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

${\displaystyle \|F'(\mathbf {x} )-F'(\mathbf {y} )\|\leq L\;\|\mathbf {x} -\mathbf {y} \|}$ 

holds. The norm on the left is the operator norm. In other words, for any vector ${\displaystyle \mathbf {v} \in \mathbb {R} ^{n}}$  the inequality

${\displaystyle \|F'(\mathbf {x} )(\mathbf {v} )-F'(\mathbf {y} )(\mathbf {v} )\|\leq L\;\|\mathbf {x} -\mathbf {y} \|\,\|\mathbf {v} \|}$ 

must hold.

Now choose any initial point ${\displaystyle \mathbf {x} _{0}\in X}$ . Assume that ${\displaystyle F'(\mathbf {x} _{0})}$  is invertible and construct the Newton step ${\displaystyle \mathbf {h} _{0}=-F'(\mathbf {x} _{0})^{-1}F(\mathbf {x} _{0}).}$ 

The next assumption is that not only the next point ${\displaystyle \mathbf {x} _{1}=\mathbf {x} _{0}+\mathbf {h} _{0}}$  but the entire ball ${\displaystyle B(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)}$  is contained inside the set ${\displaystyle X}$ . Let ${\displaystyle M}$  be the Lipschitz constant for the Jacobian over this ball (assuming it exists).

As a last preparation, construct recursively, as long as it is possible, the sequences ${\displaystyle (\mathbf {x} _{k})_{k}}$ , ${\displaystyle (\mathbf {h} _{k})_{k}}$ , ${\displaystyle (\alpha _{k})_{k}}$  according to

${\displaystyle {\begin{alignedat}{2}\mathbf {h} _{k}&=-F'(\mathbf {x} _{k})^{-1}F(\mathbf {x} _{k})\\[0.4em]\alpha _{k}&=M\,\|F'(\mathbf {x} _{k})^{-1}\|\,\|\mathbf {h} _{k}\|\\[0.4em]\mathbf {x} _{k+1}&=\mathbf {x} _{k}+\mathbf {h} _{k}.\end{alignedat}}}$ 

Now if ${\displaystyle \alpha _{0}\leq {\tfrac {1}{2}}}$  then

1. a solution ${\displaystyle \mathbf {x} ^{*}}$  of ${\displaystyle F(\mathbf {x} ^{*})=0}$  exists inside the closed ball ${\displaystyle {\bar {B}}(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)}$  and
2. the Newton iteration starting in ${\displaystyle \mathbf {x} _{0}}$  converges to ${\displaystyle \mathbf {x} ^{*}}$  with at least linear order of convergence.

A statement that is more precise but slightly more difficult to prove uses the roots ${\displaystyle t^{\ast }\leq t^{**}}$  of the quadratic polynomial

${\displaystyle p(t)=\left({\tfrac {1}{2}}L\|F'(\mathbf {x} _{0})^{-1}\|^{-1}\right)t^{2}-t+\|\mathbf {h} _{0}\|}$ ,

${\displaystyle t^{\ast /**}={\frac {2\|\mathbf {h} _{0}\|}{1\pm {\sqrt {1-2\alpha _{0}}}}}}$ 

and their ratio

${\displaystyle \theta ={\frac {t^{*}}{t^{**}}}={\frac {1-{\sqrt {1-2\alpha _{0}}}}{1+{\sqrt {1-2\alpha _{0}}}}}.}$ 

Then

1. a solution ${\displaystyle \mathbf {x} ^{*}}$  exists inside the closed ball ${\displaystyle {\bar {B}}(\mathbf {x} _{1},\theta \|\mathbf {h} _{0}\|)\subset {\bar {B}}(\mathbf {x} _{0},t^{*})}$ 
2. it is unique inside the bigger ball ${\displaystyle B(\mathbf {x} _{0},t^{*\ast })}$ 
3. and the convergence to the solution of ${\displaystyle F}$  is dominated by the convergence of the Newton iteration of the quadratic polynomial ${\displaystyle p(t)}$  towards its smallest root ${\displaystyle t^{\ast }}$ ,^[4] if ${\displaystyle t_{0}=0,\,t_{k+1}=t_{k}-{\tfrac {p(t_{k})}{p'(t_{k})}}}$ , then

   ${\displaystyle \|\mathbf {x} _{k+p}-\mathbf {x} _{k}\|\leq t_{k+p}-t_{k}.}$ 

4. The quadratic convergence is obtained from the error estimate^[5]

   ${\displaystyle \|\mathbf {x} _{n+1}-\mathbf {x} ^{*}\|\leq \theta ^{2^{n}}\|\mathbf {x} _{n+1}-\mathbf {x} _{n}\|\leq {\frac {\theta ^{2^{n}}}{2^{n}}}\|\mathbf {h} _{0}\|.}$ 

In 1986, Yamamoto proved that the error evaluations of the Newton method such as Doring (1969), Ostrowski (1971, 1973),^[6][7] Gragg-Tapia (1974), Potra-Ptak (1980),^[8] Miel (1981),^[9] Potra (1984),^[10] can be derived from the Kantorovich theorem.^[11]

There is a q-analog for the Kantorovich theorem.^[12][13] For other generalizations/variations, see Ortega & Rheinboldt (1970).^[14]

Oishi and Tanabe claimed that the Kantorovich theorem can be applied to obtain reliable solutions of linear programming.^[15]

• John H. Hubbard and Barbara Burke Hubbard: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Matrix Editions, ISBN 978-0-9715766-3-6 (preview of 3. edition and sample material including Kant.-thm.)
• Yamamoto, Tetsuro (2001). "Historical Developments in Convergence Analysis for Newton's and Newton-like Methods". In Brezinski, C.; Wuytack, L. (eds.). Numerical Analysis : Historical Developments in the 20th Century. North-Holland. pp. 241–263. ISBN 0-444-50617-9.