Abstract: In this paper we give the following results:
1. A point type of constructive theorem;
2. An interval type of constructive theorem;
3. A convergent type of constructive theorem;
4. An accelerative formula.
They have the following characteristics:1. these methods are simpler and more convenient than the methods which had been given; 2. the iterative functions which are constructed to use the methods are always convergent; 3. the methods are the new parts of constructive theorems on the iterative function.

Key words: accelerating convergence, iterative function of one point, cumputing efficiency, iterative function of multipoint order