Study of frozen Newton-like method in a banach space with dynamics

Abstract

UDC 519.6 The main objective of this work is investigation of positives and negatives of the three steps iterative frozen-type Newtonlike method for solving nonlinear equations in a Banach space. We perform a local convergence analysis by Taylor’s expansion and semilocal convergence by recurrence relations technique under the conditions of Kantorovich theorem for the Newton’s method. The convergence results are examined by comparing the proposed method with the Newton’s method and the fourth order Jarratt’s method using some test functions. We discuss the corresponding conjugacy maps for quadratic polynomials along with the extraneous fixed points. Additionally, the theoretical and numerical results are examined byusing the dynamical analysis of a selected test function. It not only confirms the theoretical and numerical results, but also reveals some drawbacks of the frozen Newton-like method.