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词汇 Hessian matrix
分类 英语词汇 英语翻译词典
释义

Hessian matrix

中文百科

海森矩阵

在数学中，海森矩阵（Hessian matrix或Hessian）是一个多变量实值函数的二阶偏导数组成的方块矩阵，此函数如下：

f(x_1, x_2, \dots, x_n),

如果f所有的二阶导数都存在，那幺f的海森矩阵即：

H(f)_{ij}(x) = D_i D_j f(x)

其中x = (x_1, x_2, \dots, x_n)，即

H(f) = \begin{bmatrix} \frac{\partial^2 f}{\partial x_1^2} & \frac{\partial^2 f}{\partial x_1\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_1\,\partial x_n} \\ \\ \frac{\partial^2 f}{\partial x_2\,\partial x_1} & \frac{\partial^2 f}{\partial x_2^2} & \cdots & \frac{\partial^2 f}{\partial x_2\,\partial x_n} \\ \\ \vdots & \vdots & \ddots & \vdots \\ \\ \frac{\partial^2 f}{\partial x_n\,\partial x_1} & \frac{\partial^2 f}{\partial x_n\,\partial x_2} & \cdots & \frac{\partial^2 f}{\partial x_n^2} \end{bmatrix}

（也有人把海森定义为以上矩阵的行列式。）

海森矩阵被应用于牛顿法解决的大规模优化问题。

英语百科

Hessian matrix 海森矩阵

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".
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