A third-order modification of Newton's method for multiple roots

Keywords: Newton's method Multiple roots Iterative methods Nonlinear equations Order of convergence Root-finding a b s t r a c t In this paper, we present a new third-order modification of Newton's method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots. Solving nonlinear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a multiple root a of multiplicity m, 1 and f ðmÞ ðaÞ – 0, of a nonlinear equation f ðxÞ ¼ 0. Newton's method is only of first order unless it is modified to gain the second order of convergence, see Schröder [1]. This modification requires a knowledge of the multiplicity. Traub [2] has suggested to use any method for f ðmÞ ðxÞ 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 one of those methods require the knowledge of the multiplicity m. In such a case, there are several other methods developed by Hansen general one does not know the multiplicity, Traub [2] suggested a way to approximate it during the iteration. The way it is done is by evaluating the quotient x nÀ2 À x n x nÀ2 À x nÀ1 and rounding the number up. For example, the quadratically convergent modified Newton's method is (see [1])