Inverse Cosine and Hyperbolic Cosine Reverse Mode Theory
We use the reverse theory
standard math function
definition for the functions \(H\) and \(G\).
In addition, we use the forward mode notation in
acos_forward for
\begin{eqnarray}
Q(t) & = & \mp ( X(t) * X(t) - 1 ) \\
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
acos
function are
\begin{eqnarray}
q^{(0)} & = & \mp ( x^{(0)} x^{(0)} - 1) \\
b^{(0)} & = & \sqrt{ q^{(0)} } \\
z^{(0)} & = & F ( x^{(0)} )
\end{eqnarray}
where \(F(x) = \R{acos} (x)\) for \(-\)
and \(F(x) = \R{acosh} (x)\) for \(+\).
For 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(
\mp j x^{(j)}
- \sum_{k=1}^{j-1} k z^{(k)} b^{(j-k)}
\right)
\end{eqnarray}
If \(j = 0\), we note that
\(F^{(1)} ( x^{(0)} ) = \mp 1 / b^{(0)}\) 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)} }
\mp \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)} }
\mp \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}