Abstract
In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency index of 1.587. Furthermore, we have extended one of the methods to sixth and twelfth order methods whose efficiency indices are 1.565 and 1.644, respectively. Some numerical examples are tested to demonstrate the performance of the proposed methods, which verifies the theoretical results. Further, we discuss the extraneous fixed points and basins of attraction for a few existing methods, such as Newton’s method and the proposed family of fourth order methods. An application problem arising from Planck’s radiation law has been verified using our methods.
Highlights
IntroductionF 00 is difficult to calculate and computationally more costly, and f 00 in Equation (2) is approximated using the finite difference; still, the convergence order and total number function evaluation are maintained [1]
One of the best root-finding methods for solving nonlinear scalar equation f ( x ) = 0 is Newton’s method
To improve the order of the above method with the same number of function evaluations leading to an optimal method, we propose the following without memory method, which includes weight functions: f
Summary
F 00 is difficult to calculate and computationally more costly, and f 00 in Equation (2) is approximated using the finite difference; still, the convergence order and total number function evaluation are maintained [1]. Such a third order method similar to Equation (2) after approximating f 00 in Halley’s iteration method is given below: yn = xn − β f ( xn ). This paper considers a new family of optimal fourth order methods, which is an improvement of the method given in [16].
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