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@(@\newcommand{\W}[1]{ \; #1 \; } \newcommand{\R}[1]{ {\rm #1} } \newcommand{\B}[1]{ {\bf #1} } \newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} } \newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} } \newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} } \newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@
Hessian: Easy Driver

hes = f.Hessian(xw)
hes = f.Hessian(xl)

We use @(@ F : B^n \rightarrow B^m @)@ to denote the AD function corresponding to f . The syntax above sets hes to the Hessian The syntax above sets h to the Hessian @[@ hes = \dpow{2}{x} \sum_{i=1}^m w_i F_i (x) @]@ The routine sparse_hessian may be faster in the case where the Hessian is sparse.

The object f has prototype
Note that the ADFun object f is not const (see Hessian Uses Forward below).

The argument x has prototype
Vector &x
(see Vector below) and its size must be equal to n , the dimension of the domain space for f . It specifies that point at which to evaluate the Hessian.

If the argument l is present, it has prototype
and is less than m , the dimension of the range space for f . It specifies the component of F for which we are evaluating the Hessian. To be specific, in the case where the argument l is present, @[@ w_i = \left\{ \begin{array}{ll} 1 & i = l \\ 0 & {\rm otherwise} \end{array} \right. @]@

If the argument w is present, it has prototype
Vector &w
and size @(@ m @)@. It specifies the value of @(@ w_i @)@ in the expression for h .

The result hes has prototype
Vector hes
(see Vector below) and its size is @(@ n * n @)@. For @(@ j = 0 , \ldots , n - 1 @)@ and @(@ \ell = 0 , \ldots , n - 1 @)@ @[@ hes [ j * n + \ell ] = \DD{ w^{\rm T} F }{ x_j }{ x_\ell } ( x ) @]@

The type Vector must be a SimpleVector class with elements of type Base . The routine CheckSimpleVector will generate an error message if this is not the case.

Hessian Uses Forward
After each call to Forward , the object f contains the corresponding Taylor coefficients . After a call to Hessian, the zero order Taylor coefficients correspond to f.Forward(0, x) and the other coefficients are unspecified.

The routines hessian.cpp and hes_lagrangian.cpp are examples and tests of Hessian. They return true, if they succeed and false otherwise.
Input File: cppad/core/hessian.hpp