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The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...
In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial ...
The following test can be applied at any critical point a for which the Hessian matrix is invertible: If the Hessian is positive definite (equivalently, has all eigenvalues positive) at a, then f attains a local minimum at a. If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a.
In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for the symmetry to hold are given by Schwarz's theorem , also called Clairaut's theorem or Young's theorem .
Finding the inverse of the Hessian in high dimensions to compute the Newton direction = (″ ()) ′ can be an expensive operation. In such cases, instead of directly inverting the Hessian, it is better to calculate the vector h {\displaystyle h} as the solution to the system of linear equations
Recover the Hessian matrix from the compact matrix. Steps one and two need only be carried out once, and tend to be costly. When one wants to calculate the Hessian at numerous points (such as in an optimization routine), steps 3 and 4 are repeated.
Newton's method requires the Jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used for finding extrema. Quasi-Newton methods, on the other hand, can be used when the Jacobian matrices or Hessian matrices are unavailable or are impractical to compute at every iteration.
As described above, some method such as quantum mechanics can be used to calculate the energy, E(r) , the gradient of the PES, that is, the derivative of the energy with respect to the position of the atoms, ∂E/∂r and the second derivative matrix of the system, ∂∂E/∂r i ∂r j, also known as the Hessian matrix, which describes the curvature of the PES at r.