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 > NonCommutativeMultiply instead: > > 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 > be happy to have a dialog about your results. > > Carl Woll > Wolfram Research > >

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