Suitable Iterated Convergent Algorithm Useful for Inspect a Local Optimum Becoming Global

Abstract

Number generation algorithms are quite formal in number computation techniques. The Golden Ratio algorithm, continued fraction algorithm and Fibonacci sequence determination are three such techniques. The present paper has five sections. The first section contains the geometrical definition of the Golden Ratio process. The second section of the paper contains a MATLAB program for evaluating and generating continued fractions up to certain terms to approximate the Golden Ratio process. In the third section of this paper we compute Fibonacci sequence using MATLAB tools with the Golden Ratio process and continued fraction algorithms. The fourth section contains the description and MATLAB code of the algorithm applying Golden Ratio process to determine the minimum of a unimodal function. In the fifth section of this paper we discuss the inference and conclusion. We observe that, there are certain ranges in finite terms’ of continued fractions where we see that the difference of Golden Ratio number and the sum of the finite terms of the continued fractions is zero. That means the Golden Ratio number equals with the sum of the finite terms of the continued fractions. More over in the third section we observe that the sum of certain terms of the continued fraction is related to Fibonacci sequence. In the continuation, we observe that the Golden Ratio also relates to Fibonacci sequence and it is also applied in determining the minimum value of funtions.