  Multichannel quantum defect theory (MQDT) provides a powerful toolkit for
describing and understanding collisions of cold alkali atoms. Various MQDT
approximations differ primarily in how they characterize the so-called
short-ranged $K$-matrix, ${\mathbf K}_{\text{sr}}$, which encapsulates the
short-ranged, high-energy physics into a handful of low-energy parameters that
exhibit simple and smooth dependence on energy and field. Here, we compare
three different methods for computing ${\mathbf K}_{\text{sr}}$ for homonuclear
collisions of alkali atoms, from lithium to cesium. The MQDT calculations are
benchmarked against numerically converged coupled-channels calculations that
use a log-derivative propagator out to the asymptotic region. We study how well
these approximations reproduce positions of $s$-wave magnetic Feshbach
resonances, comparing to experiment where possible, and identify the
limitations of various approximations.
