Question on replacementFunction
in [Mathematica]
|
From: carlos on 17 Apr 2010 06:03 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], hed = Head[expr], base, expon}, 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[Head[Evaluate[hed]] === Symbol && 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.
From: Bob Hanlon on 18 Apr 2010 05:58 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 ---- carlos(a)colorado.edu wrote: ============= 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], hed = Head[expr], base, expon}, 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[Head[Evaluate[hed]] === Symbol && 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.
From: Andrzej Kozlowski on 18 Apr 2010 05:58 I can see a bug in replacementFunction. The following code fixes it: replacementFunction[expr_, rep_, vars_] := Module[{num = Numerator[expr], den = Denominator[expr], hed = Head[expr], base, expon}, 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]], If[Head[Evaluate[hed]] === Symbol && 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 On 17 Apr 2010, at 19:04, carlos(a)colorado.edu wrote: > 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], > hed = Head[expr], base, expon}, > 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[Head[Evaluate[hed]] === Symbol && > 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. >
From: carlos on 20 Apr 2010 05:51 On Apr 18, 3:58 am, Andrzej Kozlowski <a...(a)mimuw.edu.pl> wrote: > I can see a bug in replacementFunction. The following code fixes it: > > replacementFunction[expr_, rep_, vars_] := > Module[{num = Numerator[expr], den = Denominator[expr], > hed = Head[expr], base, expon}, > 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]], > If[Head[Evaluate[hed]] === Symbol && > 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 > > On 17 Apr 2010, at 19:04, car...(a)colorado.edu wrote: > > > > > 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], > > hed = Head[expr], base, expon}, > > 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[Head[Evaluate[hed]] === Symbol && > > 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= .. 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.
|
Pages: 1 Prev: DynamicModule not saving reliably a variable between Next: Piecewise once more |