Polynomial iterations to roots of algebraic equations

(2) O(M)= t, +(8)(t)= 0 (s= 1 2,*** ,r1), then +(x) is said to define an iteration of order r to the root t. In fact, for r> 1, when xo is in a sufficiently small neighborhood of t the sequence (3) x+1 = 4(xi) converges to t with (4) = + O(xi O)r For analyticf, iterations of all orders exist and can be constructed in many ways. Domb [2]1 has shown further that for polynomial f it is always possible to make k a polynomial. The purpose of this note is to describe a simple algorithm: Let f(x) be a polynomial with no multiple factors; let p(x) and q(x) be any polynomials satisfying