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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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