初等函数和一些特殊函数的高精度快速计算

发布时间：2018-06-24 18:45

本文选题：高精度快速计算 + 基本初等函数　；参考：《山东理工大学》2017年硕士论文

【摘要】：“初等函数和一些特殊函数的高精度快速计算”来源于国家自然科学基金项目“特殊函数和基本函数的快速算法”的研究内容.本毕业论文主要研究基本函数和特殊函数中的(不完全)Beta函数和超几何函数的高精度快速计算,给出高精度(有效数字为百位甚至千位以上)快速算法.几乎所有的科学计算中需要调用基本函数进行计算,所以很多的科学计算软件中都以基本函数作为内部函数.随着科学计算精度的不断提高,这些软件中的基本函数的高精度快速算法都有改进的空间.本文中讨论了一些函数的高精度快速计算方法,给出了该算法的基本思想,分析了该算法在计算中引起的误差以及需要的运算次数等方面,从理论上对算法的可行性和高效性进行分析,最后给出了该算法的应用.在物理学特别是量子学中经常出现各种(广义)积分.解决此类积分的计算问题,常用的方法为数值积分,而基于数值积分的算法会累积误差,在高精度计算中有时真值会被误差掩盖,得不到期望精度的积分值,因此需要通过级数展开的方法,或者以特殊常数和特殊函数表示的方法来求解这类积分.(不完全)Beta函数作为最基本的一种特殊函数,很多积分形式都可以通过(不完全)Beta函数进行求解.本文中主要讨论了(不完全)Beta函数的快速计算方法,并使用该方法去解决了一些积分的运算问题.超几何函数在特殊函数中具有特殊的地位,因为许多其他类型的特殊函数都是它的特殊情况.在快速计算的研究过程中,很多地方都有它的踪迹.因此,本文中也研究了超几何函数的快速计算,并给出了应用.本论文从简到繁,逐渐深入分析和研究了函数的高精度快速计算.在研究的过程中,将本文给出的算法的计算的结果和已有的算法的计算结果进行比较.可以发现,本文中给出的算法更加优秀.
[Abstract]:"High accuracy and fast calculation of elementary functions and some special functions" comes from the research content of "Fast algorithms for Special functions and basic functions" of the National Natural Science Foundation of China. In this thesis, the (incomplete) Beta function and the hypergeometric function in the basic function and the special function are studied, and the fast algorithm of high precision (the effective number is 100 bits or more) is given. In almost all scientific calculations, basic functions are called to calculate, so many scientific computing software use basic functions as internal functions. With the improvement of scientific calculation precision, there is room for improvement of high precision fast algorithms of basic functions in these software. In this paper, some high precision and fast calculation methods of functions are discussed, the basic idea of the algorithm is given, and the errors caused by the algorithm and the number of operations required are analyzed. The feasibility and efficiency of the algorithm are analyzed theoretically. Finally, the application of the algorithm is given. In physics, especially in quantum science, there are often various (generalized) integrals. The common method to solve this kind of integral problem is numerical integration, and the algorithm based on numerical integration accumulates errors. In high precision calculation, the true value is sometimes concealed by error, and the integral value with expected accuracy is not obtained. So it is necessary to solve this kind of integrals by the method of series expansion or by the representation of special constant and special function. As a basic special function, many integral forms can be solved by (incomplete) Beta function. In this paper, we mainly discuss the fast calculation method of (incomplete) Beta function, and use this method to solve some integral problems. Hypergeometric functions play a special role in special functions because many other types of special functions are its special cases. It has been found in many places in the course of fast computing. Therefore, the fast calculation of hypergeometric functions is also studied in this paper, and its application is given. In this paper, from simplicity to complexity, the high precision and fast calculation of functions is analyzed and studied. In the course of the research, the results of the algorithms presented in this paper are compared with those of the existing algorithms. It can be found that the algorithm presented in this paper is more excellent.
【学位授予单位】：山东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174

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