第398行: |
第398行: |
| | | |
| ===二阶导数=== | | ===二阶导数=== |
− | 进一步地,我们可以求出EI这个函数的二阶导数 | + | 进一步地,我们可以求出EI这个函数的二阶导数<math>\frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}</math>,其中<math>1\leq s \leq N, 1\leq t \leq N-1 </math>。首先我们需要引入一个函数符号<math>\delta_{i,j} </math>, |
| | | |
| + | <math> |
| + | \delta_{i,j} = |
| + | \begin{cases} |
| + | 0 & \text{if } i\ne j,\\ |
| + | 1 & \text{if } i = j. |
| + | \end{cases} |
| + | </math> |
| + | |
| + | 于是我们可以来推导EI的二阶导数,当<math>i=s </math>时, |
| + | |
| + | <math> |
| + | \begin{equation} |
| + | \begin{aligned} |
| + | \frac{\partial^2 EI}{\partial p_{ij}\partial p_{it}}&=\frac{\delta_{j,t}}{N}\left(\frac{1}{p_{ij}}-\frac{1}{N\cdot \bar{p}_{\cdot j}}\right)+\frac{1}{N\cdot p_{iN}}-\frac{1}{N^2\cdot \bar{p}_{\cdot N}}\\ |
| + | &=\delta_{j,t}\frac{\sum_{k=1}^{N-1}p_{k j}-p_{ij}}{N^2\cdot p_{ij}\cdot \bar{p}_{\cdot j}}+\frac{\sum_{k=1}^{N-1}p_{k N}-p_{iN}}{N^2\cdot p_{iN}\cdot \bar{p}_{\cdot N}}\\ |
| + | &=\delta_{j,t}\frac{\sum_{k\neq i}p_{kj}}{N^2\cdot p_{ij}\cdot \bar{p}_{\cdot j}}+\frac{\sum_{k\neq i}p_{k N}}{N^2\cdot p_{iN}\cdot \bar{p}_{\cdot N}}, |
| + | \end{aligned} |
| + | \end{equation} |
| + | </math> |
| + | |
| + | 当<math>i\ne s</math>时, |
| | | |
| <math> | | <math> |
| \begin{equation} | | \begin{equation} |
− | \frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}=\frac{1}{N}\cdot\left(\frac{\delta_{i,s}\delta_{j,t}}{p_{ij}}+\frac{\delta_{i,s}}{p_{iN}}-\frac{\delta_{j,t}}{N\cdot\Bar{p}_{\cdot j}}-\frac{1}{N\cdot \Bar{p}_{\cdot N}}\right),
| + | \frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}=-\frac{\delta_{j,t}}{N^2\cdot \bar{p}_{\cdot j}}-\frac{1}{N^2\cdot \bar{p}_{\cdot N}}. |
| \end{equation} | | \end{equation} |
| </math> | | </math> |
| | | |
| + | 综上,EI的二阶导数为, |
| + | |
| + | <math> |
| + | \begin{equation} |
| + | \frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}=\frac{1}{N}\cdot\left(\frac{\delta_{i,s}\delta_{j,t}}{p_{ij}}+\frac{\delta_{i,s}}{p_{iN}}-\frac{\delta_{j,t}}{N\cdot\bar{p}_{\cdot j}}-\frac{1}{N\cdot \bar{p}_{\cdot N}}\right). |
| + | \end{equation} |
| + | </math> |
| | | |
| ===最大值=== | | ===最大值=== |