### Polynomial iterations to roots of algebraic equations

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(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

[1] Ernst Schröder,et al. Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen , 1870 .

[2] On iterative solutions of algebraic equations , 1949 .

[3] D. R. Blaskett,et al. A formula for the solution of an arbitrary analytic equation , 1945 .