Two variants of a method for finding the roots of a polynomial are described. A proof of the method for a general polynomial with complex coefficients is not based on neighbourhoods of saddle points which are sufficiently small. The speed of convergence is guaranteed since the method is a modification of the downhill method and since it can be used in combination with an arbitrary method which quickly converges in practice, but whose convergence cannot be proved. © 1969 Springer-Verlag.