Substitute expressions with FullSimplify
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From: Guido Walter Pettinari on 26 Mar 2010 06:36 Hello world! This is my first post in this group, but it has been a while since I started reading it. I always found it quite useful, therefore I wish to thank everibody for their contributions! Here is my problem. Let's say I have an expression. I would like to substitute all the occurences of a given subexpression in this expression with a symbol. I want to do it in an intelligent way, i.e. by using FullSimplify instead of ReplaceAll. If my expression is: x^2 + y^2 I know that: FullSimplify [ x^2 + y^2, x^2 + y^2 == r ] will produce 'r' as a result, which is what I want. However, if my expression is x + y , then FullSimplify [ x + y, x + y == r ] produces 'x + y' and not 'r' ! I tried to use FullSimplify [ x + y, x + y == r, ComplexityFunction -> LeafCount ] but I still get 'x+y' as a result. Do you have any idea on how to substitute x+y with r in an expression? Thank you very much, Guido
From: Andrzej Kozlowski on 27 Mar 2010 06:08 FullSimplify is neither intended for this purpose nor an effective way to achieve it. The best way, in my opinion, is to use on of a number of "replacement functions" that have been posted over the years by Daniel Lichtblau, which relies on PolynomialReduce. Here is (I think) the latest version (which has been slightly modified by me): 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]]]]]] Note that you have to write your replacement rule as a "difference" i.e. x^2+y^2====r becomes x^2+y^2-r etc. Here are some replacements: replacementFunction[Exp[x^2 + y^2], x^2 + y^2 - r, {x, y}] E^r replacementFunction[Sin[x + y], x + y - r, {x, y}] Sin[r] (The actual function posted by Daniel could not deal with the last example). Andrzej Kozlowski On 26 Mar 2010, at 11:36, Guido Walter Pettinari wrote: > Hello world! > > This is my first post in this group, but it has been a while since I > started reading it. I always found it quite useful, therefore I wish > to thank everibody for their contributions! > > Here is my problem. Let's say I have an expression. I would like to > substitute all the occurences of a given subexpression in this > expression with a symbol. I want to do it in an intelligent way, i.e. > by using FullSimplify instead of ReplaceAll. > > If my expression is: > > x^2 + y^2 > > I know that: > > FullSimplify [ x^2 + y^2, x^2 + y^2 ==== r ] > > will produce 'r' as a result, which is what I want. > > However, if my expression is > > x + y , > > then > > FullSimplify [ x + y, x + y ==== r ] > > produces 'x + y' and not 'r' ! I tried to use > > FullSimplify [ x + y, x + y ==== r, ComplexityFunction -> LeafCount ] > > but I still get 'x+y' as a result. > > Do you have any idea on how to substitute x+y with r in an expression? > > Thank you very much, > > Guido >
From: David Park on 27 Mar 2010 06:08 Guido, Your example is a little too simple. Why not just use x + y -> r? This is an area where I'm not a total expert and you may get good methods from others, but I will try to offer something. When FullSimplify is good, then it is very very good, but when it is bad it is horrible. I find it difficult to use ComplexityFunction and make it do precisely what I want. It is rather like a crude hammer, when one needs a fine screwdriver. Here us a routine that substitutes by solving equations. One secret to it use is to apply it selectively and in order to substitute in an expression. SubstituteSolve[eqns_List, eliminate_List][expr_] := Module[{result, workexpr, resultrules, neweqns}, neweqns = {result == expr, Sequence @@ eqns}; resultrules = Solve[{result == expr, Sequence @@ eqns}, result, eliminate]; result /. First(a)resultrules] Here is a somewhat complicated example: expr = x + y + Sin[(x + y)^-2] + (x + y)^3 // ExpandAll x + x^3 + y + 3 x^2 y + 3 x y^2 + y^3 + Sin[1/(x^2 + 2 x y + y^2)] We don't want to try solving with a variable inside a Sin function so we apply the routine to the argument of Sin first and then apply it to the entire remaining expression. MapAt[SubstituteSolve[{x + y == r}, {x}][#] &, expr, {{7, 1}}] SubstituteSolve[{x + y == r}, {x}][%] x + x^3 + y + 3 x^2 y + 3 x y^2 + y^3 + Sin[1/r^2] r + r^3 + Sin[1/r^2] I know that this method can have problems, especially if there are multiple solutions. But I doubt if FullSimplify will handle those cases either. And for special cases you might modify the routine, say pick a specific solution, and apply it to specific portions of the initial expression. David Park djmpark(a)comcast.net http://home.comcast.net/~djmpark/ From: Guido Walter Pettinari [mailto:coccoinomane(a)gmail.com] Hello world! This is my first post in this group, but it has been a while since I started reading it. I always found it quite useful, therefore I wish to thank everibody for their contributions! Here is my problem. Let's say I have an expression. I would like to substitute all the occurences of a given subexpression in this expression with a symbol. I want to do it in an intelligent way, i.e. by using FullSimplify instead of ReplaceAll. If my expression is: x^2 + y^2 I know that: FullSimplify [ x^2 + y^2, x^2 + y^2 == r ] will produce 'r' as a result, which is what I want. However, if my expression is x + y , then FullSimplify [ x + y, x + y == r ] produces 'x + y' and not 'r' ! I tried to use FullSimplify [ x + y, x + y == r, ComplexityFunction -> LeafCount ] but I still get 'x+y' as a result. Do you have any idea on how to substitute x+y with r in an expression? Thank you very much, Guido
From: Nasser M. Abbasi on 27 Mar 2010 06:11 "Guido Walter Pettinari" <coccoinomane(a)gmail.com> wrote in message news:hoi2n9$qj7$1(a)smc.vnet.net... > Hello world! > > This is my first post in this group, but it has been a while since I > started reading it. I always found it quite useful, therefore I wish > to thank everibody for their contributions! > > Here is my problem. Let's say I have an expression. I would like to > substitute all the occurences of a given subexpression in this > expression with a symbol. I want to do it in an intelligent way, i.e. > by using FullSimplify instead of ReplaceAll. > > If my expression is: > > x^2 + y^2 > > I know that: > > FullSimplify [ x^2 + y^2, x^2 + y^2 == r ] > > will produce 'r' as a result, which is what I want. > > However, if my expression is > > x + y , > > then > > FullSimplify [ x + y, x + y == r ] > > produces 'x + y' and not 'r' ! I tried to use > > FullSimplify [ x + y, x + y == r, ComplexityFunction -> LeafCount ] > > but I still get 'x+y' as a result. > > Do you have any idea on how to substitute x+y with r in an expression? > Good question. I do not know myself, may be a Mathematica expert can tell us. But I am just curious, why do you consider Simplify [ x + y, x + y == r ] more "intelligent" than In[39]:= x + y /. {x + y -> r} Out[39]= r ? --Nasser
From: Guido Walter Pettinari on 28 Mar 2010 05:06
Thank you all for the answers! @Andrzej Thank you for pointing me to replacementFunction! It works well in most cases, however it seems to fail when the analyzed expression contains lists. For example, this works fine: replacementFunction[HypergeometricPFQ [ a + b, d, x ], a + b - c, {a}] -> HypergeometricPFQ[c, d, x] however: replacementFunction[HypergeometricPFQ[ {a + b}, {d, e}, x ], a + b - c, {a}] returns this error: Part::partw: Part 2 of {{{1},c}} does not exist. >> HypergeometricPFQ::hdiv: HypergeometricPFQ[{{{1},c}}[[2]],{{0},e},x] does not exist. Arguments are not consistent. >> I cannot avoid using a list as an argument since I need to compute 1F2. Moreover, I am afraid I have not the expertise to fix replacementFunction by myself. As a side note, the code for replacementFunction was messed up in you post: the = symbol was always repeated twice. This is the corrected version I am using: 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]]]]]] Example: expr = x/6 + y/6 - 1/2*(3 + x + y); replacementFunction[expr, x + y - r, {x, y}] // Simplify -> 1/6 (-9 - 2 r) @Nasser My example was probably too simple, as David Park noted. Let me consider two more complex expressions: expr1 = x^2/6 + y^2/6 - 1/2*(3 + x^2 + y^2) expr2 = x/6 + y/6 - 1/2*(3 + x + y) The simple use of ReplaceAll on both of them: expr1 /. x^2 + y^2 -> r expr2 /. x + y -> r produces an unmodified output because, as far as I know, ReplaceAll works synctatically i.e. it only looks for the exact expression to substitute and will ignore something simply related to that like x^2/6 + y^2/6. On the other hand, the use of FullSimplify: FullSimplify[expr1, x^2 + y^2 == r] FullSimplify[expr2, x + y == r] works only in the first case, where I succesfully obtain as an output the expression 1/6 (-9 - 2 r) , while it is useless in the x+y == r case. Using replacementFunction, I achieve what I am looking for: replacementFunction[expr, x + y - r, {x, y}] // Simplify -> 1/6 (-9 - 2 r) Cheers, Guido On Mar 27, 10:11 am, "Nasser M. Abbasi" <n...(a)12000.org> wrote: > "Guido Walter Pettinari" <coccoinom...(a)gmail.com> wrote in messagenews:ho= i2n9$qj7$1(a)smc.vnet.net... > > > > > Hello world! > > > This is my first post in this group, but it has been a while since I > > started reading it. I always found it quite useful, therefore I wish > > to thank everibody for their contributions! > > > Here is my problem. Let's say I have an expression. I would like to > > substitute all the occurences of a given subexpression in this > > expression with a symbol. I want to do it in an intelligent way, i.e. > > by using FullSimplify instead of ReplaceAll. > > > If my expression is: > > > x^2 + y^2 > > > I know that: > > > FullSimplify [ x^2 + y^2, x^2 + y^2 == r ] > > > will produce 'r' as a result, which is what I want. > > > However, if my expression is > > > x + y , > > > then > > > FullSimplify [ x + y, x + y == r ] > > > produces 'x + y' and not 'r' ! I tried to use > > > FullSimplify [ x + y, x + y == r, ComplexityFunction -> LeafCou= nt ] > > > but I still get 'x+y' as a result. > > > Do you have any idea on how to substitute x+y with r in an expression? > > Good question. I do not know myself, may be a Mathematica expert can tell > us. But I am just curious, why do you consider > > Simplify [ x + y, x + y == r ] > > more "intelligent" than > > In[39]:= x + y /. {x + y -> r} > Out[39]= r > > ? > > --Nasser |