Kantorovich process

An iterative method for improving the approximation to the value of a root of a non-linear functional (operator) equation (a generalization of Newton's method cf. Newton method). For the equation , where is a non-linear operator from one Banach space to another, the formula for calculating the root has the form

(Here is the Fréchet derivative.) Sometimes a modified process, given by the following formula, is used:

Suppose that the operator is twice continuously differentiable and that the following conditions hold (see [2]):

1) the linear operator exists;

2) ;

3) when ;

4) ;

5) .

Then the equation has a solution such that

The sequences and converge to this solution, with

and in the case ,

The Kantorovich process always converges to a root of the equation , provided that is sufficiently smooth, exists and the initial approximation is chosen sufficiently close to . If exists and is continuous, then the convergence of the basic process is quadratic. The rate of convergence of the modified process is that of a decreasing geometric progression; the denominator of this progression tends to zero as .

The process was proposed by L.V. Kantorovich [1].

References

[1]	L.V. Kantorovich, "On Newton's method for functional equations" Dokl. Akad. Nauk SSSR , 59 : 6 (1948) pp. 1237–1240 (In Russian)
[2]	L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)
[3]	M.A. Krasnosel'skii, G.M. Vainikko, P.P. Zabreiko, et al., "Approximate solution of operator equations" , Wolters-Noordhoff (1972) (Translated from Russian)
[4]	L. Collatz, "Funktionalanalysis und numerische Mathematik" , Springer (1964)

Comments

The modified process is also called the Newton–Raphson method.

References

[a1]	J.E. Denis jr., R. Schnable, "Numerical methods for unconstrained optimization and nonlinear equations" , Prentice-Hall (1983)
[a2]	J.M. Ortega, W.C. Rheinboldt, "Iterative solution of non-linear equations in several variables" , Acad. Press (1970)