[Forget the high number] Hesse Matrix

Updated: 5 June 2016

"Multivariate function Taylor expands" n-ary function \ (y=f (X) \) A field in the \ (x_0\) point \ (B (x_0,r) \) Nei continuous micro, then \ (\forall x\in B (x_0,r), \exists \theta\in (0,1) \), Makes

\ (f (X) =f (x_0) +JF (x_0) \delta x+\dfrac{1}{2} (\delta x) ^th (X_0+\theta\delta x) \delta x\)

where \ (\delta x=x-x_0\) is n Willi Vector;

\ (Jf (x_0) \) i.e. \ (f (X) \) Jacobi matrix at \ (x_0\);

\ (H (x) \) is \ (f (x) \) the Hesse matrix at \ (x\).

The above-mentioned wording of the Lagrange-type remainder. Of course, it can also be written as the Peano form of the first-order

\ (f (X) =f (x_0) +JF (x_0) \delta x+\dfrac{1}{2} (\delta x) ^th (x_0) \delta X+\alpha (\delta x) \)

"Hesse Matrix"

\ (H (X) =\begin{bmatrix} \dfrac{\partial^2 f}{\partial x_1\partial x_1} & \dfrac{\partial^2 f}{\partial x_1\partial X _2} & \cdots &\dfrac{\partial^2 f}{\partial x_1\partial x_n} \ \dfrac{\partial^2 f}{\partial x_2\partial x_1} &am P \dfrac{\partial^2 f}{\partial x_2\partial x_2} &\cdots& \dfrac{\partial^2 f}{\partial x_2\partial x_n} \ \vdots & & & \vdots \ \dfrac{\partial^2 f}{\partial x_n\partial x_1} &\dfrac{\partial^2 f}{\partial x_n\partial X_2} & \cdots &\dfrac{\partial^2 f}{\partial x_n\partial x_n} \end{bmatrix} \)

For n-ary function, it is equivalent to n-element m=1-dimensional vector-valued function, whose Jacobi matrix is the transpose of its gradient vector (see Jacobi Matrix), which can be regarded as its first derivative;

Then the Hesse matrix of the N-ary function is equivalent to its second derivative.

The "Application" uses Taylor to expand the surface approximation of the potential energy surface (energy as a multivariate function) in computational chemistry when the Hesse matrix is computed.

[Forget the high number] Hesse Matrix