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

$\|F'(\mathbf x)-F'(\mathbf y)\|\le L\;\|\mathbf x-\mathbf y\|$

holds. The norm on the left is some operator norm that is compatible with the vector norm on the right. This inequality can be rewritten to only use the vector norm. Then for any vector $v\in\R^n$ the inequality

$\|F'(\mathbf x)(v)-F'(\mathbf y)(v)\|\le L\;\|\mathbf x-\mathbf y\|\,\|v\|$

must hold.

Now choose any initial point $\mathbf x_0\in X$. Assume that $F'(\mathbf x_0)$ is invertible and construct the Newton step $\mathbf h_0=-F'(\mathbf x_0)^{-1}F(\mathbf x_0).$

The next assumption is that not only the next point $\mathbf x_1=\mathbf x_0+\mathbf h_0$ but the entire ball $B(\mathbf x_1,\|\mathbf h_0\|)$ is contained inside the set X. Let $M\le L$ be the Lipschitz constant for the Jacobian over this ball.

As a last preparation, construct recursively, as long as it is possible, the sequences $(\mathbf x_k)_k$, $(\mathbf h_k)_k$, $(\alpha_k)_k$ according to

$\begin{alignat}{2} \mathbf h_k&=-F'(\mathbf x_k)^{-1}F(\mathbf x_k)\\[0.4em] \alpha_k&=M\,\|F'(\mathbf x_k)^{-1}\|\,\|h_k\|\\[0.4em] \mathbf x_{k+1}&=\mathbf x_k+\mathbf h_k. \end{alignat}$

Statement

Now if $\alpha_0\le\tfrac12$ then

1. a solution $\mathbf x^*$ of $F(\mathbf x^*)=0$ exists inside the closed ball $\bar B(\mathbf x_1,\|\mathbf h_0\|)$ and
2. the Newton iteration starting in $\mathbf x_0$ converges to $\mathbf x^*$ with at least linear order of convergence.

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

$p(t) =\left(\tfrac12L\|F'(\mathbf x_0)^{-1}\|^{-1}\right)t^2 -t+\|\mathbf h_0\|$,

$t^{\ast/**}=\frac{2\|\mathbf h_0\|}{1\pm\sqrt{1-2\alpha}}$

and their ratio

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

Then

1. a solution $\mathbf x^*$ exists inside the closed ball $\bar B(\mathbf x_1,\theta\|\mathbf h_0\|)\subset\bar B(\mathbf x_0,t^*)$
2. it is unique inside the bigger ball $B(\mathbf x_0,t^{*\ast})$
3. and the convergence to the solution of $F$ is dominated by the convergence of the Newton iteration of the quadratic polynomial $p(t)$ towards its smallest root $t^\ast$, if $t_0=0,\,t_{k+1}=t_k-\tfrac{p(t_k)}{p'(t_k)}$, then

   $\|\mathbf x_{k+p}-\mathbf x_k\|\le t_{k+p}-t_k.$

4. The quadratic convergence is obtained from the error estimate

   $\|\mathbf x_{n+1}-\mathbf x^*\| \le \theta^{2^n}\|\mathbf x_{n+1}-\mathbf x_n\| \le\frac{\theta^{2^n}}{2^n}\|\mathbf h_0\|.$

References

• 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.)

Literature

• Kantorowitsch, L. (1948): Functional analysis and applied mathematics (russ.). UMN3, 6 (28), 89–185.
• Kantorowitsch, L. W.; Akilow, G. P. (1964): Functional analysis in normed spaces.
• P. Deuflhard: Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms., Springer, Berlin 2004, ISBN 3-540-21099-7 (Springer Series in Computational Mathematics, Vol. 35)