Basin attractors for various methods for multiple roots

There are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss all known methods of orders two to four and present the basin of attraction for several examples. In [1], the authors investigated the basin of attraction for several well-known algorithms for the simple roots of a non-linear equation. The purpose was to propose using the basin of attraction as another method for comparing the algorithms along with such items as order of convergence and efficiency. The authors found that some algorithms have a smooth convergence pattern and others have a rather chaotic pattern, which leads the algorithm to convergence to an unwanted root. In this paper we intend to extend that investigation to algorithms for solving nonlinear equations whose solutions contain roots with multiplicity greater than one. There is a vast literature on the solution of nonlinear equations and nonlinear systems, see for example Ostrowski [2], Traub [3], Neta [4] and references therein. Here we compare several high order fixed point type methods to approximate a multiple root. Newton's method is only of first order unless it is modified to gain the second order of convergence, see Rall [5] or Schröder [6]. This modification requires a knowledge of the multiplicity. Traub [3] has suggested to use any method for f (mÀ1) (x) or f 1/m or gðxÞ ¼ f ðxÞ f 0 ðxÞ. Any such method will require higher derivatives than the corresponding one for simple zeros. Also the first two of those methods require the knowledge of the multiplicity m. In such a case, there are several other methods developed by Hansen Since in general one does not know the multiplicity, Traub [3] suggested a way to approximate it during the iteration. Here we discuss the following methods listed in increasing order of convergence: Werner: A method of order 1.5 for double roots given by Werner [14].