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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
