Re: Simplify and Noncommutativity

*To*: mathgroup at smc.vnet.net*Subject*: [mg60372] Re: Simplify and Noncommutativity*From*: "Carl K. Woll" <carlw at u.washington.edu>*Date*: Wed, 14 Sep 2005 03:27:44 -0400 (EDT)*Organization*: University of Washington*References*: <dfrhi4$g4l$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

"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