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sqrt_reverse¶
View page sourceSquare Root Function Reverse Mode Theory¶
We use the reverse theory standard math function definition for the functions \(H\) 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{eqnarray}
\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{eqnarray}
If \(j > 0\), then for \(k = 1, \ldots , j-1\)
\begin{eqnarray}
\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{eqnarray}