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 x {\displaystyle x} of an equation f ( x ) = 0 {\displaystyle f(x)=0} or a vector solution of a system of equation F ( x ) = 0 {\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.
Assumptions Let X ⊂ R n {\displaystyle X\subset \mathbb {R} ^{n}} be an open subset and F : X ⊂ R n → R n {\displaystyle F:X\subset \mathbb {R} ^{n}\to \mathbb {R} ^{n}} a differentiable function with a Jacobian F ′ ( x ) {\displaystyle F^{\prime }(\mathbf {x} )} that is locally Lipschitz continuous (for instance if F {\displaystyle F} is twice differentiable). That is, it is assumed that for any x ∈ X {\displaystyle x\in X} there is an open subset U ⊂ X {\displaystyle U\subset X} such that x ∈ U {\displaystyle x\in U} and there exists a constant L > 0 {\displaystyle L>0} such that for any x , y ∈ U {\displaystyle \mathbf {x} ,\mathbf {y} \in U}
‖ F ′ ( x ) − F ′ ( y ) ‖ ≤ L ‖ x − y ‖ {\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 v ∈ R n {\displaystyle \mathbf {v} \in \mathbb {R} ^{n}} the inequality
‖ F ′ ( x ) ( v ) − F ′ ( y ) ( v ) ‖ ≤ L ‖ x − y ‖ ‖ v ‖ {\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 x 0 ∈ X {\displaystyle \mathbf {x} _{0}\in X} . Assume that F ′ ( x 0 ) {\displaystyle F'(\mathbf {x} _{0})} is invertible and construct the Newton step h 0 = − F ′ ( x 0 ) − 1 F ( x 0 ) . {\displaystyle \mathbf {h} _{0}=-F'(\mathbf {x} _{0})^{-1}F(\mathbf {x} _{0}).}
The next assumption is that not only the next point x 1 = x 0 + h 0 {\displaystyle \mathbf {x} _{1}=\mathbf {x} _{0}+\mathbf {h} _{0}} but the entire ball B ( x 1 , ‖ h 0 ‖ ) {\displaystyle B(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)} is contained inside the set X {\displaystyle X} . Let M {\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 ( x k ) k {\displaystyle (\mathbf {x} _{k})_{k}} , ( h k ) k {\displaystyle (\mathbf {h} _{k})_{k}} , ( α k ) k {\displaystyle (\alpha _{k})_{k}} according to
h k = − F ′ ( x k ) − 1 F ( x k ) α k = M ‖ F ′ ( x k ) − 1 ‖ ‖ h k ‖ x k + 1 = x k + h k . {\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}}}
Statement Now if α 0 ≤ 1 2 {\displaystyle \alpha _{0}\leq {\tfrac {1}{2}}} then
a solution x ∗ {\displaystyle \mathbf {x} ^{*}} of F ( x ∗ ) = 0 {\displaystyle F(\mathbf {x} ^{*})=0} exists inside the closed ball B ¯ ( x 1 , ‖ h 0 ‖ ) {\displaystyle {\bar {B}}(\mathbf {x} _{1},\|\mathbf {h} _{0}\|)} and the Newton iteration starting in x 0 {\displaystyle \mathbf {x} _{0}} converges to x ∗ {\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 t ∗ ≤ t ∗ ∗ {\displaystyle t^{\ast }\leq t^{**}} of the quadratic polynomial
p ( t ) = ( 1 2 L ‖ F ′ ( x 0 ) − 1 ‖ − 1 ) t 2 − t + ‖ h 0 ‖ {\displaystyle p(t)=\left({\tfrac {1}{2}}L\|F'(\mathbf {x} _{0})^{-1}\|^{-1}\right)t^{2}-t+\|\mathbf {h} _{0}\|} ,
t ∗ / ∗ ∗ = 2 ‖ h 0 ‖ 1 ± 1 − 2 α 0 {\displaystyle t^{\ast /**}={\frac {2\|\mathbf {h} _{0}\|}{1\pm {\sqrt {1-2\alpha _{0}}}}}}
and their ratio
θ = t ∗ t ∗ ∗ = 1 − 1 − 2 α 0 1 + 1 − 2 α 0 . {\displaystyle \theta ={\frac {t^{*}}{t^{**}}}={\frac {1-{\sqrt {1-2\alpha _{0}}}}{1+{\sqrt {1-2\alpha _{0}}}}}.}
Then
a solution x ∗ {\displaystyle \mathbf {x} ^{*}} exists inside the closed ball B ¯ ( x 1 , θ ‖ h 0 ‖ ) ⊂ B ¯ ( x 0 , t ∗ ) {\displaystyle {\bar {B}}(\mathbf {x} _{1},\theta \|\mathbf {h} _{0}\|)\subset {\bar {B}}(\mathbf {x} _{0},t^{*})}
it is unique inside the bigger ball B ( x 0 , t ∗ ∗ ) {\displaystyle B(\mathbf {x} _{0},t^{*\ast })}
and the convergence to the solution of F {\displaystyle F} is dominated by the convergence of the Newton iteration of the quadratic polynomial p ( t ) {\displaystyle p(t)} towards its smallest root t ∗ {\displaystyle t^{\ast }} , if t 0 = 0 , t k + 1 = t k − p ( t k ) p ′ ( t k ) {\displaystyle t_{0}=0,\,t_{k+1}=t_{k}-{\tfrac {p(t_{k})}{p'(t_{k})}}} , then
‖ x k + p − x k ‖ ≤ t k + p − t k . {\displaystyle \|\mathbf {x} _{k+p}-\mathbf {x} _{k}\|\leq t_{k+p}-t_{k}.}
The quadratic convergence is obtained from the error estimate
‖ x n + 1 − x ∗ ‖ ≤ θ 2 n ‖ x n + 1 − x n ‖ ≤ θ 2 n 2 n ‖ h 0 ‖ . {\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}\|.}
Corollary In 1986, Yamamoto proved that the error evaluations of the Newton method such as Doring (1969), Ostrowski (1971, 1973), Gragg-Tapia (1974), Potra-Ptak (1980), Miel (1981), Potra (1984), can be derived from the Kantorovich theorem.
Generalizations There is a q-analog for the Kantorovich theorem. For other generalizations/variations, see Ortega & Rheinboldt (1970).
Applications Oishi and Tanabe claimed that the Kantorovich theorem can be applied to obtain reliable solutions of linear programming.
References
Further reading 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.