Hessian matrix

See the article on Hessian matrices for more on the definition.

Further the Hessian matrix of second derivatives will have both positive and negative eigenvalues.

We can check that this is indeed a maximum by looking at the Hessian matrix of the log-likelihood function.

No computation of gradient or Hessian matrix is needed.

To get the principal curvature, the Hessian matrix is calculated:

In quasi-Newton methods the Hessian matrix does not need to be computed.

This update maintains the symmetry and positive definiteness of the Hessian matrix.

From an initial guess and an approximate Hessian matrix the following steps are repeated until converges to the solution.

H is the Hessian matrix of V in the equilibrium .

There also exist a number of specialized methods for calculating large sparse Hessian matrices.

The frequencies are related to the eigenvalues of the Hessian matrix, which contains second derivatives.

To compute the Hessian matrix we consider the differential two-form .

If the Hessian matrix is singular, then the second derivative test is inconclusive.

See Hessian matrix for a discussion that generalizes these rules to the case of equality-constrained optimization.

This is the two-form whose matrix representation is the Hessian matrix.

The matrix H is known as the Hessian matrix.

Recover the Hessian matrix from the compact matrix.

The new off-diagonal elements of the Hessian matrix take the below form, where p is an empirical parameter:

In most "real-life" circumstances the Hessian matrix is symmetric, although there are- mathematically- a far greater number of functions that do not have this property.

In particular at a point where , the mean curvature is half the trace of the Hessian matrix of .

The Hessian matrix and the adjoint are required for 4dimensional variational data assimilation (4D-Var).

To calculate the quadratic approximation, one must first calculate its gradient and Hessian matrix.

This can be thought of as a multi-dimensional array with dimensions , which degenerates to the usual Hessian matrix for .

Let , for each the Hessian matrix is the second order derivative and is a symmetric matrix.

The following test can be applied at any critical point (a, b, ...) for which the Hessian matrix is invertible: