您可以将Hessian用于其他答案中描述的各种事物。一种基本用法是作为第二阶导数测试。
一阶微积分的二阶导数检验
The second derivative test In calculus of one variable
Do you remember first semester calculus when you learned the second derivative test? It went like this. You've got a function f， and you want to optimize it;find where it takes its maxunum value and its minimum value.
当您学习二阶导数测验时，您还记得第一学期的微积分吗？ 像这样您已经有了一个函数f:R->R，并且想要对其进行优化；找到它取最大值和最小值的位置。
For a differentiable function on an open interval that can only occur where the first derivative f' is 0, at places called critical point. So you computed the first derivative. set it to 0, and solved the equation f ' (x) = 0. That told you where you might find the extreme values of f .
对于开放区间上的微分函数，该区间只能在第一阶导数f'为0的情况下发生在临界点处。 因此，您计算了一阶导数。 将其设置为0，并求解方程f'（x）=0。这告诉您在哪里可以找到f的极值。
Then you took the second derivative f'' , and evaluated it at each of the critical points in turn. If the second derivative was negative. then you had a local maximum; if the second derivative was positive. then you had a local minimum; if the second derivative was zero. then the test was inconclusive and you had to try something else:
然后，您取二阶导数f''，并依次在每个关键点对其进行评估。 如果二阶导数为负。那么你有一个局部最大值 如果二阶导数为正。 那么你有一个局部最小值 如果二阶导数为零。 那么测试是不确定的，您必须尝试其他方法：
The second derivative test when there's more than one varlable
Now you've got a function of n variables. Let's make it three variables to make it complicated enough to see what's going on. f : R3-->R
You find the critical points. Those will be where the three partial derivatives are simultaneously 0. So you solve the three equations
一个变量以上时的二阶导数测试
现在，您有了n个变量的函数。 让我们将其设为三个变量，使其足够复杂以查看正在发生的情况。 f：R3-> R
您找到了关键点。 这些将是三个偏导数同时为0的地方。因此，您可以求解三个方程
and that will tell you where the extreme values of f could occrur
Next you take all the second partial derivatives of them. There are nine of them,but the mixed partial derivatives are going to be the same since the functions we're looking at are all nice. You put them in a matrix called the Hessian Hf.
这将告诉您f的极值可能会出现在哪里
接下来，您将使用它们的所有第二个偏导数。 它们有九个，但是混合的偏导数将是相同的，因为我们要查看的函数都很好。 您将它们放在称为Hessian Hf的矩阵中。
Evaluate this Hessian at each of your critical points. and the resulting matrix will tell you what kind of critical point it is.
Like the second derivative test for functions of one variable，sometimes it’s inconclusive.That will happen when the determinant of the Hessian is 0. If that determinant is not 0 ,you can tell. In order to tell, you have to compute the sequence of principle minors d1, d2, d3. (There are more of them if n > 3. )
The first principle minor d1 is just the upper left entry of the matrix. The second d2 is the determinant of the upper left 2 by 2 submatrix of the matrix.And so forth. (So when n = 3, d3 is the determinant of the entire matrix)
在您的每个关键点上评估此粗麻布。 然后得出的矩阵将告诉您这是哪种临界点。
就像对一个变量的函数进行二阶导数检验一样，它有时是不确定的。当Hessian的行列式为0时，就会发生这种情况。 为了说明这一点，您必须计算主辅音d1，d2，d3的顺序。 （如果n> 3，则更多。）
第一个基本次要d1只是矩阵的左上角条目。 第二个d2是矩阵左上角2×2子矩阵的行列式，依此类推。 （因此，当n = 3时，d3是整个矩阵的行列式）
If all of them, d1, d2, d3, are positive, then you've got a minimum. If they alternate_negative. positive, negative, etc.-then you've got a maximum.Othenwise you've got a saddlle point.
Saddle points can occur with n > 2. A function with a saddle is illustrated below.
So, that's one reason for a Hessian. It's used in a second derivative test to find extreme values of functions of more than one variable.
如果所有d1，d2，d3均为正数，则您有一个最小值。 如果它们为alter_negative。 正数，负数等-然后得到最大值，然后得到鞍点。
当n> 2时，可能会出现鞍点。带有鞍的功能如下所示。
因此，这就是使用海森矩阵的原因之一。 在二阶导数测试中使用它来查找多个变量的极值。
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