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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/sqrt_reverse.omh