Re: Simplify and Noncommutativity

• To: mathgroup at smc.vnet.net
• Subject: [mg60384] Re: Simplify and Noncommutativity
• From: Robert Schoefbeck <schoefbeck at hep.itp.tuwien.ac.at>
• Date: Wed, 14 Sep 2005 05:26:37 -0400 (EDT)
• References: <dfrhi4\$g4l\$1@smc.vnet.net> <dg8lfv\$r8g\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```thank you very much for your answer,
it helped a great deal.

unfortunately im busy with other things right now
but as soon as I implement your idea I will once again respond here.

your solution is very nice, it will also help a lot when doing
superspace calculus.

kind regards
robert schoefbeck

Carl K. Woll wrote:
> "Robert Schoefbeck" wrote:
>
>>I have a rather lengthy expression of abstract products of matrices
>>of the form
>>
>>myDot[M1,M2,...]
>>
>>If Inv[M] denotes the inverse matrix
>>
>>i have told mathematica that
>>
>>myDot[P1___,P2_,Inv[P2_],P3___]:=myDot[P1,P3];
>>myDot[P1___,Inv[P2_],P2_,P3___]:=myDot[P1,P3];
>>myDot[]=Unity;
>>
>>and that
>>
>>myDot[P1___,P2_+P3_,P4___]:=myDot[P1,P2,P4]+myDot[P1,P3,P4];
>>
>>and a lot more things.
>>
>>My Problem is:
>>
>>In big expressions i have huge cancellations of the form
>>
>>myDot[M1,Inv[M1+M2+M3+....]] + myDot[M2,Inv[M1+M2+M3+....]]
>> + myDot[M3 ,Inv[M1+M2+M3+....]]+...
>>
>>such that the summands M1,M2.... should be summed and then cancel
>>against the Inv[...] part.
>>
>>I have a very slow workaround,
>>
>>    myDotSimp[HoldPattern[Plus[P6___, myDot[P5___, P1_, P3___],
>>  myDot[P5___,P2_, P3___]]]] := P6 + myDot[P5, P1 + P2, P3];
>>
>>    SetOptions[Simplify, TransformationFunctions ->
>>    {Automatic,myDotSimp}];
>>
>>this thing, however, is immensly time consuming.
>>
>>On the other hand, cancellations of the type
>>Simplify[b/(b+c)+c/(b+c)]
>>are extremly fast.
>>Is there a way to combine the power of Simplify on Rational functions
>>with a noncommutative multiplication?
>>
>>kind regards
>>robert schoefbeck
>>
>
>
> One approach is to use regular multiplication, but to change your variables
> by adding ordering information. Here are the definitions:
>
> Clear[ov]
> Format[ov[ord_, var_]] := var
>
> ov /: ov[i_, x_] + ov[i_, y_] := ov[i, x + y]
> ov /: ov[i_, x_] ov[j_, Inv[x_]] := ov[Max[i, j], 1] /; Abs[i - j] == 1
> ov /: a_. ov[i_, 1] := a /. ov[j_ /; j > i, x_] -> ov[j - 2, x]
>
> Since it is cumbersome to type in ov for each variable, I create a rule that
> converts noncommutative multiplication using CenterDot into regular
> multiplication with ov variables. However, since CenterDot can't be
> displayed in plain text, for the purposes of this post I use
>
> Unprotect[NonCommutativeMultiply];
> NonCommutativeMultiply[a__] := Times @@ MapIndexed[ov[First[#2], #1] &, {a}]
> Protect[NonCommutativeMultiply];
>
> If you try it out, use CenterDot (or something else that looks nice) and you
> won't need the Unprotect/Protect statements.
>
> Now, let's try out your problem:
>
> In[50]:=
> p2**p1**Inv[p1+p3+p4]**p5+p2**p3**Inv[p1+p3+p4]**p5+p2**p4**Inv[p1+p3+p4]**p5
> Out[50]=
> p2 p1 Inv[p1+p3+p4] p5+p2 p3 Inv[p1+p3+p4] p5+p2 p4 Inv[p1+p3+p4] p5
>
> Notice that the proper order of the factors is maintained. The displayed
> form of ov[1,p2] is p2, but the ordering is based on the first argument of
> ov.
>
> Now, let's factor:
>
> In[51]:=
> Factor[%]
> Out[51]=
> p2 p5
>
> Just what you wanted. Now, for a few comments on the ov definitions. It is
> possible that the ov definitions can by very slow due to combinatorial
> explosion associated with the pattern matcher. If this happens, I have more
> complicated definitions which will avoid the combinatorial explosion and
> will hence be much quicker. Second, the rule with a_. ov[i_,1] is there to
> reindex all of the ov variables, so that stuff like
>
> x y Inv[y] Inv[x]
>
> will simplify. If you like this approach, and decided to experiment, I would