Complex Dynamics of Relaxed Newton's Method and its Application to Fourth-Degree Polynomials

الملخص

The Newton-Raphsen method is an important numerical method for finding the roots of nonlinear equations and is considered one of the best numerical methods in terms of convergence. Furthermore, the relaxed Newton method, used to find the roots of nonlinear equations with recurring roots, is of great importance. Therefore, we used this method in this research to study its complex dynamics. We also demonstrated the coupling of the relaxed Newton method, which uses a single fourth root to obtain a fourth root, via a linear fractional transformation using a Riemann sphere, with recurrence rates of the quadratic equation q(a) = a⁴ - 0.75. The fractional transformation, the Möbius transformation, was also applied to complex quadratic polynomials with recurring roots.

We demonstrate the conjugation of the relaxed Newton's approach, which uses one quartic root to get the rooted of a fourth, through a Riemann sphere linear fractional transformation, to the iterations of the quartic . Fractional transformation, the Möbius transform, has also been applied to complex quadrilateral polynomials with repeated roots.