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MathGroup Archive 2006

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Re: Simplifying equations for Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65839] Re: [mg65752] Simplifying equations for Mathematica
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Tue, 18 Apr 2006 06:56:35 -0400 (EDT)
  • References: <200604160749.DAA11245@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Yaroslav Bulatov wrote:
> [This post has been delayed due to email problems - moderator]
> 
> 
> I'm trying to solve some likelihood equations, and Mathematica will not
> 
> finish in reasonable time. I'm wondering if there is a way I can
> 
> rewrite it so that Mathematica can do them
> 
> 
> 
> The concrete example is
> 
> 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]]
> 
> 
> 
> The gradient of f defines a map R^5->R^5, and I need to invert that
> 
> map. (This particular example can be solved by hand, but I'm wondering 
> about other cases of the same form)
> 
> 
> 
> Here's the command I use to solve it (works on Mathematica 5.2 only)
> 
> Solve[Map[Apply[Equal,#]&,Thread[{D[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]],
> {{t1,t2,t3,t4,t5},1}],{m1,m2,m3,m4,m5}}]],{t1,t2,t3,t4,t5}]
> 
> 
> 
> It's been running for several days on a Pentium 2Ghz. Are there simple
> 
> transformations I can do to help Mathematica solve it?


One method is to make it explicitly polynomial by working, in effect, 
with "variables" Exp[t1], etc. This can be done as below. We work with 
expressions rather than equations.

exprs = Map[Apply[Subtract,#]&,
   Thread[{D[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]],
   {{t1,t2,t3,t4,t5},1}],{m1,m2,m3,m4,m5}}]];

Below we make exponentials into variables, keeping the same names. This 
will not work if the variables appear other than in powers of exponentials.

exprs2 = exprs /. {Exp[a_]:>a, Exp[Plus[a__]]:>Apply[Times,a]};
vars = {t1,t2,t3,t4,t5};

This we can solve.

In[75]:= Timing[soln = Solve[exprs2==0, vars];]
Out[75]= {0.344022 Second, Null}

Now take logs of results since in effect we solved for Exp[tj]'s rather 
than tj's.

In[76]:= InputForm[Log[vars] /. First[soln]]
Out[76]//InputForm=
{Log[-(5*m1 - 3*m2 - 5*m3 + 4*m4 - m5)/(4*(-2 + m1 + m2 - m3 + m5))],
  Log[-(3*m1 - 5*m2 - 3*m3 + 4*m4 + m5)/(4*(2 - m1 - m2 + m3 - m5))],
  Log[-(-5*m1 + 3*m2 + 21*m3 - 12*m4 + m5)/(4*(-2 + m1 + m2 - m3 + m5))],
  Log[-((m1 - m2 - 3*m3 + 4*m4 - m5)/(-2 + m1 + m2 - m3 + m5))],
  Log[-(-m1 - m2 + m3 - 4*m4 + 5*m5)/(4*(-2 + m1 + m2 - m3 + m5))]}


Daniel Lichtblau
Wolfram Research





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