Newton Metodundan Elde Edilen Rasyonel Fonksiyonların Dinamiği ve Geometrisi

Abstract

In this study a new iterative method is introduced, aims to find all roots of complex polynomials with help of Newton’s method. There are methods similar to this one, their degrees are introduced by Sutherland in 1989 and which is introduced by Hubbard, Schleicher and Sutherland in 2001. The worst case result obtained in this paper is . The root of polynomial is fixed point of Newton function. Moreover, these fixed points are attractive fixed points. There are attractive basins surrounding eveery attractive fixed points. If Newton’s Method is applied to any point in the attractive basin, the fixed point of Newton’s map or root of polynomial will be found. If at least one point which belongs to attractive basins of every different fixed points then Newton’s method will be applied to these points one could reach all roots of polynomial. In order to achieve this Sutherland distribute points on the disc with equal distance between them. Then he showed at least one point fall in all attractive basins. Hubbard, Schleicher and Sutherland used similar method that they distribute points on circles families with equal diatances then they showed at least one point will fall on the all attractive basins of different roots. In this thesis, we assume that all roots of polynomial is in the unit disc. Unless with an affine transformation all roots can be collected in the unit disc. In order to find roots firstly points distributed on the unit disc. Then Newton’s method is applied to them. If all roots is not found then the distributed points will be rotated. After every rotation Newton’s method will be applied to these points again. With rotation at least one point fall on the attractive basin of every roots. If Newton’s method is applied to these points, they converge to root of polynomial. Therefore, all roots are found. According to this method in the best case with points all roots can be found and in the worst case with points all roots can be found.