Re: question 1

*To*: mathgroup at smc.vnet.net*Subject*: [mg75873] Re: [mg75810] question 1*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 12 May 2007 03:11:17 -0400 (EDT)*References*: <200705110924.FAA05900@smc.vnet.net> <A52D6750-3017-4EF6-82F7-DFE4CB83812F@mimuw.edu.pl> <CB1CAE8E-3B30-4645-9279-14C936512446@mimuw.edu.pl>

On 12 May 2007, at 10:37, Andrzej Kozlowski wrote: > > On 12 May 2007, at 10:23, Andrzej Kozlowski wrote: > >> On 11 May 2007, at 18:24, dimitris wrote: >> >>> I have >>> >>> In[5]:= >>> f = (o - 8)^4 - (e + 4)^8 >>> >>> Out[5]= >>> -(4 + e)^8 + (-8 + o)^4 >>> >>> In[6]:= >>> ff = Expand[f] >>> >>> Out[6]= >>> -61440 - 131072*e - 114688*e^2 - 57344*e^3 - 17920*e^4 - 3584*e^5 - >>> 448*e^6 - 32*e^7 - e^8 - 2048*o + 384*o^2 - 32*o^3 + o^4 >>> >>> Is it possible to simplify ff to f again? >>> >>> Thanks! >>> >>> >> >> I don't think Mathematica can do it automatically, because the >> most obvious way of carrying out this simplification relies on >> transformations of the kind that that Simplify or FullSimplify >> never use (such as adding and simultaneously subtracting some >> number, then rearranging the whole expression and factoring parts >> of it). These kind of transformations could be implemented but >> they would work in only a few cases, and would considerably >> increase the time complexity of simplifying. Here is an example of >> another transformation that will work in this and some similar >> cases: >> >> transf[f_, {e_, o_}] := >> With[{a = Integrate[D[f, e], e], b = Integrate[D[f, o], o]}, >> Simplify[a] + Simplify[b] + Simplify[(f - (a + b))]] >> >> (one can easily implment a version with more than two variables). >> >> This works in your case: >> >> ff = Expand[(o - 8)^4 - (e + 4)^8]; >> >> transf[ff, {e, o}] >> (o - 8)^4 - (e + 4)^8 >> >> and in quite many cases like yours : >> >> gg = Expand[(a + 3)^5 + (x - 7)^6]; >> >> transf[gg, {a, x}] >> (x - 7)^6 + (a + 3)^5 >> >> but not in all >> >> hh = Expand[(x - 5)^4 + (y - 3)^3]; >> >> transf[hh, {x, y}] >> (x - 5)^4 + y*((y - 9)*y + 27) - 27 >> >> Even this, however, is better than the answer FullSimplify gives: >> >> FullSimplify[hh] >> >> (x - 10)*x*((x - 10)*x + 50) + y*((y - 9)*y + 27) + 598 >> >> Unfortunately transf also has very high complexity (it uses >> Integrate) and is unlikely to be useful in cases other than sums >> of polynomials in different variables (without "cross terms") so I >> doubt that it would be worth implementing some version of it in >> FullSimplify. >> >> >> Andrzej Kozlowski >> > > I forgot that Integrate already maks use of Simplify, so (probably) > the folowing version of transf will work just as well and faster: > > transf[f_, {e_, o_}] := > With[{a = Integrate[D[f, e], e], b = Integrate[D[f, o], o]}, > a + b + Simplify[(f - (a + b))]] > > Andrzej Kozlowski Sorry, that last remark just not true. Cleary Integrate uses Simplify before integration but does not Simplify its output (obviously to save time) so including Simplify does make a difference. Note also the following works much better, but is,, of course, much more tiem consuming: transf[f_, {e_, o_}] := With[{a = Integrate[D[f, e], e], b = Integrate[D[f, o], o]}, FullSimplify[Simplify[a] + Simplify[b] + Simplify[(f - (a + b))], ExcludedForms -> {(x + c_)^n_, (y + d_)^m_}]] This will deal with many cases that would not work before: hh = Expand[(x - 5)^4 + (y - 3)^3]; transf[hh, {x, y}] (x - 5)^4 + (y - 3)^3 p = Expand[(x - 2)^4 + (y + 1)^2]; transf[p, {x, y}] (x - 2)^4 + (y + 1)^2 but at least one power has to be higher than 3. This this will work q = Expand[(x - 2)^4 + (y + 1)^3]; transf[q, {x, y}] (x - 2)^4 + (y + 1)^3 but this will not (all powers are <=3) q = Expand[(x - 2)^3 + (y + 1)^3]; transf[q, {x, y}] x*((x - 6)*x + 12) + y*(y*(y + 3) + 3) - 7 Andrzej Kozlowski

**References**:**question 1***From:*dimitris <dimmechan@yahoo.com>