非线性函数三次型逼近算法研究

Research on approach algorithm of cubic form for nonlinear functions

为提高函数逼近的精度,扩展二次型的适用范围,对二次型及矩阵的概念及运算方法进行了推广,提出了三次型及体阵的概念和运算方法,给出了多元函数梯度、黑塞矩阵,特别是多元函数泰勒公式的新的矩阵表达式,为二次型的进一步深入研究提供了新的基础,为函数逼近,最优化等方面的研究提供了一个新的视角,使得已有的特征值,特征向量等有可能被进一步推广从而使二次型的研究及应用得到极大的扩展,对于控制问题、优化问题的研究具有潜在的应用价值。

To improve the accuracy of function approximation and to expand the application range of quadratic form, the concept and operation method of matrix and quadratic form are extended. The concept and operation method of cubic form and cubic array are proposed. The new matrix expressions of gradient, Hesse Matrix, Taylor rule of multivariate functions are given specially. All these not only provide a new foundation for further study quadratic form but also provide a new perspective for further study function approximation and optimization. In addition to these, the concept of eigenvalues and eigenvector can be further expanded, where by making quadratic form research and application get the extremely large expansion so as to holding the potential ap- plication value for solving the problems control optimization researches. The results have certain potential practical value for solving of control problems and optimization problems.