Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4631322 | Applied Mathematics and Computation | 2012 | 11 Pages |
Abstract
A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung-Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton's interpolatory polynomial of the second degree. The corresponding R-order of convergence is increased from 8 to 4(1+5/2)â8.472, 9, 10 and 11. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.
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Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
J. DžuniÄ, M.S. PetkoviÄ, L.D. PetkoviÄ,