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lines 6-122 of file: xrst/theory/tan_reverse.xrst
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{xrst_begin tan_reverse}

Tangent and Hyperbolic Tangent Reverse Mode Theory
##################################################

Notation
********
We use the reverse theory
:ref:`standard math function<reverse_theory@Standard Math Functions>`
definition for the functions :math:`H` and :math:`G`.
In addition, we use the forward mode notation in :ref:`tan_forward-name` for
:math:`X(t)`, :math:`Y(t)` and :math:`Z(t)`.

Eliminating Y(t)
****************
For :math:`j > 0`, the forward mode coefficients are given by

.. math::

   y^{(j-1)} = \sum_{k=0}^{j-1} z^{(k)} z^{(j-k-1)}

Fix :math:`j > 0` and suppose that :math:`H` is the same as :math:`G`
except that :math:`y^{(j-1)}` is replaced as a function of the Taylor
coefficients for :math:`Z(t)`.
To be specific, for :math:`k = 0 , \ldots , j-1`,

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ z^{(k)} }
   & = &
   \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } \D{ y^{(j-1)} }{ z^{(k)} }
   \\
   & = &
   \D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } 2 z^{(j-k-1)}
   \end{eqnarray}

Positive Orders Z(t)
********************
For order :math:`j > 0`,
suppose that :math:`H` is the same as :math:`G` except that
:math:`z^{(j)}` is expressed as a function of
the coefficients for :math:`X(t)`, and the
lower order Taylor coefficients for :math:`Y(t)`, :math:`Z(t)`.

.. math::

   z^{(j)}
   =
   x^{(j)} \pm \frac{1}{j}  \sum_{k=1}^j k x^{(k)} y^{(j-k)}

For :math:`k = 1 , \ldots , j`,
the partial of :math:`H` with respect to :math:`x^{(k)}` is given by

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(k)} } & = &
   \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
   \\
   & = &
   \D{G}{ x^{(k)} } +
   \D{G}{ z^{(j)} }
   \left[ \delta ( j - k ) \pm \frac{k}{j} y^{(j-k)} \right]
   \end{eqnarray}

where :math:`\delta ( j - k )` is one if :math:`j = k` and zero
otherwise.
For :math:`k = 1 , \ldots ,  j`
The partial of :math:`H` with respect to :math:`y^{j-k}`,
is given by

.. math::
   :nowrap:

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

Order Zero Z(t)
***************
The order zero coefficients for the tangent and hyperbolic tangent are

.. math::
   :nowrap:

   \begin{eqnarray}
   z^{(0)} & = & \left\{
      \begin{array}{c} \tan ( x^{(0)} ) \\ \tanh (  x^{(0)} ) \end{array}
   \right.
   \end{eqnarray}

Suppose that :math:`H` is the same as :math:`G` except that
:math:`z^{(0)}` is expressed as a function of the Taylor coefficients
for :math:`X(t)`.
In this case,

.. math::
   :nowrap:

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

{xrst_end tan_reverse}