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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}} }@)@ Inverse Sine and Hyperbolic Sine Reverse Mode Theory We use the reverse theory standard math function definition for the functions @(@ H @)@ and @(@ G @)@. In addition, we use the forward mode notation in asin_forward for @[@ \begin{array}{rcl} Q(t) & = & 1 \mp X(t) * X(t) \\ B(t) & = & \sqrt{ Q(t) } \end{array} @]@ We use @(@ q @)@ and @(@ b @)@ for the p-th order Taylor coefficient row vectors corresponding to these functions and replace @(@ z^{(j)} @)@ by @[@ ( z^{(j)} , b^{(j)} ) @]@ in the definition for @(@ G @)@ and @(@ H @)@. The zero order forward mode formulas for the asin function are @[@ \begin{array}{rcl} q^{(0)} & = & 1 \mp x^{(0)} x^{(0)} \\ b^{(0)} & = & \sqrt{ q^{(0)} } \\ z^{(0)} & = & F( x^{(0)} ) \end{array} @]@ where @(@ F(x) = \R{asin} (x) @)@ for @(@ - @)@ and @(@ F(x) = \R{asinh} (x) @)@ for @(@ + @)@. For the orders @(@ j @)@ greater than zero we have @[@ \begin{array}{rcl} q^{(j)} & = & \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} \\ b^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} } \left( \frac{j}{2} q^{(j)} - \sum_{k=1}^{j-1} k b^{(k)} b^{(j-k)} \right) \\ z^{(j)} & = & \frac{1}{j} \frac{1}{ b^{(0)} } \left( j x^{(j)} - \sum_{k=1}^{j-1} k z^{(k)} b^{(j-k)} \right) \end{array} @]@ If @(@ j = 0 @)@, we note that @(@ F^{(1)} ( x^{(0)} ) = 1 / b^{(0)} @)@ and hence @[@ \begin{array}{rcl} \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(0)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(0)} } \D{ q^{(0)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} } \mp \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} } \end{array} @]@ If @(@ j > 0 @)@, then for @(@ k = 1, \ldots , j-1 @)@ @[@ \begin{array}{rcl} \D{H}{ b^{(0)} } & = & \D{G}{ b^{(0)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(0)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(0)} } \\ & = & \D{G}{ b^{(0)} } - \D{G}{ z^{(j)} } \frac{ z^{(j)} }{ b^{(0)} } - \D{G}{ b^{(j)} } \frac{ b^{(j)} }{ b^{(0)} } \\ \D{H}{ x^{(0)} } & = & \D{G}{ x^{(0)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(0)} } \\ & = & \D{G}{ x^{(0)} } \mp \D{G}{ b^{(j)} } \frac{ x^{(j)} }{ b^{(0)} } \\ \D{H}{ x^{(j)} } & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(j)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(j)} } \\ & = & \D{G}{ x^{(j)} } + \D{G}{ z^{(j)} } \frac{1}{ b^{(0)} } \mp \D{G}{ b^{(j)} } \frac{ x^{(0)} }{ b^{(0)} } \\ \D{H}{ b^{(j - k)} } & = & \D{G}{ b^{(j - k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ b^{(j - k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ b^{(j - k)} } \\ & = & \D{G}{ b^{(j - k)} } - \D{G}{ z^{(j)} } \frac{k z^{(k)} }{j b^{(0)} } - \D{G}{ b^{(j)} } \frac{ b^{(k)} }{ b^{(0)} } \\ \D{H}{ x^{(k)} } & = & \D{G}{ x^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ x^{(k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ q^{(j)} } \D{ q^{(j)} }{ x^{(k)} } \\ & = & \D{G}{ x^{(k)} } \mp \D{G}{ b^{(j)} } \frac{ x^{(j-k)} }{ b^{(0)} } \\ \D{H}{ z^{(k)} } & = & \D{G}{ z^{(k)} } + \D{G}{ z^{(j)} } \D{ z^{(j)} }{ z^{(k)} } + \D{G}{ b^{(j)} } \D{ b^{(j)} }{ z^{(k)} } \\ & = & \D{G}{ z^{(k)} } - \D{G}{ z^{(j)} } \frac{k b^{(j-k)} }{ j b^{(0)} } \end{array} @]@

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Input File: omh/appendix/theory/asin_reverse.omh