\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
exp_reverse¶
View page sourceExponential Function Reverse Mode Theory¶
We use the reverse theory standard math function definition for the functions \(H\) and \(G\). The zero order forward mode formula for the exponential is
\[z^{(0)} = F ( x^{(0)} )\]
and for \(j > 0\),
\[z^{(j)} = x^{(j)} d^{(0)}
+ \frac{1}{j} \sum_{k=1}^{j} k x^{(k)} z^{(j-k)}\]
where
\[\begin{split}d^{(0)} = \left\{ \begin{array}{ll}
0 & \R{if} \; F(x) = \R{exp}(x)
\\
1 & \R{if} \; F(x) = \R{expm1}(x)
\end{array} \right.\end{split}\]
For order \(j = 0, 1, \ldots\) we note that
\begin{eqnarray}
\D{H}{ x^{(j)} }
& = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
\\
& = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } ( d^{(0)} + z^{(0)} )
\end{eqnarray}
If \(j > 0\), then for \(k = 1 , \ldots , j\)
\begin{eqnarray}
\D{H}{ x^{(k)} } & = &
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k z^{(j-k)}
\\
\D{H}{ z^{(j-k)} } & = &
\D{G}{ z^{(j-k)} } + \D{G}{ z^{(j)} } \frac{1}{j} k x^{(k)}
\end{eqnarray}