Monotonieeigenschaften des Steffensen-Verfahrens

Steffensen's method is slightly generalized by introducing a real parameter. In this way one can get different monotonicity properties, depending on the choice of this parameter. These monotonicity statements give the possibility of bracketing the solution of a given problem. In a special case they even ensure the convergence and the existence of a solution. Furthermore there are given sufficient conditions, which show that Steffensen's method converges at least as quickly as Newton's method. A numerical example shows the efficiency of the theorems and compares Steffensen's and Newton's method.