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The determinant of the Hessian matrix is called the Hessian determinant. [1] The Hessian matrix of a function is the transpose of the Jacobian matrix of the gradient of the function ; that is: (()) = (()).
The Jacobian determinant is sometimes simply referred to as "the Jacobian". The Jacobian determinant at a given point gives important information about the behavior of f near that point. For instance, the continuously differentiable function f is invertible near a point p ∈ R n if the Jacobian determinant at p is non-zero.
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.
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 software engineering, a class diagram [1] in the Unified Modeling Language (UML) is a type of static structure diagram that describes the structure of a system by showing the system's classes, their attributes, operations (or methods), and the relationships among objects. The class diagram is the main building block of object-oriented modeling.
The decomposition uses a low-rank representation for the direct and/or inverse Hessian or the Jacobian of a nonlinear system. Because of this, the compact representation is often used for large problems and constrained optimization .
Newton's method for solving f(x) = 0 uses the Jacobian matrix, J, at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden ...
The Laplace–Beltrami operator, when applied to a function, is the trace (tr) of the function's Hessian: = (()) where the trace is taken with respect to the inverse of the metric tensor. The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields , by ...