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.[1][2] 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.[3]

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.[1][2]

Assumptions

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 open subset ${\displaystyle U\subset X}$ 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 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 ${\displaystyle \mathbf {v} \in \mathbb {R} ^{n}}$ 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 ${\displaystyle M\leq L}$ be the Lipschitz constant for the Jacobian over this ball.

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

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

Statement

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

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

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

and their ratio

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

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]

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

Corollary

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]

Generalizations

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

Applications

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

References

1. Deuflhard, P. (2004). Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Series in Computational Mathematics. Vol. 35. Berlin: Springer. ISBN 3-540-21099-7.
2. Zeidler, E. (1985). Nonlinear Functional Analysis and its Applications: Part 1: Fixed-Point Theorems. New York: Springer. ISBN 0-387-96499-1.
3. Dennis, John E.; Schnabel, Robert B. (1983). "The Kantorovich and Contractive Mapping Theorems". Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs: Prentice-Hall. pp. 92–94. ISBN 0-13-627216-9.
4. Ortega, J. M. (1968). "The Newton-Kantorovich Theorem". Amer. Math. Monthly. 75 (6): 658–660. doi:10.2307/2313800. JSTOR 2313800.
5. Gragg, W. B.; Tapia, R. A. (1974). "Optimal Error Bounds for the Newton-Kantorovich Theorem". SIAM Journal on Numerical Analysis. 11 (1): 10–13. Bibcode:1974SJNA...11...10G. doi:10.1137/0711002. JSTOR 2156425.
6. Ostrowski, A. M. (1971). "La method de Newton dans les espaces de Banach". C. R. Acad. Sci. Paris. 27 (A): 1251–1253.
7. Ostrowski, A. M. (1973). Solution of Equations in Euclidean and Banach Spaces. New York: Academic Press. ISBN 0-12-530260-6.
8. Potra, F. A.; Ptak, V. (1980). "Sharp error bounds for Newton's process". Numer. Math. 34: 63–72. doi:10.1007/BF01463998.
9. Miel, G. J. (1981). "An updated version of the Kantorovich theorem for Newton's method". Computing. 27 (3): 237–244. doi:10.1007/BF02237981.
10. Potra, F. A. (1984). "On the a posteriori error estimates for Newton's method". Beiträge zur Numerische Mathematik. 12: 125–138.
11. Yamamoto, T. (1986). "A method for finding sharp error bounds for Newton's method under the Kantorovich assumptions". Numerische Mathematik. 49 (2–3): 203–220. doi:10.1007/BF01389624.
12. Rajkovic, P. M.; Stankovic, M. S.; Marinkovic, S. D. (2003). "On q-iterative methods for solving equations and systems". Novi Sad J. Math. 33 (2): 127–137.
13. Rajković, P. M.; Marinković, S. D.; Stanković, M. S. (2005). "On q-Newton–Kantorovich method for solving systems of equations". Applied Mathematics and Computation. 168 (2): 1432–1448. doi:10.1016/j.amc.2004.10.035.
14. Ortega, J. M.; Rheinboldt, W. C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. SIAM. OCLC 95021.
15. Oishi, S.; Tanabe, K. (2009). "Numerical Inclusion of Optimum Point for Linear Programming". JSIAM Letters. 1: 5–8. doi:10.14495/jsiaml.1.5.

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.