Least Squares Parameter Fit for the Catenary

 This iterative correction for the fit of the parameters for the catenary using the least squares gradient formula works quite well too. The equations of condition for x_min and L are fairly smooth over a wide range as these grid plots show.

And the iteration of the corrections converges rapidly to the optimal values of x_min and L using an  initial rough estimate of their values.

Using the formulas for the catenary we can now compute exact values for y_min and y(b).