词条 | Kantorovich theorem |
释义 |
The Kantorovich theorem (Newton-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 of an equation or a vector solution of a system of equation . 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. AssumptionsLet be an open subset and a differentiable function with a Jacobian that is locally Lipschitz continuous (for instance if is twice differentiable). That is, it is assumed that for any open subset there exists a constant such that for any 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 the inequality must hold. Now choose any initial point . Assume that is invertible and construct the Newton step The next assumption is that not only the next point but the entire ball is contained inside the set . Let be the Lipschitz constant for the Jacobian over this ball. As a last preparation, construct recursively, as long as it is possible, the sequences , , according to StatementNow if then
A statement that is more precise but slightly more difficult to prove uses the roots of the quadratic polynomial , and their ratio Then
Notes1. ^{{cite journal |first=J. M. |last=Ortega |title=The Newton-Kantorovich Theorem |journal=Amer. Math. Monthly |volume=75 |year=1968 |issue=6 |pages=658–660 |jstor=2313800 |doi=10.2307/2313800}} 2. ^{{cite journal |first=W. B. |last=Gragg |first2=R. A. |last2=Tapia |year=1974 |title=Optimal Error Bounds for the Newton-Kantorovich Theorem |journal=SIAM Journal on Numerical Analysis |volume=11 |issue=1 |pages=10–13 |jstor=2156425 |doi=10.1137/0711002}} References
Literature
4 : Functional analysis|Theorems in analysis|Optimization in vector spaces|Optimization algorithms and methods |
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