Vandermonde Matrix/Optimization question

*To*: mathgroup at smc.vnet.net*Subject*: [mg71336] Vandermonde Matrix/Optimization question*From*: scottmacd123 at gmail.com*Date*: Wed, 15 Nov 2006 06:43:43 -0500 (EST)

Hello all, I am trying to find the numerical values of x[i] variables that maximize the determinant of a Vandermonde matrix. For small N x N matrices where N <= 7, Mathematica has no problem doing this calculuation. Indeed: VandermondMatrix[n_]:=Table[x[i]^j,{i,1,n},{j,1,n}] vars[n_] := Table[x[i], {i, 1, n}] cons[n_] := Table[-1 â?¤ x[i] â?¤ 1, {i, 1, n}] NMaximize[{Det[VandermondMatrix[4], cons[4]}, vars[4], Method -> {"NelderMead","RandomSeed"->100}] gives: {0.366453, {x[1] -> 1., x[2] -> -1., x[3] -> 0.654654, x[4] -> -0.654654}} I want to do this same calculation for larger N, say N=100. Clearly the code I have above will fail because I am asking Mathematica to calculate the determinant of a 100x100 matrix symbolically, which is not feasible. Obviously the determinant must be evaluated numerically. I have searched this group and found others with similar problems but so far I have not been able to come up with a solution. I think what I want to do is something like the following, 1. Define a function that tests if lists are numeric: num[c_List]:=And @@ (NumericQ /@ c) (I don't fully understand this line, I found it in a post in this group) 2. Define a function f that takes as arguments the dimension of the Matrix and a list of variables f[n_?NumericQ, x_List?num]:=Det[VandermondMatrix[n]] 3. Call NMaximize on f While the above does not work, I hope it is clear what I want to do: have NMaximize fill in numerical values for the x[i] variables before calling Det[]. Thank you in advance for any help/suggestions. Scott MacDonald PS: $Version returns 5.2 for Mac OS X