@(@\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}} }@)@
This is cppad-20221105 documentation. Here is a link to its
current documentation
.
Inverse Cosine and Hyperbolic Cosine Forward Mode Theory
Derivatives
@[@
\begin{array}{rcl}
\R{acos}^{(1)} (x) & = & - 1 / \sqrt{ 1 - x * x }
\\
\R{acosh}^{(1)} (x) & = & + 1 / \sqrt{ x * x - 1}
\end{array}
@]@If @(@
F(x)
@)@ is @(@
\R{acos} (x)
@)@ or @(@
\R{acosh} (x)
@)@
the corresponding derivative satisfies the equation
@[@
\sqrt{ \mp ( x * x - 1 ) } * F^{(1)} (x) - 0 * F (u) = \mp 1
@]@
and in the
standard math function differential equation
,
@(@
A(x) = 0
@)@,
@(@
B(x) = \sqrt{ \mp( x * x - 1 ) }
@)@,
and @(@
D(x) = \mp 1
@)@.
We use @(@
a
@)@, @(@
b
@)@, @(@
d
@)@ and @(@
z
@)@ to denote the
Taylor coefficients for
@(@
A [ X (t) ]
@)@,
@(@
B [ X (t) ]
@)@,
@(@
D [ X (t) ]
@)@,
and @(@
F [ X(t) ]
@)@ respectively.
Taylor Coefficients Recursion
We define @(@
Q(x) = \mp ( x * x - 1 )
@)@
and let @(@
q
@)@ be the corresponding Taylor coefficients for
@(@
Q[ X(t) ]
@)@.
It follows that
@[@
q^{(j)} = \left\{ \begin{array}{ll}
\mp ( x^{(0)} * x^{(0)} - 1 ) & {\rm if} \; j = 0 \\
\mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise}
\end{array} \right.
@]@
It follows that
@(@
B[ X(t) ] = \sqrt{ Q[ X(t) ] }
@)@ and
from the equations for the
square root
that for @(@
j = 0 , 1, \ldots
@)@,
@[@
\begin{array}{rcl}
b^{(0)} & = & \sqrt{ q^{(0)} }
\\
b^{(j+1)} & = &
\frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\frac{j+1}{2} q^{(j+1) }
- \sum_{k=1}^j k b^{(k)} b^{(j+1-k)}
\right)
\end{array}
@]@
It now follows from the general
Taylor coefficients recursion formula
that for @(@
j = 0 , 1, \ldots
@)@,
@[@
\begin{array}{rcl}
z^{(0)} & = & F ( x^{(0)} )
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & \left\{ \begin{array}{ll}
\mp 1 & {\rm if} \; j = 0 \\
0 & {\rm otherwise}
\end{array} \right.
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\sum_{k=0}^j e^{(k)} (j+1-k) x^{(j+1-k)}
- \sum_{k=1}^j b^{(k)} (j+1-k) z^{(j+1-k)}
\right)
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\mp (j+1) x^{(j+1)}
- \sum_{k=1}^j k z^{(k)} b^{(j+1-k)}
\right)
\end{array}
@]@
Input File: omh/theory/acos_forward.omh
