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