Enlarging the radius of convergence for Newton–like method in which the derivative is re-evaluated after certain steps

Numerous attempts have been made to enlarge the radius of convergence for Newton–like method under the same set of conditions.  It turns out that not only the radius of convergence but the error bounds on the distances involved and the uniqueness of the solution ball  can more accurately be defined.

radius of convergence
Newton–like method
Hilbert space
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