# Newton method

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.

#### References

 [1] L.V. Kantorovich, "Functional analysis and applied mathematics" Nat. Bur. Sci. Rep. , 1509 (1952) Uspekhi Mat. Nauk , 3 : 6 (1948) pp. 89–185 MR0053389 Zbl 0034.21203 [2] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian) [3] L. Collatz, "Funktionalanalysis und numerische Mathematik" , Springer (1964) MR0165651 Zbl 0139.09802 [4] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian) MR385655 Zbl 0231.41024 [5] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001