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H
and
G
.
The forward mode formulas for the
square root
function are
\[
z^{(j)} = \sqrt { x^{(0)} }
\]
for the case
j = 0
, and for
j > 0
,
\[
z^{(j)} = \frac{1}{j} \frac{1}{ z^{(0)} }
\left(
\frac{j}{2} x^{(j) }
- \sum_{\ell=1}^{j-1} \ell z^{(\ell)} z^{(j-\ell)}
\right)
\]
If
j = 0
, we have the relation
\[
\begin{array}{rcl}
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{2 z^{(0)} }
\end{array}
\]
If
j > 0
, then for
k = 1, \ldots , j-1
\[
\begin{array}{rcl}
\D{H}{ z^{(0)} } & = &
\D{G}{ z^{(0)} } + \D{G} { z^{(j)} } \D{ z^{(j)} }{ z^{(0)} }
\\
& = &
\D{G}{ z^{(0)} } -
\D{G}{ z^{(j)} } \frac{ z^{(j)} }{ z^{(0)} }
\\
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
\\
& = &
\D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ 2 z^{(0)} }
\\
\D{H}{ z^{(k)} } & = &
\D{G}{ z^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} }
\\
& = &
\D{G}{ z^{(k)} } - \D{G}{ z^{(j)} } \frac{ z^{(j-k)} }{ z^{(0)} }
\end{array}
\]