       Re: Question on replacementFunction

• To: mathgroup at smc.vnet.net
• Subject: [mg109235] Re: Question on replacementFunction
• From: Bob Hanlon <hanlonr at cox.net>
• Date: Sun, 18 Apr 2010 05:58:53 -0400 (EDT)

```x*y - w*z -> (2*A) // FullForm

Rule[Plus[Times[x,y],Times[-1,w,z]],Times[2,A]]

-x*y + w*z -> -2*A // FullForm

Rule[Plus[Times[-1,x,y],Times[w,z]],Times[-2,A]]

The LHS of these rules are different forms and will behave differently than you expect.  To get easily understood behaviour, keep the LHS of replacement rules as simple as possible or use multiple rules to address the different forms.

expr = (x*y - w*z)^Range[-2, 2]

{1/(x*y - w*z)^2, 1/(x*y - w*z), 1,
x*y - w*z, (x*y - w*z)^2}

expr /. x -> (2 A + w*z)/y

{1/(4*A^2), 1/(2*A), 1, 2*A, 4*A^2}

expr /. {x*y - w*z -> (2*A), -x*y + w*z -> -2*A}

{1/(4*A^2), 1/(2*A), 1, 2*A, 4*A^2}

Bob Hanlon

=============
Could somebody explain why replacementFunction fails for
the simpler x*y-w*z but works for (x*y-w*z)^2?   Of course the
erratic behavior of ReplaceAll is well known.  Here are the tests
(I took replacementFunction from an earlier thread):

replacementFunction[expr_, rep_, vars_] :=
Module[{num = Numerator[expr], den = Denominator[expr],
If[PolynomialQ[num, vars] &&
PolynomialQ[den, vars] && ! NumberQ[den],
replacementFunction[num, rep, vars]/
replacementFunction[den, rep, vars],
If[hed === Power && Length[expr] == 2,
base = replacementFunction[expr[], rep, vars];
expon = replacementFunction[expr[], rep, vars];
PolynomialReduce[base^expon, rep, vars][],
MemberQ[Attributes[Evaluate[hed]], NumericFunction],
Map[replacementFunction[#, rep, vars] &, expr],
PolynomialReduce[expr, rep, vars][]]]]] ;

expr1 = x*y-w*z; res=x*y-w*z-2*A;
Print[replacementFunction[expr1,res,{x,y,w,z}]//Simplify]; (* fails *)
Print[ReplaceAll[expr1,x*y-w*z->(2*A)]];    (* OK *)
Print[ReplaceAll[expr1,-x*y+w*z->-(2*A)]];  (* fails *)
expr2 =(x*y-w*z)^2;
Print[replacementFunction[expr2,res,{x,y,w,z}]//Simplify]; (* OK *)
Print[ReplaceAll[expr2,x*y-w*z->(2*A)]];    (* OK *)
Print[ReplaceAll[expr2,-x*y+w*z->-(2*A)]];  (* fails *)

Summary: it works for expr =(x*y-w*z)^n if n=2,3,4...
also n=-2,-3,... but fails for n=1 or n=-1.  Any fix?  Thanks.

```

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