Kantorovich theorem

The Kantorovich theorem is a mathematical statement on the convergence of Newton's method. It was first stated by Leonid Kantorovich in 1940.

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.

Let ${\displaystyle X\subset \mathbb {R} ^{n}}$ be an open subset and ${\displaystyle F:\mathbb {R} ^{n}\supset X\to \mathbb {R} ^{n}}$ a differentiable function with a Jacobian ${\displaystyle F^{\prime }(x)}$ that is locally Lipschitz continuous (for instance if ${\displaystyle F}$ is twice differentiable). That is, it is assumed that for any open subset ${\displaystyle U\subset X}$ there exists a constant ${\displaystyle L>0}$ such that for any ${\displaystyle \mathbf {x} ,\mathbf {y} \in U}$

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