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Hessian: Easy Driver

Syntax
hes = f.Hessian(x, w)  hes = f.Hessian(x, l) 
Purpose
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.

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

x
The argument x has prototype       const 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.

l
If the argument l is present, it has prototype       size_t l  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, $$w_i = \left\{ \begin{array}{ll} 1 & i = l \\ 0 & {\rm otherwise} \end{array} \right.$$

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

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

Vector
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.

Example
The routines hessian.cpp and hes_lagrangian.cpp are examples and tests of Hessian. They return true, if they succeed and false otherwise.