$Id: acos_forward.omh 3757 2015-11-30 12:03:07Z bradbell $
// BEGIN SHORT COPYRIGHT
/* --------------------------------------------------------------------------
CppAD: C++ Algorithmic Differentiation: Copyright (C) 2003-15 Bradley M. Bell

CppAD is distributed under multiple licenses. This distribution is under
the terms of the
                    GNU General Public License Version 3.

A copy of this license is included in the COPYING file of this distribution.
Please visit http://www.coin-or.org/CppAD/ for information on other licenses.
-------------------------------------------------------------------------- */
// END SHORT COPYRIGHT

$begin acos_forward$$
$spell
	acos
	acosh
	Arccosine
	Taylor
$$

$section Inverse Cosine and Hyperbolic Cosine Forward Mode Theory$$
$mindex acos acosh$$

$head Derivatives$$
$latex \[
\begin{array}{rcl}
\R{acos}^{(1)} (x)  & = & - 1 / \sqrt{ 1 - x * x }
\\
\R{acosh}^{(1)} (x) & = & + 1 / \sqrt{ x * x - 1}
\end{array}
\] $$
If $latex F(x)$$ is $latex \R{acos} (x)$$ or $latex \R{acosh} (x)$$
the corresponding derivative satisfies the equation
$latex \[
	\sqrt{ \mp ( x * x - 1 ) } * F^{(1)} (x) - 0 * F (u)  = \mp 1
\] $$
and in the
$cref/standard math function differential equation
	/ForwardTheory
	/Standard Math Functions
	/Differential Equation
/$$,
$latex A(x) = 0$$,
$latex B(x) = \sqrt{ \mp( x * x - 1 ) }$$,
and $latex D(x) = \mp 1$$.
We use $latex a$$, $latex b$$, $latex d$$ and $latex z$$ to denote the
Taylor coefficients for
$latex A [ X (t) ] $$,
$latex B [ X (t) ]$$,
$latex D [ X (t) ] $$,
and $latex F [ X(t) ] $$ respectively.

$head Taylor Coefficients Recursion$$
We define $latex Q(x) = \mp ( x * x - 1 )$$
and let $latex q$$ be the corresponding Taylor coefficients for
$latex Q[ X(t) ]$$.
It follows that
$latex \[
q^{(j)} = \left\{ \begin{array}{ll}
	\mp ( x^{(0)} * x^{(0)} - 1 )      & {\rm if} \; j = 0 \\
	\mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise}
\end{array} \right.
\] $$
It follows that
$latex B[ X(t) ] = \sqrt{ Q[ X(t) ] }$$ and
from the equations for the
$cref/square root/sqrt_forward/$$
that for $latex j = 0 , 1, \ldots$$,
$latex \[
\begin{array}{rcl}
b^{(0)}   & = & \sqrt{ q^{(0)} }
\\
b^{(j+1)} & = &
	\frac{1}{j+1} \frac{1}{ b^{(0)} }
	\left(
		\frac{j+1}{2} q^{(j+1) }
		- \sum_{k=1}^j k b^{(k)} b^{(j+1-k)}
	\right)
\end{array}
\] $$
It now follows from the general
$xref/
	ForwardTheory/
	Standard Math Functions/
	Taylor Coefficients Recursion Formula/
	Taylor coefficients recursion formula/
	1
/$$
that for $latex j = 0 , 1, \ldots$$,
$latex \[
\begin{array}{rcl}
z^{(0)} & = & F ( x^{(0)} )
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & \left\{ \begin{array}{ll}
	\mp 1 & {\rm if} \; j = 0 \\
	0 & {\rm otherwise}
\end{array} \right.
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
	\sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)}
	- \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)}
\right)
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
	\mp (j+1) x^{(j+1)}
	- \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
\right)
\end{array}
\] $$


$end
