lines 6-76 of file: lib/python/cppad_py/fun_hessian.py

# {xrst_begin py_fun_hessian}
# {xrst_comment_ch #}
#
# Hessian of an AD Function
# #########################
#
# Syntax
# ******
# *H* = *f*\ ``.hessian`` ( *x* , *w* )
#
# f
# *
# This is either a
# :ref:`d_fun<py_fun_ctor@Syntax@d_fun>` or
# :ref:`a_fun<py_fun_ctor@Syntax@a_fun>` function object.
# Upon return, the zero order
# :ref:`taylor_coefficients<py_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<py_fun_ctor@ax>`
# and *m* is the size of :ref:`ay<py_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`` ,
# *x* is a numpy vector with ``float`` ( ``a_double`` ) elements
# and size *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`` ,
# *w* is a numpy vector with ``float`` ( ``a_double`` ) elements
# and size *m* .
# It specifies the vector *w* in the definition of :math:`g(x)` above.
#
# H
# *
# The result is a numpy matrix with ``float`` ( ``a_double`` ) elements,
# *n* rows and ``n`` columns.
# For *i* between zero and *n* -1
# and *j* between zero and *n* -1 ,
#
# .. math::
#
#    H [ i, j ] = \frac{ \partial^2 g }{ \partial x_i \partial x_j } (x)
#
# {xrst_toc_hidden
#  example/python/core/fun_hessian_xam.py
# }
# Example
# *******
# :ref:`fun_hessian_xam.py-name`
#
# {xrst_end py_fun_hessian}