lines 6-113 of file: xrst/theory/pow_reverse.xrst

{xrst_begin pow_reverse}

Power Function Reverse Mode Theory
##################################

We use the reverse theory
:ref:`standard math function<reverse_theory@Standard Math Functions>`
definition for the functions :math:`H` and :math:`G`.
The zero order forward mode formula for the
:ref:`power<pow_forward-name>` function is

.. math::

   z^{(0)}  =  F ( x^{(0)} )

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(0)} }
   & = & \D{G}{ x^{(0)} }  + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} }
   \\
   \D{ z^{(0)} }{ x^{(0)} } & = & y [ x^{(0)} ]^{y - 1}  = y z^{(0)} / x{(0)}
   \end{eqnarray}

All the equations below apply to the case where :math:`j > 0`.
For this case, the equation for :math:`z^{(j)}` is

.. math::

   z^{(j)}
   =
   \left. \left(
      y z^{(0)} x^{(j)}
      +
      \frac{1}{j} \sum_{k=1}^{j-1} k ( y x^{(k)} z^{(j-k)} - z^{(k)}  x^{(j-k)} )
   \right) \right/ x^{(0)}

x^j
***

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(j)} }
   & = & \D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
   \\
   \D{ z^{(j)} }{ x^{(j)} } & = & y z^{(0)} / x^{(0)}
   \end{eqnarray}

x^k
***
For :math:`k = 1 , \ldots , j-1`

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(k)} }
   & = & \D{G}{ x^{(k)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
   \\
   \D{ z^{(j)} }{ x^{(k)} } & = &
   \frac{1}{j} ( k y - (j-k) ) z^{(j-k)} / x^{(0)}
   \end{eqnarray}

z^k
***
For :math:`k = 1 , \ldots , j-1`

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ z^{(k)} }
   & = & \D{G}{ z^{(k)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} }
   \\
   \D{ z^{(j)} }{ z^{(k)} } & = &
   \frac{1}{j} ( (j-k) y - k ) x^{(j-k)} / x^{(0)}
   \end{eqnarray}

x^0
***

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(0)} }
   & = & \D{G}{ x^{(0)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
   \\
   \D{ z^{(j)} }{ x^{(0)} } & = & - z^{(j)} / x^{(0)}
   \end{eqnarray}

z^0
***

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ z^{(0)} }
   & = & \D{G}{ z^{(0)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(0)} }
   \\
   \D{ z^{(j)} }{ z^{(0)} } & = &  y x^{(j)} / x^{(0)}
   \end{eqnarray}

{xrst_end pow_reverse}