On Constructing a Cubically Convergent Numerical Method for Multiple Roots

We propose the numerical method defined by xn+1 = xn − λ f(xn − μh(xn)) f ′(xn) , n ∈ N, and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability. 1.. INTRODUCTION THE iteration methods to find the roots of nonlinear equations have various applications in many science problems[1,2,3,4]. Among them, the Newton’s method is one of the most well-known iteration schemes and is modified by many researchers[5,6,7]. Assume that a function f : C → C has a multiple root α with integer multiplicity m ≥ 1 and is analytic in a small neighborhood of α. We find an approximated α by a scheme xn+1 = g(xn), n = 0, 1, 2, · · · , (1) where g : C → C is an iteration function and x0 ∈ C is given. Then we find an approximated α using an iterative method. The roots of the equation are obtained using the following scheme: g(x) = x− λ f(x − μh(x)) f ′(x) (2)