New Sixth-Order Improvements of the Jarratt Method

Abstract

Abstract. In this paper, we construct some improvements of the Jarratt method forsolving non-linear equations. A new sixth-order method are developed and numericalexamples are given to support that the method obtained can compete with other sixth-order iterative methods. 1. IntroductionA large number of problems in engineering, applied mathematics, economicsand also in the physical sciences are solved by finding the solution of nonlinearequation f ( x ) = 0. We consider iterative methods to find a simple root x ∗ , i.e., f ( x ∗ ) = 0 and f ′ ( x ∗ ) = 0, of a nonlinear equation f ( x ) = 0 that uses f and f ′ butnot the higher derivatives of f .The best known iterative method for the calculation of x ∗ is Newton’s methoddefined by x n +1 = x n −f ( x n ) f ′ ( x n ) . where x 0 is an initial approximation sufficiently close to x ∗ . This method is quadrat-ically convergent [7].To improve the local order of convergence, many modified methods have beenproposed. The Jarratt method [2] is given by