Noncommutative Multiplication and Negative coeffcients at the Beginning of an Expression in Mathematica

问题

With the help of some very gracious stackoverflow contributors in this post, I have the following new definition for NonCommutativeMultiply (**) in Mathematica:

Unprotect[NonCommutativeMultiply];
ClearAll[NonCommutativeMultiply]
NonCommutativeMultiply[] := 1
NonCommutativeMultiply[___, 0, ___] := 0
NonCommutativeMultiply[a___, 1, b___] := a ** b
NonCommutativeMultiply[a___, i_Integer, b___] := i*a ** b
NonCommutativeMultiply[a_] := a
c___ ** Subscript[a_, i_] ** Subscript[b_, j_] ** d___ /; i > j :=
c ** Subscript[b, j] ** Subscript[a, i] ** d
SetAttributes[NonCommutativeMultiply, {OneIdentity, Flat}]
Protect[NonCommutativeMultiply];

This multiplication is great, however, it does not deal with negative values at the beginning of an expression, i.e.,
a**b**c + (-q)**c**a
should simplify to
a**b**c - q**c**a
and it will not.

In my multiplication, the variable q (and any integer scaler) is commutative; I am still trying to write a SetCommutative function, without success. I am not in desperate need of SetCommutative, it would just be nice.

It would also be helpful if I were able to pull all of the q's to the beginning of each expression, i.e.,:
a**b**c + a**b**q**c**a
should simplify to:
a**b**c + q**a**b**c**a
and similarly, combining these two issues:
a**b**c + a**c**(-q)**b
should simplify to:
a**b**c - q**a**c**b

At the current time, I would like to figure out how to deal with these negative variables at the beginning of an expression and how to pull the q's and (-q)'s to the front as above. I have tried to deal with the two issues mentioned here using ReplaceRepeated (\\.), but so far I have had no success.

All ideas are welcome, thanks...

回答1:

The key to doing this is to realize that Mathematica represents a-b as a+((-1)*b), as you can see from

In[1]= FullForm[a-b]
Out[2]= Plus[a,Times[-1,b]]

For the first part of your question, all you have to do is add this rule:

NonCommutativeMultiply[Times[-1, a_], b__] := - a ** b

or you can even catch the sign from any position:

NonCommutativeMultiply[a___, Times[-1, b_], c___] := - a ** b ** c

Update -- part 2. The general problem with getting scalars to front is that the pattern _Integer in your current rule will only spot things that are manifestly integers. It wont even spot that q is an integer in a construction like Assuming[{Element[q, Integers]}, a**q**b].
To achieve this, you need to examine assumptions, a process that is probably to expensive to be put in the global transformation table. Instead I would write a transformation function that I could apply manually (and maybe remove the current rule form the global table). Something like this might work:

NCMScalarReduce[e_] := e //.  {
    NonCommutativeMultiply[a___, i_ /; Simplify@Element[i, Reals],b___] 
    :> i a ** b
}

The rule used above uses Simplify to explicitly query assumptions, which you can set globally by assigning to $Assumptions or locally by using Assuming:

Assuming[{q \[Element] Reals},
  NCMScalarReduce[c ** (-q) ** c]] 

returns -q c**c.

HTH

回答2:

Just a quick answer that repeats some of the comments from the previous question. You can remove a couple of the definitions and solve all of the parts of this question using the rule that acts on Times[i,c] where i is commutative and c has the default of Sequence[]

Unprotect[NonCommutativeMultiply];
ClearAll[NonCommutativeMultiply]
NonCommutativeMultiply[] := 1
NonCommutativeMultiply[a___, (i:(_Integer|q))(c_:Sequence[]), b___] := i a**Switch[c, 1, Unevaluated[Sequence[]], _, c]**b
NonCommutativeMultiply[a_] := a
c___**Subscript[a_, i_]**Subscript[b_, j_] ** d___ /; i > j := c**Subscript[b, j]**Subscript[a, i]**d
SetAttributes[NonCommutativeMultiply, {OneIdentity, Flat}]
Protect[NonCommutativeMultiply];

This then works as expected

In[]:= a**b**q**(-c)**3**(2 a)**q
Out[]= -6 q^2 a**b**c**a

Note that you can generalize (_Integer|q) to work on more general commutative objects.

来源：https://stackoverflow.com/questions/5038338/noncommutative-multiplication-and-negative-coeffcients-at-the-beginning-of-an-ex