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log_reverse¶
View page sourceLogarithm 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 logarithm is
\[z^{(0)} = F( x^{(0)} )\]
and for \(j > 0\),
\[z^{(j)}
= \frac{1}{ \bar{b} + x^{(0)} } \frac{1}{j}
\left(
j x^{(j)}
- \sum_{k=1}^{j-1} k z^{(k)} x^{(j-k)}
\right)\]
where
\[\begin{split}\bar{b}
=
\left\{ \begin{array}{ll}
0 & \R{if} \; F(x) = \R{log}(x)
\\
1 & \R{if} \; F(x) = \R{log1p}(x)
\end{array} \right.\end{split}\]
We note that for \(j > 0\)
\begin{eqnarray}
\D{ z^{(j)} } { x^{(0)} }
& = &
-
\frac{1}{ \bar{b} + x^{(0)} }
\frac{1}{ \bar{b} + x^{(0)} }
\frac{1}{j}
\left(
j x^{(j)}
- \sum_{k=1}^{j-1} k z^{(k)} x^{(j-k)}
\right)
\\
& = &
-
\frac{z^{(j)}}{ \bar{b} + x^{(0)} }
\end{eqnarray}
Removing the zero order partials are given by
\begin{eqnarray}
\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)} } \frac{1}{ \bar{b} + x^{(0)} }
\end{eqnarray}
For orders \(j > 0\) and for \(k = 1 , \ldots , j-1\)
\begin{eqnarray}
\D{H}{ x^{(0)} }
& = &
\D{G}{ x^{(0)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(0)} } - \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ \bar{b} + x^{(0)} }
\\
\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)} } \frac{1}{ \bar{b} + x^{(0)} }
\\
\D{H}{ x^{(j-k)} } & = &
\D{G}{ x^{(j-k)} } -
\D{G}{ z^{(j)} } \frac{1}{ \bar{b} + x^{(0)} } \frac{k}{j} z^{(k)}
\\
\D{H}{ z^{(k)} } & = &
\D{G}{ z^{(k)} } -
\D{G}{ z^{(j)} } \frac{1}{ \bar{b} + x^{(0)} } \frac{k}{j} x^{(j-k)}
\end{eqnarray}