Problems with FindRoot

*To*: mathgroup at smc.vnet.net*Subject*: [mg66235] Problems with FindRoot*From*: kerim.suruliz at gmail.com*Date*: Fri, 5 May 2006 05:02:14 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi all, I'm trying to find minima of the function V[x, y] = 12.3/x^3 + 12*Sqrt[2]*Exp[-4*Pi*y]*Pi^2*Sqrt[y]/x - 10*Exp[-2*Pi*y]*Pi*y/x^2 using FindRoot. The numerical coefficients are such that there is a minimum for exponentially large x, but Mathematica has trouble finding it - presumably because the derivatives of V are extremely small in that region. I tried playing with MaxAccuracy, WorkingPrecision and PrecisionGoal, but without any success. Now, I know analytically where the minimum is so I look for it near there: soln = FindRoot[{D[V[x, y], x] == 0, D[V[x, y], y] == 0}, {x, 3*10^10}, {y, 4.10}, MaxIterations -> 10000] and Mathematica finds it, {{x -> 4.4553238320849495`*^10, y -> 4.170188563200261`}. Even tiny perturbations about this point result in Mathematica failing to give the solution, though. Also, I need to solve a more complicated problem with three variables, but as soon as I add another one, even trivially, as in g[x, y, z] = 12.3/x^3 + 12*Sqrt[2]*Exp[-4*Pi*y]*Pi^2*Sqrt[y]/x - 10*Exp[-2*Pi*y]*Pi*y/x^2 + z^2 (just added z^2 to V[x,y]), minimisation via soln = FindRoot[{D[g[x, y, z], x] == 0, D[g[x, y, z], y] == 0, D[g[x, y, z], z] == 0}, {x, 3*10^10}, {y, 4.10}, {z, 0.4}, MaxIterations -> 10000] fails! The output is: {x -> 3.`*^10, y -> 4.1`, z -> 0.`} Note that the initial condition used for x and y is the same as in the 2D case. I'm using Mathematica 5. The function FindMinimum didn't seem to improve on the situation. Any advice on how to solve these equations/find minima of V[x,y]/g[x,y,z] reliably would be much appreciated. Cheers, Kerim