# Newton method

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method of tangents

A method for the approximation of the location of the roots of a real equation (1)

where is a differentiable function. The successive approximations of Newton's method are computed by the formulas (2)

If is twice continuously differentiable, is a simple root of (1) and the initial approximation lies sufficiently close to , then Newton's method has quadratic convergence, that is, where is a constant depending only on and the initial approximation .

Frequently, for the solution of (1) one applies instead of (2) the so-called modified Newton method: (3)

Under the same assumptions under which Newton's method has quadratic convergence, the method (3) has linear convergence, that is, it converges with the rate of a geometric progression with denominator less than 1.

In connection with solving a non-linear operator equation with an operator , where and are Banach spaces, a generalization of (2) is the Newton–Kantorovich method. Its formulas are of the form where is the Fréchet derivative of at , which is an invertible operator acting from to . Under special assumptions the Newton–Kantorovich method has quadratic convergence, and the corresponding modified method has linear convergence (cf. also Kantorovich process).

I. Newton worked out his method in 1669.