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cables -- iterative techniques

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Anyone out there work much with cables?

I'm curious about the iterative solutions I have seen, such as on pages 4.23
to 4.27 of  the Structural Steel Designer's Handbook, 2nd ed. by Brockenbrough
& Merritt.

(I like to use these simpler methods to get close before using a more exact
analysis such as can be found in "Analysis of cable & catenary structures" by
Broughton & Ndumbaro.)

Sometimes the Merritt method converges, and sometimes it doesn't.  With low
initial sags, the method doesn't converge.  If the initial sag is increased,
the method then converges.  Has anyone found this to be the case, and does
anyone have an explanation?

Brian Veit, P.E.