(i) F is differentiable at least up to order 3,

(ii) F has a simple root z in (a,b) i.e. F(z)=0, and F'(z) is different from zero.

Then there is an r>0 such that for every start-value x

x

converges to z.

The convergence is at least of order two.

We say that the

The picture above illustrates Newton's method in the case of a quadratic function. x

This method can be applied to obtain quickly the square root of a number. In fact taking F(x) = x

More general, the sequence with x

NewtonIterative2.html

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