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