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Inverse Sine and Hyperbolic Sine Forward Mode Theory

Derivatives

\begin{eqnarray} \R{asin}^{(1)} (x) & = & 1 / \sqrt{ 1 - x * x } \\ \R{asinh}^{(1)} (x) & = & 1 / \sqrt{ 1 + x * x } \end{eqnarray}

If \(F(x)\) is \(\R{asin} (x)\) or \(\R{asinh} (x)\) the corresponding derivative satisfies the equation

\[\sqrt{ 1 \mp x * x } * F^{(1)} (x) - 0 * F (u) = 1\]

and in the standard math function differential equation , \(A(x) = 0\), \(B(x) = \sqrt{1 \mp x * x }\), and \(D(x) = 1\). 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 \(Q(x) = 1 \mp x * x\) and let \(q\) be the corresponding Taylor coefficients for \(Q[ X(t) ]\). It follows that

\[\begin{split}q^{(j)} = \left\{ \begin{array}{ll} 1 \mp x^{(0)} * x^{(0)} & {\rm if} \; j = 0 \\ \mp \sum_{k=0}^j x^{(k)} x^{(j-k)} & {\rm otherwise} \end{array} \right.\end{split}\]

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{eqnarray} 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{eqnarray}

It now follows from the general Taylor Coefficients Recursion Formula that for \(j = 0 , 1, \ldots\),

\begin{eqnarray} z^{(0)} & = & F ( x^{(0)} ) \\ e^{(j)} & = & d^{(j)} + \sum_{k=0}^{j} a^{(j-k)} * z^{(k)} \\ & = & \left\{ \begin{array}{ll} 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( (j+1) x^{(j+1)} - \sum_{k=1}^j k z^{(k)} b^{(j+1-k)} \right) \end{eqnarray}