Improvements of the efficiency of some three-step iterative like-Newton methods. (English) Zbl 1123.65037
To solve iteratively a nonlinear equation, the classical Newton method can be improved from the viewpoint of convergence order and efficiency. This can be made by an additional evaluation of the function appearing in the left hand side of the equation, an additional evaluation of the first derivative, or by changing the points of evaluation.
The author discusses some new three-step iterative methods obtained from a two-step variant of Newton’s method, where the last evaluation of the derivative is replaced by a linear combination of evaluations of the functions previously computed. The new iterative methods are of third and fourth order of convergence and it seems that they work better than the known variants of Newton’s method in terms of order and of computational efficiency for multi-precision arithmetic. Numerical experiments and comparisons are performed for seven concrete equations.
The author discusses some new three-step iterative methods obtained from a two-step variant of Newton’s method, where the last evaluation of the derivative is replaced by a linear combination of evaluations of the functions previously computed. The new iterative methods are of third and fourth order of convergence and it seems that they work better than the known variants of Newton’s method in terms of order and of computational efficiency for multi-precision arithmetic. Numerical experiments and comparisons are performed for seven concrete equations.
Reviewer: Iulian Coroian (Baia Mare)
MSC:
| 65H05 | Numerical computation of solutions to single equations |
Keywords:
Newton’s method; three-step iterative methods; order of convergence; computational efficiency; numerical experimentsReferences:
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