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@(@\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 Reverse Mode Theory
We use the reverse theory standard math function definition for the functions @(@ H @)@ and @(@ G @)@. The forward mode formulas for the square root function are @[@ z^{(j)} = \sqrt { x^{(0)} } @]@ for the case @(@ j = 0 @)@, and for @(@ j > 0 @)@, @[@ z^{(j)} = \frac{1}{j} \frac{1}{ z^{(0)} } \left( \frac{j}{2} x^{(j) } - \sum_{\ell=1}^{j-1} \ell z^{(\ell)} z^{(j-\ell)} \right) @]@ If @(@ j = 0 @)@, we have the relation @[@ \begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{2 z^{(0)} } \end{array} @]@ If @(@ j > 0 @)@, then for @(@ k = 1, \ldots , j-1 @)@ @[@ \begin{array}{rcl} \D{H}{ z^{(0)} } & = & \D{G}{ z^{(0)} } + \D{G} { z^{(j)} } \D{ z^{(j)} }{ z^{(0)} } \\ & = & \D{G}{ z^{(0)} } - \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ z^{(0)} } \\ \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ 2 z^{(0)} } \\ \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } - \D{G}{ z^{(j)} } \frac{ z^{(j-k)} }{ z^{(0)} } \end{array} @]@
Input File: omh/theory/sqrt_reverse.omh