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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 of the gradient of a scalar function of several variables has a special name: the Hessian matrix, which in a sense is the "second derivative" of the function in question. Jacobian determinant
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.
If the Hessian is negative definite (equivalently, has all eigenvalues negative) at a, then f attains a local maximum at a. If the Hessian has both positive and negative eigenvalues then a is a saddle point for f (and in fact this is true even if a is degenerate). In those cases not listed above, the test is inconclusive. [2]
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]
There also exist various quasi-Newton methods, where an approximation for the Hessian (or its inverse directly) is built up from changes in the gradient. If the Hessian is close to a non-invertible matrix, the inverted Hessian can be numerically unstable and the solution may diverge. In this case, certain workarounds have been tried in the past ...
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 matrix H is known as the Hessian matrix. Although this model has better convergence properties near to the minimum, it is much worse when the parameters are far from their optimal values. Calculation of the Hessian adds to the complexity of the algorithm. This method is not in general use. Davidon–Fletcher–Powell method. This method, a ...