lines 564-647 of file: lib/cplusplus/fun.cpp

{xrst_begin cpp_fun_hessian}


Hessian of an AD Function
#########################

Syntax
******
*H* = *f*\ ``.hessian`` ( *x* , *w* )

f
*
This is either a
:ref:`d_fun<cpp_fun_ctor@Syntax@d_fun>` or
:ref:`a_fun<cpp_fun_ctor@Syntax@a_fun>` function object.
Upon return, the zero order
:ref:`taylor_coefficients<cpp_fun_forward@Taylor Coefficient>` in *f*
correspond to the value of *x* .
The other Taylor coefficients in *f* are unspecified.

f(x)
****
We use the notation :math:`f: \B{R}^n \rightarrow \B{R}^m`
for the function corresponding to *f* .
Note that *n* is the size of :ref:`ax<cpp_fun_ctor@ax>`
and *m* is the size of :ref:`ay<cpp_fun_ctor@ay>`
in to the constructor for *f* .

g(x)
****
We use the notation :math:`g: \B{R}^n \rightarrow \B{R}`
for the function defined by

.. math::

   g(x) = \sum_{i=0}^{n-1} w_i f_i (x)

x
*
If *f* is a ``d_fun`` or ``a_fun`` ,
this argument has prototype

| |tab| ``const vec_double&`` *x*
| |tab| ``const vec_a_double&`` *x*

and its size must be *n* .
It specifies the argument value at we are computing the Hessian
:math:`g^{(2)}(x)`.

w
*
If *f* is a ``d_fun`` or ``a_fun`` ,
this argument has prototype

| |tab| ``const vec_double&`` *w*
| |tab| ``const vec_a_double&`` *w*

and its size must be *m* .
It specifies the vector *w* in the definition of :math:`g(x)` above.

H
*
If *f* is a ``d_fun`` or ``a_fun`` ,
the result has prototype

| |tab| ``vec_double`` *H*
| |tab| ``vec_a_double`` *H*

and its size is *n* ``*`` *n* .
For *i* between zero and *n* -1
and *j* between zero and *n* -1 ,

.. math::

   H [ i * n + j ] = \frac{ \partial^2 g }{ \partial x_i \partial x_j } (x)

{xrst_toc_hidden
   example/cplusplus/fun_hessian_xam.cpp
}
Example
*******
:ref:`fun_hessian_xam.cpp-name`

{xrst_end cpp_fun_hessian}