lines 6-72 of file: xrst/theory/sqrt_reverse.xrst

{xrst_begin sqrt_reverse}

Square Root 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 forward mode formulas for the
:ref:`square root<sqrt_forward-name>`
function are

.. math::

   z^{(j)}  =  \sqrt { x^{(0)} }

for the case :math:`j = 0`, and for :math:`j > 0`,

.. math::

   z^{(j)}  =  \frac{1}{j} \frac{1}{ z^{(0)} }
   \left(
      \frac{j}{2} x^{(j) }
      - \sum_{\ell=1}^{j-1} \ell z^{(\ell)} z^{(j-\ell)}
   \right)

If :math:`j = 0`, we have the relation

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ x^{(j)} } & = &
   \D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
   \\
   & = &
   \D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \frac{1}{2 z^{(0)} }
   \end{eqnarray}

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

.. math::
   :nowrap:

   \begin{eqnarray}
   \D{H}{ z^{(0)} } & = &
   \D{G}{ z^{(0)} }  + \D{G} { z^{(j)} } \D{ z^{(j)} }{ z^{(0)} }
   \\
   & = &
   \D{G}{ z^{(0)} }  -
   \D{G}{ z^{(j)} }  \frac{ z^{(j)} }{ z^{(0)} }
   \\
   \D{H}{ x^{(j)} } & = &
   \D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
   \\
   & = &
   \D{G}{ x^{(j)} }  + \D{G}{ z^{(j)} } \frac{1}{ 2 z^{(0)} }
   \\
   \D{H}{ z^{(k)} } & = &
   \D{G}{ z^{(k)} }  + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} }
   \\
   & = &
   \D{G}{ z^{(k)} }  - \D{G}{ z^{(j)} } \frac{ z^{(j-k)} }{ z^{(0)} }
   \end{eqnarray}

{xrst_end sqrt_reverse}