Use Theorem 1.11 or 1.12 to estimate the error  in terms of the previous error  as Newton’s Method converges to the given roots. Is the convergence linear or quadratic?

THEOREM 1.11

Let f be twice continuously differentiable and f(r) = 0. If f’(r) ≠ 0, then Newton’s Method is locally and quadratic ally convergent to r. The error  at step i satisfies

THEOREM 1.12

Assume that the (m + 1)-times continuously differentiable function f on [a,b] has a multiplicity m root at r. Then Newton’s Method is locally convergent to r, and the error  at step i satisfies

 

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