A modification of the Kantorovich conditions for the secant method (Q2743170)
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scientific article; zbMATH DE number 1651227
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A modification of the Kantorovich conditions for the secant method |
scientific article; zbMATH DE number 1651227 |
Statements
26 September 2001
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secant method
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Kantorovich conditions
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recurrence relations
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A modification of the Kantorovich conditions for the secant method (English)
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For a nonlinear operator \(F\) between Banach spaces the authors prove a convergence theorem of Kantorovich--type for the abstract secant method \(x_{n+1} = x_n - [x_{n-1},x_n;F]^{-1} F(x_n)\) where the usual Lipschitz or Hölder condition for the divided difference operator is replaced by \(\|[x,y;F] - [v,w;F] \|\leq \omega(\|x-v \|, \|y-w \|)\) with a suitable nondecreasing function \(\omega\). As an example, the result is applied to a second order ODE boundary value problem.
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