Search results
Results from the WOW.Com Content Network
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 ...
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.
A multivariate polynomial is SOS-convex (or sum of squares convex) if its Hessian matrix H can be factored as H(x) = S T (x)S(x) where S is a matrix (possibly rectangular) which entries are polynomials in x. [1] In other words, the Hessian matrix is a SOS matrix polynomial.
The graph colouring techniques explore sparsity patterns of the Hessian matrix and cheap Hessian vector products to obtain the entire matrix. Thus these techniques are suited for large, sparse matrices. The general strategy of any such colouring technique is as follows. Obtain the global sparsity pattern of
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. [1] [2]
A critical point at which the Hessian matrix is nonsingular is said to be nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In the case of a function of a single variable, the Hessian is simply the second derivative, viewed as a 1×1-matrix, which is nonsingular if and only if it is not ...
When is a convex quadratic function with positive-definite Hessian , one would expect the matrices generated by a quasi-Newton method to converge to the inverse Hessian =. This is indeed the case for the class of quasi-Newton methods based on least-change updates.
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.