Fitting implicit function using lsqnonlin

Hi,

I am trying to fit an implicit function to data. As an example I tried to fit:

x = ((D - B*log(y/D) - y)/A)+C.

I get it to fit but the coeffs do not converge to the values I generate the data with, even when I set the initial values very close to the ones I used initially. Is there are better way or is this a problem with the large number of local minima that the routine can run into.

Thanks!

%generate random 20 points between 0.5 and 5.5 (b + (a-b)*rand(1,20)) coeff = [] n = 20; y = 0.5+5*rand(1,n);

%set the parameters we want and generate corresponding x values % we now have data to fit to [x,y]

A = 1.5 B = 2.0 C = 3.5 D = 4.5 x = ((D - B*log(y/D) - y)/A)+C

%put some random error on the y values

err = randn(1,n)/100; ye = y + err;

%plot generated data figure; plot(x, ye, '*r'); hold on;

%set up the function

fun = @(p) (((p(4) - p(2)*log(ye/p(4)) - ye)/p(1))+p(3)) - x

% and the options for the least squares non-linear fit

options = optimset('lsqnonlin'); options.Display = 'iter'; options.MaxFunEvals = 10000;

%inital guesses for parameters abstart = [1.5 2.0 3.4 4.5]; coeff = lsqnonlin(fun,abstart,[],[],options)

%coeff = [1.5 2.0 3.5 4.5]; x_fit = ((coeff(4) - coeff(2)*log(ye./coeff(4)) - ye)/coeff(1))+coeff(3); [x_fit I_fit] = sort(x_fit);

ye_fit = ye(I_fit); plot(x_fit, ye_fit,'-ob');