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).
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