Abstract:
A general efficient family of adaptive methods with memory are developed to solving a nonlinear equation such that all previous information
are applied. This technique enables us to achieve the highest efficiency index theoretically and practically.It is proved that this adaptive method
with memory have efficiency index 2, hence competes all the existing without and with memory in the literature. The improvement
of the rate of convergence of these familys are obtained by using two,three and four self-accelerating parameters, in which the orders of
convergence are increased from 2, 4 and 8 to 4, 8 and 16
without any new function evaluation, respectively. It means that without any new function calculations the order of convergence can be improved
up to 100%. Numerical examples and comparisons with existing one-, two-, three-, and four-point methods are included to
demonstrate exceptional convergence speed of the proposed method and confirm theoretical results.
Key Words: Nonlinear equations, Newton's interpolatory polynomial, adaptive method with memory, R-order convergence, self accelerating parameter.
2010 Mathematics Subject Classification: Primary 65H05.
