1Blurb:: 2Hessians are needed and will be approximated by finite differences 3 4Description:: 5The \c numerical_hessians specification means that Hessian information 6is needed and will be computed with finite differences using either 7first-order gradient differencing (for the cases of \c 8analytic_gradients or for the functions identified by \c 9id_analytic_gradients in the case of \c mixed_gradients) or first- or 10second-order function value differencing (all other gradient 11specifications). In the former case, the following expression 12\f[ 13\nabla^2 f ({\bf x})_i \cong 14\frac{\nabla f ({\bf x} + h {\bf e}_i) - \nabla f ({\bf x})}{h} 15\f] 16estimates the \f$i^{th}\f$ Hessian column, and in the latter case, the 17following expressions 18\f[ 19\nabla^2 f ({\bf x})_{i,j} \cong \frac{f({\bf x} + h_i {\bf e}_i + h_j {\bf e}_j) - 20f({\bf x} + h_i {\bf e}_i) - 21f({\bf x} - h_j {\bf e}_j) + 22f({\bf x})}{h_i h_j} 23\f] 24and 25\f[ 26\nabla^2 f ({\bf x})_{i,j} \cong \frac{f({\bf x} + h {\bf e}_i + h {\bf e}_j) - 27f({\bf x} + h {\bf e}_i - h {\bf e}_j) - 28f({\bf x} - h {\bf e}_i + h {\bf e}_j) + 29f({\bf x} - h {\bf e}_i - h {\bf e}_j)}{4h^2} 30\f] 31provide first- and second-order estimates of the \f$ij^{th}\f$ Hessian term. 32Prior to %Dakota 5.0, %Dakota always used second-order estimates. 33In %Dakota 5.0 and newer, the default is to use first-order estimates 34(which honor bounds on the variables and 35require only about a quarter as many function evaluations 36as do the second-order estimates), but specifying <tt>central</tt> 37after <tt>numerical_hessians</tt> causes %Dakota to use the old second-order 38estimates, which do not honor bounds. In optimization algorithms that 39use Hessians, there is little reason to use second-order differences in 40computing Hessian approximations. 41 42<!-- 43The \c fd_hessian_step_size specifies the relative finite difference 44step size to be used in these differences. Either a single value may 45be entered for use with all parameters, or a list of step sizes may be 46entered, one for each parameter. When the interval scaling type is \c 47absolute, the differencing intervals are \c fd_hessian_step_size. 48When the interval scaling type is \c bounds, the differencing 49intervals are computed by multiplying the \c fd_hessian_step_size with 50the range of the parameter. When the interval scaling type is \c 51relative, the differencing intervals are computed by multiplying the 52\c fd_hessian_step_size with the current parameter value. A minimum 53absolute differencing interval of <tt>.01*fd_hessian_step_size</tt> is 54used when the current parameter value is close to zero. 55--> 56 57 58Topics:: 59Examples:: 60Theory:: 61Faq:: 62See_Also:: responses-no_hessians, responses-quasi_hessians, responses-analytic_hessians, responses-mixed_hessians 63