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H
and
G
.
The forward mode formulas for the
logarithm
function are
\[
z^{(j)} = \log ( x^{(0)} )
\]
for the case
j = 0
, and for
j > 0
,
\[
z^{(j)}
= \frac{1}{ x^{(0)} } \frac{1}{j}
\left(
j x^{(j)}
- \sum_{k=1}^{j-1} k z^{(k)} x^{(j-k)}
\right)
\]
otherwise.
If
j = 0
, we have the relation
\[
\D{H}{ x^{(j)} } =
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} }
\]
If
j > 0
, then for
k = 1 , \ldots , j-1
\[
\begin{array}{rcl}
\D{H}{ x^{(0)} } & = &
\D{G}{ x^{(0)} } - \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} }
\frac{1}{ x^{(0)} } \frac{1}{j}
\left(
j x^{(j)}
- \sum_{m=1}^{j-1} m z^{(m)} x^{(j-m)}
\right)
\\
& = &
\D{G}{ x^{(0)} } - \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } z^{(j)}
\\
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ x^{(0)} }
\\
\D{H}{ x^{(j-k)} } & = &
\D{G}{ x^{(j-k)} } -
\D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } \frac{1}{j} k z^{(k)}
\\
\D{H}{ z^{(k)} } & = &
\D{G}{ z^{(k)} } -
\D{G}{ z^{(j)} } \frac{1}{ x^{(0)} } \frac{1}{j} k x^{(j-k)}
\end{array}
\]