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\(\newcommand{\W}[1]{ \; #1 \; }\)
\(\newcommand{\R}[1]{ {\rm #1} }\)
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\(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\)
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asin_reverse
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Inverse Sine and Hyperbolic Sine Reverse Mode Theory
We use the reverse theory
standard math function \(H\) and \(G\) .
In addition, we use the forward mode notation in
asin_forward
\begin{eqnarray}
Q(t) & = & 1 \mp X(t) * X(t) \\
B(t) & = & \sqrt{ Q(t) }
\end{eqnarray}
We use \(q\) and \(b\)
for the p -th order Taylor coefficient
row vectors corresponding to these functions
and replace \(z^{(j)}\) by
\[( z^{(j)} , b^{(j)} )\]
in the definition for \(G\) and \(H\) .
The zero order forward mode formulas for the
asin
\begin{eqnarray}
q^{(0)} & = & 1 \mp x^{(0)} x^{(0)} \\
b^{(0)} & = & \sqrt{ q^{(0)} } \\
z^{(0)} & = & F( x^{(0)} )
\end{eqnarray}
where \(F(x) = \R{asin} (x)\) for \(-\)
and \(F(x) = \R{asinh} (x)\) for \(+\) .
For the orders \(j\) greater than zero we have
\begin{eqnarray}
q^{(j)} & = &
\mp \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{eqnarray}
If \(j = 0\) , we note that
\(F^{(1)} ( x^{(0)} ) = 1 / b^{(0)}\) and hence
\begin{eqnarray}
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} }
+ \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)} }
\mp \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} }
\end{eqnarray}
If \(j > 0\) , then for \(k = 1, \ldots , j-1\)
\begin{eqnarray}
\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)} }
\mp \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)} }
\mp \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)} }
\mp \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{eqnarray}