@(@\newcommand{\W}[1]{ \; #1 \; }
\newcommand{\R}[1]{ {\rm #1} }
\newcommand{\B}[1]{ {\bf #1} }
\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }
\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }
\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }
\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }@)@
This is cppad-20221105 documentation. Here is a link to its
current documentation
.
Power Function Reverse Mode Theory
We use the reverse theory
standard math function
definition for the functions @(@
H
@)@ and @(@
G
@)@.
The zero order forward mode formula for the
power
function is
@[@
z^{(0)} = F ( x^{(0)} )
@]@
@[@
\begin{array}{rcl}
\D{H}{ x^{(0)} }
& = & \D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} }
\\
\D{ z^{(0)} }{ x^{(0)} } & = & y [ x^{(0)} ]^{y - 1} = y z^{(0)} / x{(0)}
\end{array}
@]@
All the equations below apply to the case where @(@
j > 0
@)@.
For this case, the equation for @(@
z^{(j)}
@)@ is
@[@
z^{(j)}
=
\left. \left(
y z^{(0)} x^{(j)}
+
\frac{1}{j} \sum_{k=1}^{j-1} k ( y x^{(k)} z^{(j-k)} - z^{(k)} x^{(j-k)} )
\right) \right/ x^{(0)}
@]@
