Re: Full expansion with a mixture of Times and NonCommutativeMultiply
- To: mathgroup at smc.vnet.net
- Subject: [mg103592] Re: Full expansion with a mixture of Times and NonCommutativeMultiply
- From: ChrisL <chris.ladroue at gmail.com>
- Date: Tue, 29 Sep 2009 07:39:38 -0400 (EDT)
- References: <h9kpqa$n0g$1@smc.vnet.net> <h9nigh$503$1@smc.vnet.net>
Thank you everyone for your swift and very helpful replies. I look
forward to making use of them ans solving my problem.
Cheers!
ChrisL
On Sep 27, 12:32 pm, David Bailey <d... at removedbailey.co.uk> wrote:
> 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 Baileyhttp://www.dbaileyconsultancy.co.uk