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.
|||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|
|||L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1964) (Translated from Russian)|
|||L. Collatz, "Funktionalanalysis und numerische Mathematik" , Springer (1964) MR0165651 Zbl 0139.09802|
|||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|
|||N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) MR0362811 Zbl 0524.65001|
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) MR0354951 Zbl 0264.26002|
|[a2]||J.E., jr. Dennis, R. Schnable, "Least change secant updates for quasi-Newton methods" SIAM Review , 21 (1979) pp. 443–459 MR0545880 Zbl 0424.65020|
|[a3]||J.M. Ortega, W.C. Rheinboldt, "Iterative solution of nonlinear equations" , Acad. Press (1970) MR0273810 Zbl 0241.65046|
|[a4]||F.B. Hildebrand, "Introduction to numerical analysis" , Dover, reprint (1987) pp. Chapt. 8 MR0895822 Zbl 0641.65001|
Newton method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Newton_method&oldid=28253