On the global convergence of Schröder's iterative formulae for real roots of algebraic equations

Schröder's formulae of the first (S1) and second (S2) kind of order m of convergence are generalizations of Newton's (m=2) and Halley's (S2, m=3) iterative formulae for finding zeros of functions. Davies and Dawson show that for entire functions with only real zeros, Halley's formula converges globally and monotonically to their zeros, independently of the initial value on the real line. We show that the S2 formulae of odd order ≥5 enjoy the same convergence feature for polynomials with only real zeros. Numerical examples illustrate this. We illustrate no monotonic convergence of the S1 formulae and of the S2 formulae of even order.