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H
and
G
.
In addition,
we use
b
for the p-th order Taylor coefficient
row vectors corresponding to
1 + X(t) * X(t)
and replace
z^{(j)}
by
\[
( z^{(j)} , b^{(j)} )
\]
in the definition for
G
and
H
.
The forward mode formulas for the
arctangent
function are
\[
\begin{array}{rcl}
z^{(j)} & = & \arctan ( x^{(0)} ) \\
b^{(j)} & = & 1 + x^{(0)} x^{(0)}
\end{array}
\]
for the case
j = 0
, and for
j > 0
,
\[
\begin{array}{rcl}
b^{(j)} & = &
\sum_{k=0}^j x^{(k)} x^{(j-k)}
\\
z^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} }
\left(
j x^{(j)}
- \sum_{k=1}^{j-1} k z^{(k)} b^{(j-k)}
\right)
\end{array}
\]
If
j = 0
, we have the relation
\[
\begin{array}{rcl}
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(j)} }
+ \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} }
+ \D{G}{ b^{(j)} } 2 x^{(0)}
\end{array}
\]
If
j > 0
, then for
k = 1, \ldots , j-1
\[
\begin{array}{rcl}
\D{H}{ b^{(0)} } & = &
\D{G}{ b^{(0)} }
- \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(0)} }
\\
& = &
\D{G}{ b^{(0)} }
- \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ b^{(0)} }
\\
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(j)} }
\\
& = &
\D{G}{ x^{(j)} }
+ \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} }
+ \D{G}{ b^{(j)} } 2 x^{(0)}
\\
\D{H}{ x^{(0)} } & = &
\D{G}{ x^{(0)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(0)} }
\\
& = &
\D{G}{ x^{(0)} } +
\D{G}{ b^{(j)} } 2 x^{(j)}
\\
\D{H}{ x^{(k)} } & = &
\D{G}{ x^{(k)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ x^{(k)} }
\\
& = &
\D{G}{ x^{(k)} }
+ \D{G}{ b^{(j)} } 2 x^{(j-k)}
\\
\D{H}{ z^{(k)} } & = &
\D{G}{ z^{(k)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ z^{(k)} }
\\
& = &
\D{G}{ z^{(k)} }
- \D{G}{ z^{(j)} } \frac{k b^{(j-k)} }{ j b^{(0)} }
\\
\D{H}{ b^{(j-k)} } & = &
\D{G}{ b^{(j-k)} }
+ \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(j-k)} }
+ \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(j-k)} }
\\
& = &
\D{G}{ b^{(j-k)} }
- \D{G}{ z^{(j)} } \frac{k z^{(k)} }{ j b^{(0)} }
\end{array}
\]