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MathGroup Archive 2007

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Re: Re: Trouble with a system of equations


Yaroslav Bulatov wrote:

>As Ray Koopman and some others pointed out, the system of equations
>becomes linear with a proper choice of variables. But now getting the
>original variables is problematic.
>
>In particular, if I replace all terms of the form Exp[a+b+c]/1+Exp[a+b
>+c..] with single variables, I will have to solve the following to get
>a,b,c..back
>
>mapping[k_] := Module[{mm, params, tuples},
>  mm = KroneckerProduct @@ Table[{{1, 1}, {1, 0}}, {i, 1, k}];
>  params = (Subscript[t, #1] & ) /@ Range[1, 2^k];
>  tuples = (Plus @@ (params*#1) & ) /@ mm /. a_Plus :> Times @@ a;
>  terms = (#1/(#1 + 1) & ) /@ tuples;
>  {MapIndexed[#1 == Subscript[m, #2[[1]]] & , terms], params}]
>
>
>Inverting it for 2 variables works
>Solve @@ mapping[2]
>
>But for 3 variables it takes too long
>Solve @@ mapping[3]
>
>Making equations polynomial and solving them gives solution in terms
>of "InverseFunction" for 3 variables and takes too long for 4
>
>  
>
The mapping[k] equations are trivial to solve if one solves them for 
t_1, t_2, etc in order. For example:

tsol =  MapThread[Solve[#1, #2] &, {Reverse[mapping[3][[1]]], 
mapping[3][[2]]}] //Flatten

tsol solves for the t_i in terms of m_i and lower order t_i. Next, we 
replace the t_i in the right hand side with their known values:

solution = Thread[Rule[tsol[[All, 1]], tsol[[All, 2]] //. tsol]];

Finally, let's check that the solution works:

In[41]:= mapping[3][[1]] /. solution // Simplify

Out[41]= {True,True,True,True,True,True,True,True}

I tried this procedure for mapping[7], and it took about 12 seconds.

Carl Woll
Wolfram Research


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