The iteration formula to find the reciprocal of a given number N by Newton’s method is

Concept:

Newton-Raphson method: It has order of convergence 2 and number of guesses required is 1.

Iteration formula, \({{\rm{x}}_{{\rm{n}} + 1}} = {\rm{\;}}{{\rm{x}}_{\rm{n}}} - \frac{{{\rm{f}}\left( {{{\rm{x}}_{\rm{n}}}} \right)}}{{{\rm{f'}}\left( {{{\rm{x}}_{\rm{n}}}} \right)}}\)

Calculation:

Given, \({\rm{x}} = \frac{1}{{\rm{N}}}\)

Let \({\rm{f}}\left( {\rm{x}} \right) = \frac{1}{{\rm{x}}} - {\rm{N}}\)

\({\rm{f'}}\left( {\rm{x}} \right) = - \frac{1}{{{{\rm{x}}^2}}}\)

\({{\rm{x}}_{{\rm{i}} + 1}} = {{\rm{x}}_{\rm{i}}} - \left( {\frac{{\frac{1}{{{{\rm{x}}_{\rm{i}}}}} - {\rm{N}}}}{{ - \frac{1}{{{\rm{x}}_{\rm{i}}^2}}}}} \right) = {{\rm{x}}_{\rm{i}}} + \left( {\frac{{1 - {\rm{N}}{{\rm{x}}_{\rm{i}}}}}{{{{\rm{x}}_{\rm{i}}}}}} \right)\left( {{\rm{x}}_{\rm{i}}^2} \right)\)

\(\therefore {\rm{\;}}{{\rm{x}}_{{\rm{i}} + 1}} = {{\rm{x}}_{\rm{i}}}\left( {2 - {\rm{N}}{{\rm{x}}_{\rm{i}}}} \right)\)