毕业论文-利用二次型理论求二次函数的极值.doc

-1- 利用二次型理论求二次函数的极值,数学计算机科学学院摘要: 二次型是线性代数的重要内容之一,本文用二次型理论给出了多元二次函数极值的求法。首先,沿用得到一般多元函数存在极值的必要条件的思想给出并论证了多元二次函数存在极值的必要条件;其次,为了进一步判定极值的存在以及当极值存在时是极大值还是极小值,利用多元二次函数对应矩阵的正定性给出其充分条件;最后基于必要条件和充分条件总结了求极值的一般方法。这一方法最终体现在对多元二次函数本身的系数对应线性方程组的解和对应矩阵的正定性进行讨论,并且在极值存在时,给出了求极值的明确的表达式。关键词:二次型;矩阵;二次函数;极值。 Make U se ofQ uadratic F orm toS eek Q uadratic F unction E xtreme Shi Fangfang , College of Mathematics puter Science Abstract : Quadratic isone of the important contents of linear algebra theory, this paper presents with quadratic multiple quadratic function of extrem e method. First, continue to get general multivariate function of the necessary conditions for the existence of the ideological extreme are demonstrated and a multiple quadratic function of the necessary conditions for the extreme . Secondly, in order to determine the existence of extreme value and when there is great value when extreme or minimum, using multiple quadratic function of the corresponding matrix is given its qualitative sufficient conditions . Final ly based on the necessary and sufficient conditions for the extrem e summarizes the general method. This method is embodied in the final of the coefficient ofa multiple quadratic function itself of linear equations corresponding solution and the corresponding matrix are discussed, and the qualitative extrem e, given the explicit expression of the extreme. Key words : Q uadratic form; M atrix; Q uadratic function; E xtreme 1 引言-2- 求多元函数的极值是实际中常常遇到的重要问题。求一般多元函数的极值, 可以应用二次型的理论。我们已经知道: (1 )多元函数存在极值的必要条件:若点?? 002 010,,, nxxxX??是函数?? nxxxfy,,, 21??的极值点,并且偏导数?? nix f i,,2,1????存在,则函数?? nxxxfy,,, 21??在该点的梯度必然为零,即?? 0 0?X gradf .(文[9] ) (2 )多元函数极值存在的充分条件:设函数?? nxxxfy,,, 21??在点? 0X?? 002 01,,, nxxx?的某个邻域内具有直到二阶的连续偏导数,且?? nxxxf,,, 21?在该点的梯度?? 0 0?X gradf ,则①当黑塞矩阵?? 0X Hf 为正定矩阵时,?? 002 01,,, nxxxf?为?? nxxxf,,, 21?的极小值, ②当黑塞矩阵?? 0X Hf 为负定矩阵时,?? 002 01,,, nxxxf?为?? nxxxf,,, 21?的极大值, ③当黑塞矩阵?? 0X Hf 为不定矩阵时,?? 002 01,,, nxxxf?不是?? nxxxf,,, 21?的极值。(文[9]) 由此可见当多元函数在驻点 0X 处的黑塞矩阵