@(@\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
.
Square Root Function Forward Mode Theory
If @(@
F(x) = \sqrt{x}
@)@
@[@
F(x) * F^{(1)} (x) - 0 * F (x) = 1/2
@]@
and in the
standard math function differential equation
,
@(@
A(x) = 0
@)@,
@(@
B(x) = F(x)
@)@,
and @(@
D(x) = 1/2
@)@.
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.
It now follows from the general
Taylor coefficients recursion formula
that for @(@
j = 0 , 1, \ldots
@)@,
@[@
\begin{array}{rcl}
z^{(0)} & = & \sqrt { x^{(0)} }
\\
e^{(j)}
& = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)}
\\
& = & \left\{ \begin{array}{ll}
1/2 & {\rm if} \; j = 0 \\
0 & {\rm otherwise}
\end{array} \right.
\\
z^{(j+1)} & = & \frac{1}{j+1} \frac{1}{ b^{(0)} }
\left(
\sum_{k=1}^{j+1} k x^{(k)} e^{(j+1-k)}
- \sum_{k=1}^j k z^{(k)} b^{(j+1-k)}
\right)
\\
& = & \frac{1}{j+1} \frac{1}{ z^{(0)} }
\left(
\frac{j+1}{2} x^{(j+1) }
- \sum_{k=1}^j k z^{(k)} z^{(j+1-k)}
\right)
\end{array}
@]@
Input File: omh/theory/sqrt_forward.omh
