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The Hessian matrix is the second partial derivatives matrix of f at the point x. That is, the (i,j)th component of H is the second partial derivative of f with respect to x i and x j , The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test if a critical point x is a local maximum, local minimum, or a saddle point, as follows: If the Hessian is positive definite at x , then f attains a local minimum at x .
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Errata for the paper entitled “Second-order stagewise backpropagation for Hessian-matrix analyses and investigation of negative curvature” by Eiji Mizutani and Stuart E. Dreyfus. Appeared in Neural Networks, pages 193–203, Vol.21, 2008. — 2008 Special Issue: Advances in Neural Networks Research: IJCNN’07.
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A Hessian matrix is the matrix of partial derivatives. If there are d variables, the Hessian matrix is a set of d*d partial derivatives. An example of this is borrowed from David Joyce ’s answer, where the Hessian matrix is the set of partial derivatives in three variables x,y and z.
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(c) for the bordered Hessian matrix H the last n ¡ k leading principal minors H2k+1; H2k+2; ::: ;Hn+k = H evaluated at (x⁄ 1;:::;x ⁄ n;„ ⁄ 1;:::;„ ⁄ k) alternate in sign where the last minor Hn+k = H has the sign as (¡1)n. Then x⁄ is a local max in C h. If instead of (c) we have the condition
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If you just need to compute the hessian matrix of a general function (often called a black box: you pass stuff into the black box and stuff comes out, based on some unknown algorithm.) As long as your function is reasonably well behaved, you can use my derivest toolbox, found on the file exchange. I'll show it here, using a function handle.