The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point.
42.
Therefore, to find the local maxima and minima, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
43.
Given a point p = ( x, y ) in an image I, the Hessian matrix H ( p, ? ) at point p and scale ?, is:
44.
Existing methods based on the Hessian matrix of the system have been reported to converge to desired \ Delta x values using fewer iterations, though, in some cases more computational resources.
45.
In statistical estimation problems ( such as maximum likelihood or Bayesian inference ), credible intervals or confidence intervals for the solution can be estimated from the inverse of the final Hessian matrix.
46.
More specifically, a Hessian equation is the " k "-trace, or the " k " th elementary symmetric polynomial of eigenvalues of the Hessian matrix.
47.
A non-degenerate critical point is a local maximum if and only if the index is " n ", or, equivalently, if the Hessian matrix is positive definite.
48.
Defined mathematically, an " n " th order saddle point is characterized by the following : and the Hessian matrix,, has exactly " n " negative Eigenvalues.
49.
More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.
50.
Adaptation of the covariance matrix amounts to learning a second order model of the underlying objective function similar to the approximation of the inverse Hessian matrix in the Quasi-Newton method in classical optimization.
