# 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 $P ( x) = 0$, where $P$ is a non-linear operator from one Banach space to another, the formula for calculating the root has the form

$$x _ {n+} 1 = x _ {n} - [ P ^ { \prime } ( x _ {n} ) ] ^ {-} 1 P ( x _ {n} ) .$$

(Here $P ^ { \prime }$ is the Fréchet derivative.) Sometimes a modified process, given by the following formula, is used:

$$\widetilde{x} _ {n+} 1 = \ \widetilde{x} _ {n} - [ P ^ { \prime } ( x _ {0} ) ] ^ {-} 1 P ( \widetilde{x} _ {n} ) .$$

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

1) the linear operator $\Gamma _ {0} = [ P ^ { \prime } ( x _ {0} ) ] ^ {-} 1$ exists;

2) $\| \Gamma _ {0} P ( x _ {0} ) \| \leq \eta$;

3) $\| \Gamma _ {0} P ^ { \prime\prime } ( x) \| \leq K$ when $\| x - x _ {0} \| \leq r$;

4) $h = K \eta \leq 1 / 2$;

5) $r \geq r _ {0} = ( 1 - \sqrt 1- 2h ) \eta / h$.

Then the equation $P ( x) = 0$ has a solution $x ^ {*}$ such that

$$\| x ^ {*} - x _ {0} \| \leq r _ {0} .$$

The sequences $x _ {n}$ and $\widetilde{x} _ {n}$ converge to this solution, with

$$\| x ^ {*} - x _ {n} \| \leq \frac{( 2 h ) ^ {2 ^ {n} } \eta }{2 ^ {n} h } ,$$

and in the case $h < 1 / 2$,

$$\| x ^ {*} - \widetilde{x} _ {n} \| \leq \frac{( 1 - \sqrt 1- 2h ) ^ {n+} 1 \eta }{h} .$$

The Kantorovich process always converges to a root $x ^ {*}$ of the equation $P ( x) = 0$, provided that $P$ is sufficiently smooth, $[ P ^ { \prime } ( x ^ {*} ) ] ^ {-} 1$ exists and the initial approximation $x _ {0}$ is chosen sufficiently close to $x ^ {*}$. If $P ^ { \prime\prime } ( x)$ 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 $x _ {0} \rightarrow x ^ {*}$.

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)