer
On Mon, Feb 27, 2017 at 3:39 PM, Aaron Meurer <asmeu...@gmail.com> wrote:
> The function that does the simplification you want is factor():
>
> In [22]: var('a b c d')
> Out[22]: (a, b, c, d)
>
> In [23]: factor(a*c + b*d - a*d - b*c)
> Out[23]: (a - b)⋅(c - d)
>
> However, I'm not sure how to apply it here. You can't just convert
> your dot products to multiplications because it isn't true that <a,
> b>*<c, d> = <a, c>*<b, d>.
>
> You might need to write a naive factor that recursively collects terms
> with the same coefficient. For instance
>
> <a, c> + <b,d> - <b,c> - <a, d>
>
> -> <a, c - d> + <b, d - c>
> -> <a - b, c - d>
>
> This also needs to recognize that c - d = -(d - c).
> could_extract_minus_sign is useful for this.
>
> I don't recall if something like this is already written in SymPy.
>
> Aaron Meurer
>
>
> On Mon, Feb 27, 2017 at 12:44 PM, Nico Schlömer
> <nico.schloe...@gmail.com> wrote:
>> Thanks for the reply.
>>
>>> I assume e0, e1, and e2 are arbitrary vectors.
>>
>> Indeed, they can be anything. (I'm looking at 3 dimensions here but given
>> the fact that everything is a dot product I assume that doesn't play much of
>> a role.)
>>
>> Cheers,
>> Nico
>>
>>
>>
>> On Monday, February 27, 2017 at 6:37:59 PM UTC+1, brombo wrote:
>>>
>>> How the expression zeta obtained. Do input the expression you show or is
>>> it obtained by vector algebraic operations on vector expressions. I assume
>>> e0, e1, and e2 are arbitrary vectors.
>>>
>>> On Mon, Feb 27, 2017 at 12:04 PM, Nico Schlömer <nico.sc...@gmail.com>
>>> wrote:
>>>>
>>>> I have a somewhat large expression in inner products,
>>>> ```
>>>> zeta = (
>>>> - <e0, e0> * <e1, e1> * <e2, e2>
>>>> + 4 * <e0, e1> * <e1, e2> * <e2, e0>
>>>> + (
>>>> + <e0, e0> * <e1, e2>
>>>> + <e1, e1> * <e2, e0>
>>>> + <e2, e2> * <e0, e1>
>>>> ) * (
>>>> + <e0, e0> + <e1, e1> + <e2, e2>
>>>> - <e0, e1> - <e1, e2> - <e2, e0>
>>>> )
>>>> - <e0, e0>**2 * <e1, e2>
>>>> - <e1, e1>**2 * <e2, e0>
>>>> - <e2, e2>**2 * <e0, e1>
>>>> )
>>>> ```
>>>> and the symmetry in the expression has me suspect that it can be further
>>>> simplified. Is sympy capable of simplifying vector/dot product expressions?
>>>> A small example that, for example, takes
>>>> ```
>>>> <a, c> + <b,d> - <b,c> - <a, d>
>>>> ```
>>>> and spits out
>>>> ```
>>>> <a-b, c-d>
>>>> ```
>>>> would be great.
>>>>
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>>>
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