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多元函数极值问题的分析与研究

发表时间:2009-01-15  浏览量:3414  下载量:1428
全部作者: 郭常予,徐玲,杨淑,易慧,郑学安
作者单位: 北京师范大学数学科学学院
摘 要: 主要讨论Hessian 判别法失效的情况下,如何判定多元数值函数(特别是二元函数)极值问题。首先,列举了二元函数极值问题的判别准则,并简要介绍了多元函数极值问题对应的几何意义。在判别法失效的情况下,从几何方面引入了判别二元函数极值的一些必要条件,然后在分析方面利用Taylor公式,将展式化为一元函数,进行了比较细致的讨论。在几何方面,运用二次曲面理论中通过正交变换化标准型的办法,同样进行了比较细致的讨论。其次,针对二元函数,在一种特殊情形下,运用多项式的惯性理论,得出了极值判别的一个漂亮结果。最后,对一般多元函数情形,给出了特殊情形下的推广。
关 键 词: 微积分;函数极值;Taylor公式; 多项式惯性理论
Title: Analysis and research on the extreme value of multivariable functions
Author: GUO Changyu, XU Ling, YANG Shu, YI Hui, ZHENG Xuean
Organization: Department of Mathematic Science, Beijing Normal University
Abstract: This article mainly discusses how to discriminate the extreme value of multivariable functions, especially binary functions in case that the Hessian criterion fails. First, it enumerates some discriminant criterions for the extreme value of binary functions and makes a brief introduction to the corresponding geometric meaning for the extreme value, and introduces some necessary conditions for discriminating the extreme value of binary functions from the geometric aspect when discriminance fails. After that, analytically using the Taylor formula, it reduces the expansion to one-variable function and makes a specific discussion. By using the theory of quadric surface geometrically, it is also discussed in detail. And then, for binary function, by applying the inertia theory of polynomials, it has obtained an elegant result for extreme value discrimination in a special condition. At last, it gives a simple generalization of the multivariable case.
Key words: calculus; extreme value of function; Taylor formula; the inertia theory of polynomials
发表期数: 2009年1月第1期
引用格式: 郭常予,徐玲,杨淑,等. 多元函数极值问题的分析与研究[J]. 中国科技论文在线精品论文,2009,2(1):15-24.
 
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