本文關(guān)鍵詞： 特殊函數(shù) 驗(yàn)證賦值 誤差分析 逼近 遞推鏈　出處：《華東師范大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

【摘要】：特殊函數(shù)是指一類在科學(xué)研究的眾多領(lǐng)域,如物理,工程,化學(xué),計(jì)算機(jī)科學(xué)以及統(tǒng)計(jì)學(xué)中有著廣泛應(yīng)用的函數(shù),它們往往具有特殊的性質(zhì),因其重要性,許多學(xué)者都致力于特殊函數(shù)的賦值,應(yīng)用等相關(guān)研究.由于特殊函數(shù)的表現(xiàn)形式復(fù)雜,如何對特殊函數(shù)進(jìn)行可靠的賦值已成為一項(xiàng)具有挑戰(zhàn)性的任務(wù).在實(shí)際應(yīng)用中,占主導(dǎo)的方法是使用基于浮點(diǎn)運(yùn)算的數(shù)值方法對特殊函數(shù)進(jìn)行近似計(jì)算.目前,在知名的符號計(jì)算軟件和數(shù)值計(jì)算系統(tǒng)中,不乏有效的數(shù)值方法對特殊函數(shù)進(jìn)行賦值,但程序庫遠(yuǎn)不夠豐富,高效,且在當(dāng)前廣泛使用的IEEE浮點(diǎn)系統(tǒng)中,由于計(jì)算機(jī)字長和存儲空間的限制,進(jìn)行數(shù)值計(jì)算時(shí)會累計(jì)大量誤差,無法保證賦值結(jié)果的準(zhǔn)確性.圍繞浮點(diǎn)系統(tǒng)中的自動誤差分析,特殊函數(shù)完整的賦值分析,賦值結(jié)果相對誤差較大,以及賦值過程耗時(shí)較長等相關(guān)問題,本文主要研究內(nèi)容和方法如下:1.借助浮點(diǎn)系統(tǒng)的基本理論,誤差累計(jì)規(guī)則等知識,我們在計(jì)算機(jī)代數(shù)系統(tǒng)Maple中實(shí)現(xiàn)了一個(gè)自動誤差分析工具,能夠?qū)Π舅阈g(shù)運(yùn)算和復(fù)合函數(shù)的表達(dá)式進(jìn)行誤差分析,并自動給出浮點(diǎn)運(yùn)算的最小誤差界.最后,使用特殊函數(shù)的逼近式通項(xiàng)對工具的可靠性進(jìn)行了驗(yàn)證.2.以反三角函數(shù),誤差函數(shù)以及Polygamma函數(shù)為例,對其進(jìn)行了完整的賦值分析.針對自變量的不同取值區(qū)間,比較了使用不同逼近方法和數(shù)值軟件計(jì)算的誤差結(jié)果,給出了函數(shù)的最優(yōu)逼近方法.與此同時(shí),對于某些賦值效果較差的自變量區(qū)間,結(jié)合函數(shù)性質(zhì),提出了改進(jìn)的賦值方法,能夠?qū)①x值結(jié)果的相對誤差控制在10-100以下,提高賦值結(jié)果的準(zhǔn)確性.3.研究遞推鏈的核心方法,并進(jìn)行擴(kuò)展應(yīng)用.對Trigamma函數(shù)改進(jìn)后的賦值逼近式進(jìn)行改寫,將式中兩個(gè)級數(shù)表示成遞推鏈的形式,避免了大量重復(fù)和耗時(shí)的運(yùn)算.在保證賦值結(jié)果準(zhǔn)確性的基礎(chǔ)上,相較于直接計(jì)算,將賦值速度提高了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.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O174.6
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本文編號：1481552