$Id: sqrt_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.
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// END SHORT COPYRIGHT

$begin sqrt_forward$$
$spell
	sqrt
	Arctangent
	Taylor
$$

$section Square Root Function Forward Mode Theory$$
$mindex sqrt$$

If $latex F(x) = \sqrt{x} $$
$latex \[
	F(x) * F^{(1)} (x) - 0 * F (x)  = 1/2
\] $$
and in the
$cref/standard math function differential equation
	/ForwardTheory
	/Standard Math Functions
	/Differential Equation
/$$,
$latex A(x) = 0$$,
$latex B(x) = F(x)$$,
and $latex D(x) = 1/2$$.
We use $latex a$$, $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) ] $$ respectively.
It now follows from the general
$cref/Taylor coefficients recursion formula
	/ForwardTheory
	/Standard Math Functions
	/Taylor Coefficients Recursion Formula
/$$
that for $latex j = 0 , 1, \ldots$$,
$latex \[
\begin{array}{rcl}
z^{(0)} & = & \sqrt { x^{(0)} }
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & \left\{ \begin{array}{ll}
	1/2 & {\rm if} \; j = 0 \\
	0   & {\rm otherwise}
\end{array} \right.
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
	\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
	- \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
\right)
\\
& = & \frac{1}{j+1} \frac{1}{ z^{(0)} }
\left(
	\frac{j+1}{2} x^{(j+1) }
	- \sum_{k=1}^j k z^{(k)} z^{(j+1-k)}
\right)
\end{array}
\] $$

$end
