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Arcsine Function Reverse Mode Theory
We use the reverse theory standard math function definition for the functions  H and  G . In addition, we use  q and  b for the p-th order Taylor coefficient row vectors corresponding to functions  \[
\begin{array}{rcl}
     Q(t) & = & 1 - X(t) * X(t) \\
     B(t) & = & \sqrt{ Q(t) }
\end{array}
\] 
and replace  z^{(j)} by  \[
     ( z^{(j)} , b^{(j)} )
\] 
in the definition for  G and  H . The forward mode formulas for the asin function are  \[
\begin{array}{rcl}
     q^{(0)}  & = & 1 - x^{(0)} x^{(0)} \\
     b^{(j)}  & = & \sqrt{ q^{(0)} }    \\
     z^{(j)}  & = & \arcsin ( x^{(0)} )
\end{array}
\] 
for the case  j = 0 , and for  j > 0 ,  \[
\begin{array}{rcl}
q^{(j)} & = &  
     - \sum_{k=0}^j x^{(k)} x^{(j-k)} 
\\
b^{(j)} & = &
     \frac{1}{j} \frac{1}{ b^{(0)} } 
     \left(
          \frac{j}{2} q^{(j)}
          - \sum_{k=1}^{j-1} k b^{(k)} b^{(j-k)}  
     \right)
\\
z^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} } 
\left(
     j x^{(j)}
     - \sum_{k=1}^{j-1} k z^{(k)}  b^{(j-k)} 
\right)
\end{array}
\] 
If  j = 0 , we have the relation  \[
\begin{array}{rcl}
\D{H}{ x^{(j)} } & = & 
\D{G}{ x^{(0)} }  
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(0)} } \D{ q^{(0)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(j)} }  
+ \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} }
- \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} }
\end{array}
\] 
If  j > 0 , then for  k = 1, \ldots , j-1  \[
\begin{array}{rcl}
\D{H}{ b^{(0)} } & = & 
\D{G}{ b^{(0)} } 
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(0)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(0)} }
\\
& = &
\D{G}{ b^{(0)} } 
- \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ b^{(0)} } 
- \D{G}{ b^{(j)} } \frac{ b^{(j)} }{ b^{(0)} }
\\
\D{H}{ x^{(0)} } & = & 
\D{G}{ x^{(0)} } 
+
\D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(0)} }
\\
& = & 
\D{G}{ x^{(0)} } 
- \D{G}{ b^{(j)} } \frac{ x^{(j)} }{ b^{(0)} }
\\
\D{H}{ x^{(j)} } & = & 
\D{G}{ x^{(j)} } 
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(j)} }
\\
& = & 
\D{G}{ x^{(j)} } 
+ \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} } 
- \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} }
\\
\D{H}{ b^{(j - k)} } & = & 
\D{G}{ b^{(j - k)} } 
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(j - k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(j - k)} }
\\
& = &
\D{G}{ b^{(j - k)} } 
- \D{G}{ z^{(j)} } \frac{k z^{(k)} }{j b^{(0)} }
- \D{G}{ b^{(j)} } \frac{ b^{(k)} }{ b^{(0)} }
\\
\D{H}{ x^{(k)} } & = & 
\D{G}{ x^{(k)} } 
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(k)} }
\\
& = & 
\D{G}{ x^{(k)} } 
- \D{G}{ b^{(j)} } \frac{ x^{(j-k)} }{ b^{(0)} }
\\
\D{H}{ z^{(k)} } & = & 
\D{G}{ z^{(k)} } 
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ z^{(k)} }
\\
& = &
\D{G}{ z^{(k)} } 
- \D{G}{ z^{(j)} } \frac{k b^{(j-k)} }{ j b^{(0)} } 
\end{array}
\] 

Input File: omh/asin_reverse.omh