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MathGroup Archive 2006

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Re: Re: Simplifying equations for Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65875] Re: [mg65863] Re: Simplifying equations for Mathematica
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Thu, 20 Apr 2006 05:15:01 -0400 (EDT)
  • References: <200604160749.DAA11245@smc.vnet.net> <e22h8k$e56$1@smc.vnet.net> <200604190854.EAA05819@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Maxim wrote:
> On Tue, 18 Apr 2006 11:07:00 +0000 (UTC), Daniel Lichtblau  
> <danl at wolfram.com> wrote:
> [tripe expunged...dl]
> 
> I have my doubts about this method, 

Me too, now that you point it out.


> because the rule (Exp[a_] :> a) will  
> be tried first and so it will replace E^(t1 + t2) with (t1 + t2).
> Numerical tests show that after we replace ti with Log[ti] and get rid of  
> denominators one of the solutions is t1 == t2 == -1, t5 == 0. This  
> suggests that the ordering of the variables can matter:
> 
> In[1]:= f[t1_, t2_, t3_, t4_, t5_] :=
>    Log[Exp[0] + Exp[t1] + Exp[t2] + Exp[t1 + t2] + Exp[t5] +
>      Exp[t5 + t1 + t2] + Exp[t5 + t2 + t4] + Exp[t1 + t2 + t3 + t4 + t5]]
> 
> In[2]:= Lvar = {t1, t2, t3, t4, t5};
>    Lvar2 = {t4, t3, t2, t1, t5};
>    Lm = {m1, m2, m3, m4, m5};
>    Lexpr = D[f @@ Lvar, {Lvar}] - Lm;
>    Lexpr2 = Lexpr /. Thread[Lvar -> Log[Lvar]] // Together // Numerator;
> 
> In[7]:= SetOptions[Roots, Cubics -> False, Quartics -> False];
>    (Lsol = Solve[GroebnerBasis[Lexpr2, Lvar2] == 0, Lvar2] //
>      Simplify) // ByteCount // Timing
> 
> Out[8]= {10.922*Second, 204216}
> 
> The last three seem to be generic solutions:
> 
> In[9]:= Lexpr /. Thread[Lvar -> Log[Lvar /. #]]& /@ Rest@ Lsol /.
>      Thread[Lm -> Array[Random[Real, {-10, 10}, 20]&, Length@ Lm]] //
>    MatrixQ[#, # == 0&]&
> 
> Out[9]= True
> 
> Maxim Rytin
> m.r at inbox.ru

Good catch. There may be some defects in the supply of sacrificial 
virgins. That, or I caught something contagious from an unusual Limit 
thread.

As to the order dependency, one thing I noticed is that variable order 
makes a considerable difference for the Groebner basis computation even 
to complete. A way to encourage it to do so is to use Sort->True. Then 
one can call Solve and hope it discerns the "correct" order (use the one 
for which we already have a Groebner basis). Or else mess with 
Internal`DistributedTermsList to figure out the order used by 
GroebnerBasis, and pass that to Solve with Sort->False.


Daniel Lichtblau
Wolfram Research


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