  We analyze numerically the existence of regular stationary rotating hairy
black holes within the framework of general relativity, which are the result of
solving the Einstein-Klein-Gordon system for a complex-valued scalar field
under suitable boundary (regularity and asymptotically flat) conditions. To
that aim we solve the corresponding system of elliptic partial differential
equations using spectral methods which are specially suited for such a
numerical task. In order to obtain such system of equations we employ a
parametrization for the metric that corresponds to quasi-isotropic coordinates
(QIC) that have been used in the past for analyzing different kinds of
stationary rotating relativistic systems. Our findings are in agreement with
those reported originally by Herdeiro \& Radu ite{Herdeiro2014,Herdeiro2015}.
The method is submitted to several analytic and numerical tests, which include
the recovery of the Kerr solution in QIC and the cloud solutions in the Kerr
background. We report different global quantities that allow us to determine
the contribution of the boson hair to the spacetime, as well as relevant
quantities at the horizon, like the surface gravity. The latter indicates to
what extent the hairy solutions approach the extremal limit, noting that for
this kind of solutions the ratio of the angular momentum per squared mass
$J_\infty/M^2_{\rm ADM}$ can be larger than unity due to the contribution of
the scalar hair, a situation which differs from the Kerr metric where this
parameter is bounded according to $0\leq |J/M^2| \leq 1$ with the upper bound
corresponding to the extremal case.
