Hessian Matrix

In mathematics, the Hessian matrix (or simply the Hessian) is the square matrix of second-order partial derivatives of a function; that is, it describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse himself had used the term "functional determinants".

Given the real-valued function

if all second partial derivatives of f exist, then the Hessian matrix of f is the matrix

where x = (x1, x2, ..., xn) and Di is the differentiation operator with respect to the ith argument and the Hessian becomes

$H(f) = begin{bmatrix} dfrac{partial^2 f}{partial x_1^2} & dfrac{partial^2 f}{partial x_1,partial x_2} & cdots & dfrac{partial^2 f}{partial x_1,partial x_n} \ dfrac{partial^2 f}{partial x_2,partial x_1} & dfrac{partial^2 f}{partial x_2^2} & cdots & dfrac{partial^2 f}{partial x_2,partial x_n} \ vdots & vdots & ddots & vdots \ dfrac{partial^2 f}{partial x_n,partial x_1} & dfrac{partial^2 f}{partial x_n,partial x_2} & cdots & dfrac{partial^2 f}{partial x_n^2} end{bmatrix}.$

Because f is often clear from context, is frequently shortened to simply .

The Hessian matrix is related to the Jacobian matrix by, = .

Some mathematicians define the Hessian as the determinant of the above matrix.

Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,

where J is the Jacobian matrix, which is a vector (the gradient) for scalar-valued functions. The full Hessian matrix can be difficult to compute in practice; in such situations, quasi-Newton algorithms have been developed that use approximations to the Hessian. The best-known quasi-Newton algorithm is the BFGS algorithm.

Other articles related to "hessian matrix, matrix, hessian":

Symmetry Of Second Derivatives - Hessian Matrix
... This matrix of second-order partial derivatives of f is called the Hessian matrix of f ... In most circumstances the Hessian matrix is symmetric ...
Hessian Matrix - Generalizations To Riemannian Manifolds
... We may define the Hessian tensor by , where we have taken advantage of the first covariant derivative of a function being the same as ordinary derivative ... Choosing local coordinates we obtain the local expression for the Hessian as where are the Christoffel symbols of the connection ... Other equivalent forms for the Hessian are given by and ...
BFGS Method - Algorithm
... From an initial guess and an approximate Hessian matrix the following steps are repeated until converges to the solution ... more refined by, the approximation to the Hessian ... The first step of the algorithm is carried out using the inverse of the matrix, which is usually obtained efficiently by applying the Shermanâ€“Morrison formula to the fifth line of the ...
Classification of Critical Points and Extrema - Calculus of Optimization
... conditions See also Critical point (mathematics), Differential calculus, Gradient, Hessian matrix, Positive definite matrix, Lipschitz continuity, Rademacher's ... Further, critical points can be classified using the definiteness of the Hessian matrix If the Hessian is positive definite at a critical point, then the point is a local ...
Affine Focal Set - Singularity Theory Approach
... A function has degenerate singularity if both the Jacobian matrix of first order partial derivatives and the Hessian matrix of second order partial derivatives have zero ... To discover if the Jacobian matrix has zero determinant we differentiate the equation x - p = Z + Î”A ... The Jacobian matrix will have zero determinant if, and only if, is degenerate as a one-form, i.e ...

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