本文選題：高精度快速計(jì)算 + 基本初等函數(shù)��； 參考：《山東理工大學(xué)》2017年碩士論文

【摘要】：“初等函數(shù)和一些特殊函數(shù)的高精度快速計(jì)算”來(lái)源于國(guó)家自然科學(xué)基金項(xiàng)目“特殊函數(shù)和基本函數(shù)的快速算法”的研究?jī)?nèi)容.本畢業(yè)論文主要研究基本函數(shù)和特殊函數(shù)中的(不完全)Beta函數(shù)和超幾何函數(shù)的高精度快速計(jì)算,給出高精度(有效數(shù)字為百位甚至千位以上)快速算法.幾乎所有的科學(xué)計(jì)算中需要調(diào)用基本函數(shù)進(jìn)行計(jì)算,所以很多的科學(xué)計(jì)算軟件中都以基本函數(shù)作為內(nèi)部函數(shù).隨著科學(xué)計(jì)算精度的不斷提高,這些軟件中的基本函數(shù)的高精度快速算法都有改進(jìn)的空間.本文中討論了一些函數(shù)的高精度快速計(jì)算方法,給出了該算法的基本思想,分析了該算法在計(jì)算中引起的誤差以及需要的運(yùn)算次數(shù)等方面,從理論上對(duì)算法的可行性和高效性進(jìn)行分析,最后給出了該算法的應(yīng)用.在物理學(xué)特別是量子學(xué)中經(jīng)常出現(xiàn)各種(廣義)積分.解決此類積分的計(jì)算問(wèn)題,常用的方法為數(shù)值積分,而基于數(shù)值積分的算法會(huì)累積誤差,在高精度計(jì)算中有時(shí)真值會(huì)被誤差掩蓋,得不到期望精度的積分值,因此需要通過(guò)級(jí)數(shù)展開(kāi)的方法,或者以特殊常數(shù)和特殊函數(shù)表示的方法來(lái)求解這類積分.(不完全)Beta函數(shù)作為最基本的一種特殊函數(shù),很多積分形式都可以通過(guò)(不完全)Beta函數(shù)進(jìn)行求解.本文中主要討論了(不完全)Beta函數(shù)的快速計(jì)算方法,并使用該方法去解決了一些積分的運(yùn)算問(wèn)題.超幾何函數(shù)在特殊函數(shù)中具有特殊的地位,因?yàn)樵S多其他類型的特殊函數(shù)都是它的特殊情況.在快速計(jì)算的研究過(guò)程中,很多地方都有它的蹤跡.因此,本文中也研究了超幾何函數(shù)的快速計(jì)算,并給出了應(yīng)用.本論文從簡(jiǎn)到繁,逐漸深入分析和研究了函數(shù)的高精度快速計(jì)算.在研究的過(guò)程中,將本文給出的算法的計(jì)算的結(jié)果和已有的算法的計(jì)算結(jié)果進(jìn)行比較.可以發(fā)現(xiàn),本文中給出的算法更加優(yōu)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.
【學(xué)位授予單位】：山東理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O174

【參考文獻(xiàn)】

相關(guān)期刊論文 前3條

1 蔣亞萍;孫中鋒;秦惠增;;關(guān)于不完全Beta函數(shù)的注記[J];山東理工大學(xué)學(xué)報(bào)(自然科學(xué)版);2016年01期

2 商妮娜;秦惠增;;基于冪級(jí)數(shù)展開(kāi)的基本初等函數(shù)的高精度快速計(jì)算[J];數(shù)值計(jì)算與計(jì)算機(jī)應(yīng)用;2015年01期

3 商妮娜;秦惠增;;基于Beta函數(shù)及其偏導(dǎo)數(shù)的廣義積分的高精度快速計(jì)算[J];數(shù)學(xué)的實(shí)踐與認(rèn)識(shí);2014年01期

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本文編號(hào)：2062556