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# Question on replacementFunction

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Apr 17, 2010, 6:03:54 AM4/17/10
to
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[[1]], rep, vars];
expon = replacementFunction[expr[[2]], rep, vars];
PolynomialReduce[base^expon, rep, vars][[2]],
MemberQ[Attributes[Evaluate[hed]], NumericFunction],
Map[replacementFunction[#, rep, vars] &, expr],
PolynomialReduce[expr, rep, vars][[2]]]]]] ;

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.

### Bob Hanlon

Apr 18, 2010, 5:58:45 AM4/18/10
to
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

=============

### Andrzej Kozlowski

Apr 18, 2010, 5:58:24 AM4/18/10
to
I can see a bug in replacementFunction. The following code fixes it:

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[[1]], rep, vars];
expon = replacementFunction[expr[[2]], rep, vars];
PolynomialReduce[base^expon, rep, vars][[2]],

If[PolynomialQ[expr, vars],
PolynomialReduce[expr, rep, vars][[2]],

MemberQ[Attributes[Evaluate[hed]], NumericFunction],
Map[replacementFunction[#, rep, vars] &, expr],

PolynomialReduce[expr, rep, vars][[2]]]]]]]

replacementFunction[x*y - w*z, x*y - w*z - 2*A, {x, y, z, w}]

2 A

Andrzej Kozlowski

Apr 20, 2010, 5:51:39 AM4/20/10
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> > also n=-2,-3,... but fails for n=1 or n=-1. Any fix? Thanks=
.

Thanks - this fix solved that problem. Now replacementFunction works even for
expr=(x*y-w*z)^n, with symbolic n. Hopefully this will eventually become a
built-in function that implements algebraic substitution instead of
pattern replacement.

I plan to test it in a more ambitious setting: a 12 x 12 matrix, each
entry of which has about 5000 leaves. The idea is to inject geometric
invariants through repeated replacements, finally ending up with shorter
expressions (about 100 leaves) that can be finished up with Simplify.

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