Re: Full expansion with a mixture of Times and NonCommutativeMultiply
- To: mathgroup at smc.vnet.net
- Subject: [mg103542] Re: Full expansion with a mixture of Times and NonCommutativeMultiply
- From: David Bailey <dave at removedbailey.co.uk>
- Date: Sun, 27 Sep 2009 07:32:29 -0400 (EDT)
- References: <h9kpqa$n0g$1@smc.vnet.net>
ChrisL wrote:
> Dear all,
> I am using a non-associative, non-commutative product **
> (NonCommutativeMultiply[]). I have a procedure which builds long
> polynomials that use ** and the usual Times. What I need eventually is
> to extract each of the mononials (parts of the final expression that
> do not contain any Plus[]) for some further processing.
> I defined NonCommutativeMultiply[] very simply like this:
> Unprotect[NonCommutativeMultiply];
> ClearAttributes[NonCommutativeMultiply, Flat]; (* forcing non-
> associativity *)
> 0 ** x_ := 0;
> x_ ** 0 := 0;
> 1 ** x_ := x;
> x_ ** 1 := x;
> (m_Integer*x_) ** y_ := m*(x ** y);
> x_ ** (m_Integer*y_) := m (x ** y);
> Protect[NonCommutativeMultiply];
>
> And getting the full expansion seems to work fine:
> Distribute[2 (3 a1) ** (5 a2)]
> Distribute[a1 ** (2 a2 + a3 ** a4)]
> yields
> 30 a1 ** a2
> 2 a1 ** a2 + a1 ** (a3 ** a4)
> This is great: I can pick up each mononial with Table[expr[[i]],
> {i,Length[expr]}]
>
> Unfortunately, the expansion seems to stop at the second level. Thus:
> Distribute[((2 a2 + a3 ** a4 ) ** a6) ** a7]
> Distribute[(2 a2 + a3 ** a4 ) ** (3 a6)]
> yields
> ((2 a2 + a3 ** a4) ** a6) ** a7
> 3 (2 a2 + a3 ** a4) ** a6
>
> when I need:
> 2 (a2**a6)**a7 + ((a3**a4)**a6)**a7
> and
> 6 a2**a6 + 3 (a3**a4)**a6
>
> Is there any way achieve this? Or do I need to write the full
> expansion algorithm myself? Note that the final expression will be
> much longer - about 30'000 mononials.
>
> thank you very much in advance!
> Cheers.
>
Forget about Distribute, and perform the transformation using a set of
replacement rules exhaustively applied:
expansionRules = {(a_ + b_) ** c_ -> a ** c + b ** c,
c_ ** (a_ + b_) -> c ** a + c ** b, c_ (a_ + b_) -> c a + c b}
In[26]:= (2 a2 + a3 ** a4) ** (3 a6) //. expansionRules
Out[26]= 6 a2 ** a6 + 3 a3 ** a4 ** a6
In[25]:= ((2 a2 + a3 ** a4) ** a6) ** a7 //. expansionRules
Out[25]= 2 a2 ** a6 ** a7 + a3 ** a4 ** a6 ** a7
I have not tried this on the size of problem you suggest, but I would
certainly start with this approach and test if it is adequate.
Once you have the expression fully expanded - say
ans = 2 a2 ** a6 ** a7 + a3 ** a4 ** a6 ** a7
just replace the Plus with List, taking care to deal with the degenerate
single item case:
If[Head[ans] === Plus, ans = List @@ ans]
BTW NonCommutativeMultiply is a relic from the days when Mathematica
only used the ASCII character set, \[CircleTimes] will look a lot more
readable!
David Bailey
http://www.dbaileyconsultancy.co.uk