The Study Of Accelerating Convergence Factors Of Limit Periodic Continued Fraction

Continued fraction is important numerical calculation method. We often focus on the convergence of continued fraction. This paper has introduced several famous and important conclusions about the convergence theory of continued fraction. It is needed accelerating the convergence when the convergence speed of the continued fraction was too slowly. The study of accelerating convergence has greatly developed during 1970's and 1980's. On the one hand , it is that lots of new conclusions, algorithm and applications have introduced in the research of the old problems; On the other hand, the change of new angle of view has provided new chance to the study of accelerating convergence.This paper provides an overview of the method often used in accelerating convergence of continued fractions, which contain the method of sequences transformation(see[1][2][3][4]) and the method of accelerating convergence factor (see[2][5][6]). This paper presents the development of sequence transformation, especially introduces some famous sequences transformation.Limit periodic continued fraction is very important kind continued fraction. By the influence of the idea of composite sequences transformation , this paper obtained a new sequence of accelerating convergence factors by applying composite sequence transformations to some accelerating factors of limit periodic continued fractions an.It can be proved that the new construct sequence is also accelerating convergence factors of limit periodic continued fraction an and has more advantages than no composition applied to the question of qualitative and quantitative analysis for the convergence of the limit periodic continued fraction under some hypotheses. For the obtained accelerating factors, this paper presents the method of error control.