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This is cppad-20221105 documentation. Here is a link to its
current documentation
.
Trigonometric and Hyperbolic Sine and Cosine Reverse Theory
We use the reverse theory
standard math function
definition for the functions @(@
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
We use the value
@[@
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}
@]@
Input File: omh/theory/sin_cos_reverse.omh
