# SPDX-License-Identifier: EPL-2.0 OR GPL-2.0-or-later
# SPDX-FileCopyrightText: Bradley M. Bell <bradbell@seanet.com>
# SPDX-FileContributor: 2003-24 Bradley M. Bell
# ----------------------------------------------------------------------------

{xrst_begin acos_forward}
{xrst_spell
   acosh
}

Inverse Cosine and Hyperbolic Cosine Forward Mode Theory
########################################################

Derivatives
***********

.. math::
   :nowrap:

   \begin{eqnarray}
   \R{acos}^{(1)} (x)  & = & - 1 / \sqrt{ 1 - x * x }
   \\
   \R{acosh}^{(1)} (x) & = & + 1 / \sqrt{ x * x - 1}
   \end{eqnarray}

If :math:`F(x)` is :math:`\R{acos} (x)` or :math:`\R{acosh} (x)`
the corresponding derivative satisfies the equation

.. math::

   \sqrt{ \mp ( x * x - 1 ) } * F^{(1)} (x) - 0 * F (u)  = \mp 1

and in the
:ref:`standard math function differential equation<forward_theory@Standard Math Functions@Differential Equation>` ,
:math:`A(x) = 0`,
:math:`B(x) = \sqrt{ \mp( x * x - 1 ) }`,
and :math:`D(x) = \mp 1`.
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.

Taylor Coefficients Recursion
*****************************
We define :math:`Q(x) = \mp ( x * x - 1 )`
and let :math:`q` be the corresponding Taylor coefficients for
:math:`Q[ X(t) ]`.
It follows that

.. math::

   q^{(j)} = \left\{ \begin{array}{ll}
      \mp ( x^{(0)} * x^{(0)} - 1 )      & {\rm if} \; j = 0 \\
      \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise}
   \end{array} \right.

It follows that
:math:`B[ X(t) ] = \sqrt{ Q[ X(t) ] }` and
from the equations for the
:ref:`square root<sqrt_forward-name>`
that for :math:`j = 0 , 1, \ldots`,

.. math::
   :nowrap:

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

It now follows from the general
:ref:`forward_theory@Standard Math Functions@Taylor Coefficients Recursion Formula`
that for :math:`j = 0 , 1, \ldots`,

.. math::
   :nowrap:

   \begin{eqnarray}
   z^{(0)} & = & F ( x^{(0)} )
   \\
   e^{(j)}
   & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
   \\
   & = & \left\{ \begin{array}{ll}
      \mp 1 & {\rm if} \; j = 0 \\
      0 & {\rm otherwise}
   \end{array} \right.
   \\
   z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
   \left(
      \sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)}
      - \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)}
   \right)
   \\
   z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
   \left(
      \mp (j+1) x^{(j+1)}
      - \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
   \right)
   \end{eqnarray}

{xrst_end acos_forward}
