High Precision And Fast Calculation Of Elementary Functions And Some Special Functions

“High precision and fast calculation of elementary functions and some special functions” is derived from the research of the National Natural Science Foundation of China.In this thesis,we mainly study high precision and fast calculation of the basic function and special function,especially,(incomplete)Beta function and hypergeometric function.The high accuracy(valid number for more than one hundred and even one thousand)fast algorithm will be given.Almost all of the scientific computing needs basic function,and so a number of the scientific methods of calculating basic function exist in the software.With the continuous improvement of the accuracy of scientific computing,the fast algorithms of the basic functions of these software can be improved better.We discuss the fast calculation method of basic functions,the basic idea of the method,the existence of errors from the calculation and the number of operations needed is given.Not only analyzing the theoretical of the algorithm,but also giving the concrete realization of the algorithm.In physics,especially in Quantum Science,the integral form is often used.Commonly,we use numerical integration to solve the calculation problem,but the algorithm based on numerical integration will cumulate error.Sometimes,the true value will be masked by error in high precision calculation,and we cannot get the expected integral value.So it is necessary to solve this kind of integral by the method of series expansion,or by the special constant and special function.(incomplete)Beta functions as a kind of special function are the most basic,a lot of points can be achieved by(incomplete)Beta function solution.In this paper,we discuss the fast calculation method of(incomplete)Beta function,and use this method to solve some problems of integration.Hypergeometric functions have a special place in special functions,since some ty pes of special functions are special cases of it.In this paper we also study the fast ca lculation of hypergeometric functions,and give some applications.In this paper,we study of the fast and high accuracy calculation of function,fro m simple to complex,gradually in-depth.In the process of the study,we compared th e algorithm presented in this paper and the existing algorithms.The results can be fou nd that the algorithm in this paper is more excellent.