# 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 ):

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 .