What is the median of the non-central F ratio distribution I am looking for a simple approximation to the median of the (simply) non-central F distribution with parameters dlnum, dldenominator, and ncp, the non-centrality parameter. Clearly, there is no closed-form expression; an approximation if fine. Thanks.
 A: Here is an approach (as opposed to a complete answer unfortunately) to obtain a good approximation with maybe a limited amount of programming.
It appears that for any fixed values of the two degrees of freedom parameters (call these $\nu_1$ and $\nu_2$), the median for the noncentral F distribution is a close to a linear function of $\lambda$ (the noncentrality parameter).  To support that observation consider the Mathematica code below.  (And from your profile I see that you use Mathematica.)
 Manipulate[
 Plot[N[InverseCDF[NoncentralFRatioDistribution[v1, v2, \[Lambda]], 1/2]], 
   {\[Lambda], 0, max\[Lambda]}, 
   PlotLabel -> Style["\[Nu]1=" <> ToString[v1] <> "  \[Nu]2=" <> ToString[v2], Bold, 24], 
   Frame -> True, 
   FrameLabel -> (Style[#, Bold, 18] &) /@ {"Noncentrality parameter (\[Lambda])", "Median"}, 
   ImagePadding -> {{50, 10}, {50, 10}}, 
   PlotRange -> {{0, max\[Lambda]}, {0, 5}}],
 {{v1, 1, "\[Nu]1"}, 1, 100, 1, Appearance -> "Labeled"}, 
 {{v2, 1, "\[Nu]2"}, 1, 100, 1, Appearance -> "Labeled"}, 
 {{max\[Lambda], 20, "Maximum \[Lambda]"}, 1, 100, 1, Appearance -> "Labeled"}, 
 TrackedSymbols :> {v1, v2, max\[Lambda]}]


Moving the sliders will show that given $\nu_1$ and $\nu_2$ the relationship remains fairly linear.  So it would seem that one could generate a grid of values for $\nu_1$, $\nu_2$, and $\lambda$ and then interpolate/extrapolate where desired.
Alternatively, one could probably come up with a function that estimates the slope and intercept from $\nu_1$ and $\nu_2$ which would be a much more compact approximation.  Such a function will look something like that for the mean which is a linear function of $\lambda$:
$$\mu={{\nu_2}\over{\nu_2-2}}+\lambda {{\nu_2}\over{\nu_1(\nu_2-2)}}$$
(at least for large values of $\nu_1$ and $\nu_2$).
A: Why code your own if a function with a lot of testing is available?
R:
qf(0.5, 3, 12, 1)

Mathematica:
N[InverseCDF[NoncentralFRatioDistribution[3, 12, 1], 1/2]]

My suggestion about using what's available besides being well-tested, is that many of the algorithms that you'll find will likely work only for large values of the degrees of freedom.  So that means multiple algorithms depending on the size of the degrees of freedom.  Also, many of the algorithms will be for the cumulative distribution function when means you'll need to make a guess as to the median and have an iterative routine to pin that down.
Maybe the best places to start are Abramowitz and Stegun "Handbook of Mathematical Functions" or Johnson and Kotz "Univariate Distributions".
