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Consider the equation x-1- a = 0 , (1) where a > 0 is a given...

Consider the equation
x-1- a = 0 , (1)
where a > 0 is a given number.
( I ) Write the Newton iteration applied to the equation (1), and argue that the inverse of a can be computed without performing any divisions (only multiplications, additions, and subtractions). Note: this idea lies at the basis of iterative methods for solving linear systems.

( II ) Use your iteration formula from (i) to derive a recursive formula for the error en = x*−xn (that is, write en+1 as a function of en), where x* = a-1.

( III ) Find the set of points x0 which, if used as initial guess for the Newton iteration in (i), lead to the convergence xn → x*.

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