$Id:$
/* --------------------------------------------------------------------------
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.
-------------------------------------------------------------------------- */

$begin erf_reverse$$
$spell
	erf
	Taylor
$$


$section Error Function Reverse Mode Theory$$
$mindex erf$$

$head Notation$$
We use the reverse theory
$cref/standard math function/ReverseTheory/Standard Math Functions/$$
definition for the functions $latex H$$ and $latex G$$.

$head Positive Orders Z(t)$$
For order $latex j > 0$$,
suppose that $latex H$$ is the same as $latex G$$.
$latex \[
z^{(j)}
=
\frac{1}{j}  \sum_{k=1}^j k x^{(k)} y^{(j-k)}
\] $$
For $latex k = 1 , \ldots , j$$,
the partial of $latex H$$ with respect to $latex x^{(k)}$$ is given by
$latex \[
\D{H}{ x^{(k)} }
=
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
=
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{k}{j} y^{(j-k)}
\] $$
For $latex k = 1 , \ldots ,  j$$
The partial of $latex H$$ with respect to $latex y^{j-k}$$,
is given by
$latex \[
\D{H}{ y^{(j-k)} }
=
\D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} }
=
\D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \frac{k}{j} x^{k}
\] $$


$head Order Zero Z(t)$$
The $latex z^{(0)}$$ coefficient
is expressed as a function of the Taylor coefficients
for $latex X(t)$$ and $latex Y(t)$$ as follows:
In this case,
$latex \[
\D{H}{ x^{(0)} }
=
\D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} }
=
\D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } y^{(0)}
\] $$

$end
