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Frozen iterative methods using divided differences “à la Schmidt-Schwetlick”. (English) Zbl 1305.90383

The article of Miguel Grau-Sanchez, Miquel Noguera and José M. Gutiérrez is a valuable contribution to the calculus on one of the core questions of mathematics and, by this, of science, at all: To find a zero (or zeroes) of an m-dimensional function F on an m-dimensional convex domain D. That function is assumed to be at least 3 times Fréchet differentiable with continuity on D. That problem is very wide and vast classes of problems can be equivalently represented by them, or approximated or “relaxed” by them. This, and its analytic depth and numerical-computational meaningfulness, give a great importance to this paper.
In fact, the major purpose of this article is to investigate the convergence order and the efficiency of considered 4 families of iterative methods that are involving “frozen divided differences”. The first 2 families correspond to a generalization of the secant method and an implementation introduced by Schmidt and Schwetlick. The other 2 methods are a generalization of Kurchatov method and an improvement of this method applying the technique used by Schmidt and Schwetlick before. The authors approximate the local order of convergence by their examples, and this numerically confirms that the methods order is well deduced. One example is a Hammerstein integral equation. Furthermore, they present and compute the 4 algorithms’ computational efficiency indexes, to compare their efficiencies.
This paper is very well written, structured, documented, illustrated and exemplified,
The six sections of the article are as follows: 1. Introduction, 2. Preliminaries, 3. Main Results, 4. Optimal Computational Efficiency, 5. Numerical Results, and 6. Concluding Remarks.
In the future, further strong analytical results and numerical techniques could be expected, initialized by the present research paper. Those advances might foster and initiate emerging achievements in science and engineering, economics, finance and Operational Research, in medicine and healthcare, etc.

MSC:

90C30 Nonlinear programming

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MPFR
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