lines 6-117 of file: xrst/theory/pow_forward.xrst

{xrst_begin pow_forward}

Power Function Forward Mode Theory
##################################
We consider the operation :math:`F(x) = x^y` where :math:`x`
is a variable and :math:`y` is a parameter.

Derivatives
***********
The corresponding derivative satisfies the equation

.. math::

   x * F^{(1)} (x) -  y F(x) = 0

This is the
:ref:`standard math function differential equation<forward_theory@Standard Math Functions@Differential Equation>` ,
where
:math:`A(x) = y`,
:math:`B(x) = x`,
and :math:`D(x) = 0`.
We use :math:`a`, :math:`b`, :math:`d`,
and :math:`z` to denote the
Taylor coefficients for
:math:`A [ X (t) ]`,
:math:`B [ X (t) ]`,
:math:`D [ X (t) ]`,
and :math:`F [ X(t) ]` respectively.
It follows that
:math:`b^j = x^j`, :math:`d^j = 0`,

.. math::

   a^{(j)} = \left\{ \begin{array}{ll}
      y & \R{if} \; j = 0
      \\
      0 & \R{otherwise}
   \end{array} \right.

Taylor Coefficients Recursion
*****************************

z^(0)
=====

.. math::

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

e^(j)
=====

.. math::
   :nowrap:

   \begin{eqnarray}
      e^{(j)} & = & d^{(j)} + \sum_{k=0}^j a^{(j-k)} * z^{(k)}
      \\
      e^{(j)} & = & y * z^{(j)}
   \end{eqnarray}

z^j
===
For :math:`j = 0, \ldots , p-1`

.. math::
   :nowrap:

   \begin{eqnarray}
   z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
   \left(
      \sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
      - \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
   \right)
   \\
   & = & \frac{1}{j+1} \frac{1}{ x^{(0)} }
   \left(
      y \sum_{k=1}^{j+1} k x^{(k)} z^{(j+1-k)}
      - \sum_{k=1}^j k z^{(k)}  x^{(j+1-k)}
   \right)
   \\
   & = &
   \frac{1}{j+1} \frac{1}{ x^{(0)} }
   \left(
      y (j+1) x^{(j+1)} z^{(0)}
      +
      \sum_{k=1}^j k ( y x^{(k)} z^{(j+1-k)} - z^{(k)}  x^{(j+1-k)} )
   \right)
   \\
   & = &
   y z^{(0)} x^{(j+1)} / x^{(0)}
   +
   \frac{1}{j+1} \frac{1}{ x^{(0)} }
   \sum_{k=1}^j k ( y x^{(k)} z^{(j+1-k)} - z^{(k)}  x^{(j+1-k)} )
   \end{eqnarray}

For :math:`j = 1, \ldots , p`

.. math::
   :nowrap:

   \begin{eqnarray}
   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)}
   \end{eqnarray}

{xrst_end pow_forward}