Re: Solving nonlinear equations with mathematica
- To: mathgroup at smc.vnet.net
- Subject: [mg35456] Re: Solving nonlinear equations with mathematica
- From: Selwyn Hollis <slhollis at earthlink.net>
- Date: Sat, 13 Jul 2002 03:48:33 -0400 (EDT)
- References: <ag3f2k$749$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Sebastian, You're dealing with a system that has a singular Jacobian at the root---much like a single function with a zero derivative at a root. You're probably observing _extremely_ slow convergence. It's also possible that the system you constructed doesn't have a root anywhere near where you think it does. Try perturbing the system with the known solution _much_ less than with Random[]---perhaps by just rounding some of the X[[i,j,3]] values to 2 decimal places. You'll see that FindRoot does find the solution. Hope that helps. tschu Selwyn Hollis slhollis at mac.com http://www.math.armstrong.edu/faculty/hollis Sebastian Pokutta wrote: > Hi together, > > I try to solve a system of 9 nonlinear Equations of the following forms: > > Sum[(1/Sqrt[( S[[i, 1]] - X_j)^2 + ( S[[i, 2]] - Y_j)^2])*S[[i, 3]], > {i, 1, 3}] = Z_j ( 1 <= j <= 9 ) > > I tried this with "findroot" and 9 Points (X_j,Y_j,Z_j) ( 1 <= j <= 9 ) > searching the S[[i,k]] ( a 3 x 3 matrix ) in the following way: > > X = {{127, 17, 5.0316}, {118, 54, 1.9392}, {5, 10, 1.60571}, {67, 54, > 1.04543}, {7, 15, 1.19647}, {24, 11, 1.10665}, {17, 21, 1.85287}, {31, > 18, 1.59878}, {35, 7, 1.10325}}; > > FindRoot[{(1/Sqrt[(s11 - X[[1, 1]])^2 + > (s12 - X[[1, 2]])^2])*s13 + (1/Sqrt[(s21 - X[[1, 1]])^2 + (s22 - > X[[1, 2]])^2])* s23 + (1/Sqrt[(s31 - X[[1, 1]])^2 + (s32 - X[[1, <snip> > (1/Sqrt[(s11 - X[[9, 1]])^2 + (s12 - X[[9, 2]])^2])* s13 + >(1/Sqrt[(s21 - X[[9, 1]])^2 + (s22 -X[[9, 2]])^2])*s23 + >(1/Sqrt[(s31 - X[[9, 1]])^2 + (s32 - X[[9, 2]])^2])*s33 - X[[9, 3]] == 0}, >{s11, 130}, {s12, 54}, {s13, 11},{s21, 125}, {s22, 10}, {s23, 34}, >{s31, 5}, {s32, 33}, {s33, 8}, > DampingFactor -> 2, AccuracyGoal -> 4, MaxIterations -> 15] > > but "findroot" exits with: > > FindRoot::"cvnwt": "Newton's method failed to converge to the prescribed > accuracy after \!\(15\) iterations." > > But THERE'S a solution because I constructed the 9 Points using a given S > and just disturbed them with a random-value between 0 and 1. So is there a > sensible way to calculate the S sucht that the the error is minimal? > > perhaps the way I constructed them: > > given arbitrary X and Y the following equation calculates the Z. > > ll[a_, b_] = Sum[(1/Sqrt[( S[[i, 1]] - a)^2 + ( S[[i, 2]] - > b)^2])*S[[i, 3]], {i, 1, 3}] + Random[]; > > with the following S > > S = {{130, 54, 11}, {125, 10, 34}, {5, 33, 8}}; > > > Help would be greatly appreciated. > > Cu, > Sebastian >