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Zbl 1204.65049
Argyros, Ioannis K.; Hilout, Sa{\"\i}d
A convergence analysis for directional two-step Newton methods. (English)
[J] Numer. Algorithms 55, No. 4, 503-528 (2010). ISSN 1017-1398; ISSN 1572-9265/e

The authors are concerned with the problem of approximating a zero $x^{*}$ of a differentiable function $F$ defined on a convex subset $\mathcal{D}\subset \mathcal{H}$ with values in $\mathbb{R}$, $\mathcal{H}$ being Hilbert. They provide a semilocal convergence analysis of the two step directional Newton method (TSDNM) starting from $x_{0}\in\mathcal{D}:$ $$\aligned &y_{n}=x_{n}-\frac{F(x_{n})}{\nabla F(x_{n})\cdot d_{n}}d_{n} \\ &x_{n+1}=y_{n}-\frac{F(y_{n})}{\nabla F(x_{n})\cdot d_{n}}d_{n}. \endaligned$$ Here $d_{n}$ are directions sufficiently close to the gradient, satisfying e.g., a condition: $$\text{For } \| d_{n}\|=1, \quad \exists\gamma\in[0,1] \quad \text{such that } |\nabla F(x_{n})\cdot d_{n}|\geq\gamma\|\nabla F(x_{n})\|.$$ Two different techniques to generate sufficient semilocal convergence conditions are provided together withe corresponding error bounds for the cubically convergent (TSDNM). The first technique uses the concept of recurrent sequences, the second uses a new idea of the authors' on recurrent functions. It is also shown that the order of convergence is three.
[Erwin Schechter (Moers)]

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)
65H10 Systems of nonlinear equations (numerical methods)
49M15 Methods of Newton-Raphson, Galerkin and Ritz types

Keywords: directional two-step Newton method; recurrent sequences; Newton-Kantorovitch hypotheses; Hilbert space; nonlinear equation; Lipschitz/center-Lipschitz condition; recurrent functions; semilocal convergence; error bounds

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