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H
and
G
.
In addition,
we use the following definitions for
s
and
c
and the integer
\ell
Coefficients |
s
|
c
|
\ell
| |||
Trigonometric Case |
\sin [ X(t) ]
|
\cos [ X(t) ]
| 1 | |||
Hyperbolic Case |
\sinh [ X(t) ]
|
\cosh [ X(t) ]
| -1 |
\[
z^{(j)} = ( s^{(j)} , c^{(j)} )
\]
in the definition for
G
and
H
.
The forward mode formulas for the
sine and cosine
functions are
\[
\begin{array}{rcl}
s^{(j)} & = & \frac{1 + \ell}{2} \sin ( x^{(0)} )
+ \frac{1 - \ell}{2} \sinh ( x^{(0)} )
\\
c^{(j)} & = & \frac{1 + \ell}{2} \cos ( x^{(0)} )
+ \frac{1 - \ell}{2} \cosh ( x^{(0)} )
\end{array}
\]
for the case
j = 0
, and for
j > 0
,
\[
\begin{array}{rcl}
s^{(j)} & = & \frac{1}{j}
\sum_{k=1}^{j} k x^{(k)} c^{(j-k)} \\
c^{(j)} & = & \ell \frac{1}{j}
\sum_{k=1}^{j} k x^{(k)} s^{(j-k)}
\end{array}
\]
If
j = 0
, we have the relation
\[
\begin{array}{rcl}
\D{H}{ x^{(j)} } & = &
\D{G}{ x^{(j)} }
+ \D{G}{ s^{(j)} } c^{(0)}
+ \ell \D{G}{ c^{(j)} } s^{(0)}
\end{array}
\]
If
j > 0
, then for
k = 1, \ldots , j-1
\[
\begin{array}{rcl}
\D{H}{ x^{(k)} } & = &
\D{G}{ x^{(k)} }
+ \D{G}{ s^{(j)} } \frac{1}{j} k c^{(j-k)}
+ \ell \D{G}{ c^{(j)} } \frac{1}{j} k s^{(j-k)}
\\
\D{H}{ s^{(j-k)} } & = &
\D{G}{ s^{(j-k)} } + \ell \D{G}{ c^{(j)} } k x^{(k)}
\\
\D{H}{ c^{(j-k)} } & = &
\D{G}{ c^{(j-k)} } + \D{G}{ s^{(j)} } k x^{(k)}
\end{array}
\]