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Square 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 $$\begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{2 z^{(0)} } \end{array}$$ If $j > 0$, then for $k = 1, \ldots , j-1$ $$\begin{array}{rcl} \D{H}{ z^{(0)} } & = & \D{G}{ z^{(0)} } + \D{G} { z^{(j)} } \D{ z^{(j)} }{ z^{(0)} } \\ & = & \D{G}{ z^{(0)} } - \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ z^{(0)} } \\ \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ 2 z^{(0)} } \\ \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } - \D{G}{ z^{(j)} } \frac{ z^{(j-k)} }{ z^{(0)} } \end{array}$$
Input File: omh/appendix/theory/sqrt_reverse.omh