Extracting Uncertainty from Numerical Solution

Question

I am performing numerical and uncertainty analysis for an oblique shockwave angle function:
$$
tan(delta)=frac{2}{tan(theta)}frac{M^2sin^2(theta)-1}{M^2(gamma+cos(2theta))+2}
$$
where $delta$ is deflection angle, $theta$ is shock angle, M is mach number, and $gamma$ is the ratio of specific heats. In our observation, $delta$, $theta$, and $M$ all have uncertainty. $gamma$ is assumed to be exact.
Our unknown quantity is $theta pm sigma_{theta}$. If it was possible to solve for $theta$ analytically, it would be trivial to propagate uncertainty (excluding covariances):
$$
sigma_{theta}^2 = left( frac{partialtheta}{partial M}sigma_M right)^2+ left( frac{partialtheta}{partialdelta}sigma_{delta} right)^2
$$
The question is: how do I define $theta$ so that I can take a derivative of it with respect to $M$ and $delta$ for uncertainty analysis? Or maybe there is a way to approximate $theta$ and take a numerical derivative? I am probably overthinking this.

Biswajit Banerjee · Answer

I'll try to address the question in bold.  The expression for the angle function can be expressed as $f(delta, M; theta)$.  You can compute partial derivatives of this function as
$$
  frac{partial f}{partial delta} = frac{partial f}{partial theta} frac{partial theta}{partial delta}
$$
and
$$
  frac{partial f}{partial M} = frac{partial f}{partial theta} frac{partial theta}{partial M}
$$
That will give you the two partial derivatives of $theta$ that you seek.

Extracting Uncertainty from Numerical Solution

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