Kantorovich theorem

The Kantorovich theorem, or Newton–Kantorovich theorem, is a mathematical statement on the semi-local convergence of Newton's method. It was first stated by Leonid Kantorovich in 1948. It is similar to the form of the Banach fixed-point theorem, although it states existence and uniqueness of a zero rather than a fixed point.

Newton's method constructs a sequence of points that under certain conditions will converge to a solution ${\displaystyle x}$ of an equation ${\displaystyle f(x)=0}$ or a vector solution of a system of equation ${\displaystyle F(x)=0}$. The Kantorovich theorem gives conditions on the initial point of this sequence. If those conditions are satisfied then a solution exists close to the initial point and the sequence converges to that point.