View source for 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|Newton method]]). For the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551001.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551002.png" /> is a non-linear operator from one Banach space to another, the formula for calculating the root has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551003.png" /></td> </tr></table>

(Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551004.png" /> is the [[Fréchet derivative|Fréchet derivative]].) Sometimes a modified process, given by the following formula, is used:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551005.png" /></td> </tr></table>

Suppose that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551006.png" /> is twice continuously differentiable and that the following conditions hold (see [[#References|[2]]]):

1) the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551007.png" /> exists;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551008.png" />;

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k0551009.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510010.png" />;

4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510011.png" />;

5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510012.png" />.

Then the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510013.png" /> has a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510014.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510015.png" /></td> </tr></table>

The sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510017.png" /> converge to this solution, with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510018.png" /></td> </tr></table>

and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510019.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510020.png" /></td> </tr></table>

The Kantorovich process always converges to a root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510021.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510022.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510023.png" /> is sufficiently smooth, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510024.png" /> exists and the initial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510025.png" /> is chosen sufficiently close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510027.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055100/k05510028.png" />.

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

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



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

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