Trouble with a system of equations

*To*: mathgroup at smc.vnet.net*Subject*: [mg77488] Trouble with a system of equations*From*: Yaroslav Bulatov <yaroslavvb at gmail.com>*Date*: Sun, 10 Jun 2007 07:20:40 -0400 (EDT)

Hi, I'm trying to solve a certain kind of system of equations, and while they are solvable by hand, Mathematica 6.0 has problems solving it Here's an example eqns = {a + b + c + d == 4*m0, b + d == 4*m1, c + d == 4*m2, d == 4*m3} /. {a -> t0/(1 + t0), b -> (t0*t1)/(1 + t0*t1), c -> (t0*t2)/(1 + t0*t2), d -> (t0*t1*t2*t3)/(1 + t0*t1*t2*t3)} Solve[eqns, {t0, t1, t2, t3}] The solution can be found by hand and verified below sol = {t0 -> a/(1/4 - a), t1 -> (b/(1/4 - b))*((1/4 - a)/a), t2 -> (c/ (1/4 - c))*((1/4 - a)/a), t3 -> (m3/(1/4 - m3))*(a/(1/4 - a))*((1/4 - b)/b)*((1/4 - c)/c)} /. {a -> m0 - m1 - m2 + m3, b -> m1 - m3, c -> m2 - m3} eqns /. sol // Simplify This is an example of estimating equations for a saturated logistic regression model with 2 independent variables. I'd like to see if formulas also exist for more variables, but they are too cumbersome to solve by hand. Are there any Mathematica tricks I can use to answer this question? Here's the procedure that generates the system of equations for d variables (d=2 produces the system above) logeq[d_] := Module[{bounds, monomials, params, partition,derivs,sums}, xs = (Subscript[x, #1] & ) /@ Table[i, {i, 1, d}]; monomials = Subsets[xs]; monomials = (Prepend[#1, 1] & ) /@ monomials; monomials = (Times @@ #1 & ) /@ monomials; params = (Subscript[th, #1] & ) /@ Table[i, {i, 0, 2^d - 1}]; monomials = (Times @@ #1 & ) /@ Thread[{params, monomials}]; partition = Log[1 + Exp[Plus @@ monomials]]; derivs = (D[partition, Subscript[th, #1]] & ) /@ Table[i, {i, 0, 2^d - 1}]; bounds = ({#1, 0, 1} & ) /@ xs; sums = (Table[#1, Evaluate[Sequence @@ bounds]] & ) /@ derivs; sums = (Plus @@ #1 & ) /@ (Flatten[#1] & ) /@ sums; Thread[sums == Table[Subscript[m, i], {i, 0, 2^d - 1}]]]

**Follow-Ups**:**Re: Trouble with a system of equations***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>