Iterated function system(IFS) is an effective method to define and describe fractals. An IFS determines only one fractal which is called attractor. Although random iterated algorithm proposed by Barnsley can display easily and quickly an attractor of IFS on computer screen

it is not sure to generate all points of an attractor within any limited steps. In order to overcome the shortcoming of the algorithm

a new algorithm of gradually computing IFS attractor from one fixed point of an invertible affine transformation is presented. Because of self_similarity of an attractor of IFS there exists similarity between different regions of an IFS attractor

With this property different parts of an attractor can be showed one by one. The experimental results prove this method is feasible. A whole attractor can be computed through limited steps by using this algorithm

and unlike random iterated algorithm probability is not necessary.