几类特殊函数的快速验证赋值研究

发布时间：2018-02-01 09:40

  本文关键词： 特殊函数 验证赋值 误差分析 逼近 递推链　出处：《华东师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：特殊函数是指一类在科学研究的众多领域,如物理,工程,化学,计算机科学以及统计学中有着广泛应用的函数,它们往往具有特殊的性质,因其重要性,许多学者都致力于特殊函数的赋值,应用等相关研究.由于特殊函数的表现形式复杂,如何对特殊函数进行可靠的赋值已成为一项具有挑战性的任务.在实际应用中,占主导的方法是使用基于浮点运算的数值方法对特殊函数进行近似计算.目前,在知名的符号计算软件和数值计算系统中,不乏有效的数值方法对特殊函数进行赋值,但程序库远不够丰富,高效,且在当前广泛使用的IEEE浮点系统中,由于计算机字长和存储空间的限制,进行数值计算时会累计大量误差,无法保证赋值结果的准确性.围绕浮点系统中的自动误差分析,特殊函数完整的赋值分析,赋值结果相对误差较大,以及赋值过程耗时较长等相关问题,本文主要研究内容和方法如下:1.借助浮点系统的基本理论,误差累计规则等知识,我们在计算机代数系统Maple中实现了一个自动误差分析工具,能够对包含基本算术运算和复合函数的表达式进行误差分析,并自动给出浮点运算的最小误差界.最后,使用特殊函数的逼近式通项对工具的可靠性进行了验证.2.以反三角函数,误差函数以及Polygamma函数为例,对其进行了完整的赋值分析.针对自变量的不同取值区间,比较了使用不同逼近方法和数值软件计算的误差结果,给出了函数的最优逼近方法.与此同时,对于某些赋值效果较差的自变量区间,结合函数性质,提出了改进的赋值方法,能够将赋值结果的相对误差控制在10-100以下,提高赋值结果的准确性.3.研究递推链的核心方法,并进行扩展应用.对Trigamma函数改进后的赋值逼近式进行改写,将式中两个级数表示成递推链的形式,避免了大量重复和耗时的运算.在保证赋值结果准确性的基础上,相较于直接计算,将赋值速度提高了10倍以上.
[Abstract]:Special functions are a class of functions which are widely used in many fields of scientific research, such as physics, engineering, chemistry, computer science and statistics. They often have special properties because of their importance. Many scholars have devoted themselves to the assignment and application of special functions, etc. Because of the complexity of the representation of special functions. How to assign a special function reliably has become a challenging task. In practical application, the dominant method is to approximate the special function by using the numerical method based on floating-point operation. In the well-known symbolic computing software and numerical calculation system, there are many effective numerical methods to assign the value of special functions, but the library is far from rich, efficient, and widely used in the current IEEE floating point system. Due to the limitation of computer word length and storage space, a large number of errors will be accumulated in the numerical calculation, which can not guarantee the accuracy of the evaluation results, and the automatic error analysis in the floating point system can not be guaranteed. The whole assignment analysis of special function, the relative error of assignment result is big, and the process of assignment takes a long time. The main contents and methods of this paper are as follows: 1. With the help of the basic theory of floating-point system. We implement an automatic error analysis tool in the computer algebraic system Maple, which can analyze the error of the expression which includes the basic arithmetic operation and the compound function. The minimum error bound of floating-point operation is given automatically. Finally, the reliability of the tool is verified by the approximation general term of special function. The error function and Polygamma function are taken as examples, and the error results calculated by using different approximation methods and numerical software are compared according to the different value ranges of independent variables. The optimal approximation method of functions is given. At the same time, an improved assignment method is proposed for the interval of some independent variables with poor assignment effect, combined with the properties of functions. The relative error of the assignment results can be controlled below 10-100 to improve the accuracy of the assignment results. 3. The core method of recursive chain is studied. The modified assignment approximation of Trigamma function is rewritten, and the two series are expressed as the form of recursive chain. On the basis of ensuring the accuracy of the assignment results, the assignment speed is increased by more than 10 times compared with the direct calculation.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O174.6
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本文编号：1481552