On the global convergence of Schröder’s iteration formula for real zeros of entire functions

Abstract Schroder’s formula of the second kind of order m of convergence ( S2- m formula ) is a generalization of Newton’s ( m = 2 ) and Halley’s ( m = 3 ) iterative formulae for finding zeros of functions. The authors showed that the S2- m formula of every odd order m ≥ 3 converges globally and monotonically to real zeros of polynomials on the real line. For Halley’s formula, such the convergence property for real zeros of entire functions had been shown by Davies and Dawson. In this paper, we extend both results by showing that the S2- m formula of every odd order m ≥ 5 has the same convergence property for real zeros of entire functions . By numerical examples we illustrate the monotonic convergence of the formula of odd order m = 3 , 5 and 7 and the non-monotonic convergence of even order m = 2 , 4 and 6. Further, we compare several formulae of both first and second kinds in performance.