\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
.
Tangent and Hyperbolic Tangent Reverse Mode Theory
Notation
We use the reverse theory
standard math function
definition for the functions
H
and
G
.
In addition, we use the forward mode notation in tan_forward
for
X(t)
,
Y(t)
and
Z(t)
.
Eliminating Y(t)
For
j > 0
, the forward mode coefficients are given by
y^{(j-1)} = \sum_{k=0}^{j-1} z^{(k)} z^{(j-k-1)}
Fix
j > 0
and suppose that
H
is the same as
G
except that
y^{(j-1)}
is replaced as a function of the Taylor
coefficients for
Z(t)
.
To be specific, for
k = 0 , \ldots , j-1
,
\begin{array}{rcl}
\D{H}{ z^{(k)} }
& = &
\D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } \D{ y^{(j-1)} }{ z^{(k)} }
\\
& = &
\D{G}{ z^{(k)} } + \D{G}{ y^{(j-1)} } 2 z^{(j-k-1)}
\end{array}
Positive Orders Z(t)
For order
j > 0
,
suppose that
H
is the same as
G
except that
z^{(j)}
is expressed as a function of
the coefficients for
X(t)
, and the
lower order Taylor coefficients for
Y(t)
,
Z(t)
.
z^{(j)}
=
x^{(j)} \pm \frac{1}{j} \sum_{k=1}^j k x^{(k)} y^{(j-k)}
For
k = 1 , \ldots , j
,
the partial of
H
with respect to
x^{(k)}
is given by
\begin{array}{rcl}
\D{H}{ x^{(k)} } & = &
\D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
\\
& = &
\D{G}{ x^{(k)} } +
\D{G}{ z^{(j)} }
\left[ \delta ( j - k ) \pm \frac{k}{j} y^{(j-k)} \right]
\end{array}
where
\delta ( j - k )
is one if
j = k
and zero
otherwise.
For
k = 1 , \ldots , j
The partial of
H
with respect to
y^{j-k}
,
is given by
\begin{array}{rcl}
\D{H}{ y^{(j-k)} } & = &
\D{G}{ y^{(j-k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ y^{(j-k)} }
\\
& = &
\D{G}{ y^{(j-k)} } \pm \D{G}{ z^{(j)} }\frac{k}{j} x^{k}
\end{array}
Order Zero Z(t)
The order zero coefficients for the tangent and hyperbolic tangent are
\begin{array}{rcl}
z^{(0)} & = & \left\{
\begin{array}{c} \tan ( x^{(0)} ) \\ \tanh ( x^{(0)} ) \end{array}
\right.
\end{array}
Suppose that
H
is the same as
G
except that
z^{(0)}
is expressed as a function of the Taylor coefficients
for
X(t)
.
In this case,
\begin{array}{rcl}
\D{H}{ x^{(0)} }
& = &
\D{G}{ x^{(0)} }
+ \D{G}{ z^{(0)} } \D{ z^{(0)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(0)} } + \D{G}{ z^{(0)} } ( 1 \pm y^{(0)} )
\end{array}
Input File: omh/theory/tan_reverse.omh
