Schröder iteration functionsSm(z){S_m}(z), a generalization of Newton’s method (for whichm=2m = 2), are constructed so that the sequencezn+1=Sm(zn){z_{n + 1}} = {S_m}({z_n})converges locally to a rootz∗{z^\ast }ofg(z)=0g(z) = 0asO(|zn−z∗|m)O(|{z_n} - {z^\ast }{|^m}). Forg(z)g(z)a polynomial, this involves the iteration of rational functions over the complex Riemann sphere, which is described by the classical theory of Julia and Fatou and subsequent developments. The Julia sets for theSm(z){S_m}(z), as applied to the simple casesgn(z)=zn−1{g_n}(z) = {z^n} - 1, are examined for increasingmwith the help of microcomputer plots. The possible types of behavior ofzn{z_n}iteration sequences are catalogued by examining the orbits of free critical points of theSm(z){S_m}(z), as applied to a one-parameter family of cubic polynomials.