\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

sin_cos_reverse

View page source

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{eqnarray} 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{eqnarray}

for the case \(j = 0\), and for \(j > 0\),

\begin{eqnarray} 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{eqnarray}

If \(j = 0\), we have the relation

\begin{eqnarray} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ s^{(j)} } c^{(0)} + \ell \D{G}{ c^{(j)} } s^{(0)} \end{eqnarray}

If \(j > 0\), then for \(k = 1, \ldots , j-1\)

\begin{eqnarray} \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{eqnarray}