$Id: log_forward.omh 3683 2015-05-10 02:24:16Z 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
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                    Eclipse Public License Version 1.0.

A copy of this license is included in the COPYING file of this distribution.
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// END SHORT COPYRIGHT

$begin log_forward$$
$spell
	Taylor
$$

$section Logarithm Function Forward Mode Theory$$
$mindex log, log1p$$

$head Derivatives$$
If $latex F(x)$$ is $latex \R{log} (x)$$ or $latex \R{log1p} (x)$$
the corresponding derivative satisfies the equation
$latex \[
	( \bar{b} + x ) * F^{(1)} (x) - 0 * F (x)  = 1
\] $$
where
$latex \[
\bar{b}
=
\left\{ \begin{array}{ll}
	0 & \R{if} \; F(x) = \R{log}(x)
	\\
	1 & \R{if} \; F(x) = \R{log1p}(x)
\end{array} \right.
\] $$
In the
$cref/standard math function differential equation
	/ForwardTheory
	/Standard Math Functions
	/Differential Equation
/$$,
$latex A(x) = 0$$,
$latex B(x) = \bar{b} + x$$,
and $latex D(x) = 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$$
For orders $latex j = 0 , 1, \ldots$$,
$latex \[
\begin{array}{rcl}
z^{(0)} & = & F ( x^{(0)} )
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & \left\{ \begin{array}{ll}
	1 & {\rm if} \; j = 0 \\
	0 & {\rm otherwise}
\end{array} \right.
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
	\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
	- \sum_{k=1}^j k z^{(k)}  b^{(j+1-k)}
\right)
\\
& = & \frac{1}{j+1} \frac{1}{ \bar{b} + x^{(0)} }
\left(
	(j+1) x^{(j+1) }
	- \sum_{k=1}^j k z^{(k)} x^{(j+1-k)}
\right)
\end{array}
\] $$

$end