Newton method

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method of tangents

A method for the approximation of the location of the roots of a real equation


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


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:


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.


[1] L.V. Kantorovich, "Functional analysis and applied mathematics" Nat. Bur. Sci. Rep. , 1509 (1952) Uspekhi Mat. Nauk , 3 : 6 (1948) pp. 89–185
[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)
[4] M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[5] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)


The Newton method is also known as the Newton–Raphson method, cf., e.g., [a4], Sect. (10.11) for single equations and Sect. (10.13) for systems of equations.


[a1] P. Lax, S. Burstein, A. Lax, "Calculus with applications and computing" , New York Univ. Inst. Math. Mech. (1972)
[a2] J.E., jr. Dennis, R. Schnable, "Least change secant updates for quasi-Newton methods" SIAM Review , 21 (1979) pp. 443–459
[a3] J.M. Ortega, W.C. Rheinboldt, "Iterative solution of nonlinear equations" , Acad. Press (1970)
[a4] F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. Chapt. 8
How to Cite This Entry:
Newton method. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by Yu.A. Kuznetsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article