lines 6-155 of file: xrst/theory/acos_reverse.xrst

{xrst_begin acos_reverse}

Inverse Cosine and Hyperbolic Cosine 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`.
In addition, we use the forward mode notation in
:ref:`acos_forward-name` for

.. math::
   :nowrap:

   \begin{eqnarray}
      Q(t) & = & \mp ( X(t) * X(t) - 1 ) \\
      B(t) & = & \sqrt{ Q(t) }
   \end{eqnarray}

We use :math:`q` and :math:`b`
for the *p*-th order Taylor coefficient
row vectors corresponding to these functions
and replace :math:`z^{(j)}` by

.. math::

   ( z^{(j)} , b^{(j)} )

in the definition for :math:`G` and :math:`H`.
The zero order forward mode formulas for the
:ref:`acos<acos_forward-name>`
function are

.. math::
   :nowrap:

   \begin{eqnarray}
      q^{(0)}  & = & \mp ( x^{(0)} x^{(0)} - 1) \\
      b^{(0)}  & = & \sqrt{ q^{(0)} }    \\
      z^{(0)}  & = & F ( x^{(0)} )
   \end{eqnarray}

where :math:`F(x) = \R{acos} (x)` for :math:`-`
and :math:`F(x) = \R{acosh} (x)` for :math:`+`.
For orders :math:`j` greater than zero we have

.. math::
   :nowrap:

   \begin{eqnarray}
   q^{(j)} & = &
      \mp \sum_{k=0}^j x^{(k)} x^{(j-k)}
   \\
   b^{(j)} & = &
      \frac{1}{j} \frac{1}{ b^{(0)} }
      \left(
         \frac{j}{2} q^{(j)}
         - \sum_{k=1}^{j-1} k b^{(k)} b^{(j-k)}
      \right)
   \\
   z^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} }
   \left(
      \mp j x^{(j)}
      - \sum_{k=1}^{j-1} k z^{(k)}  b^{(j-k)}
   \right)
   \end{eqnarray}

If :math:`j = 0`, we note that
:math:`F^{(1)} ( x^{(0)} ) =  \mp 1 / b^{(0)}` and hence

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(j)} } & = &
   \D{G}{ x^{(j)} }
   + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
   + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(0)} } \D{ q^{(0)} }{ x^{(0)} }
   \\
   & = &
   \D{G}{ x^{(j)} }
   \mp \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} }
   \mp \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} }
   \end{eqnarray}

If :math:`j > 0`, then for :math:`k = 1, \ldots , j-1`

.. math::
   :nowrap:

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

{xrst_end acos_reverse}