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

**References**:**Simplifying equations for Mathematica***From:*Yaroslav Bulatov <yaroslavvb@gmailnospa.com>