  This paper presents novel numerical approaches to finding the secrecy
capacity of the multiple-input multiple-output (MIMO) wiretap channel subject
to multiple linear transmit covariance constraints, including sum power
constraint, per antenna power constraints and interference power constraint. An
analytical solution to this problem is not known and existing numerical
solutions suffer from slow convergence rate and/or high per-iteration
complexity. Deriving computationally efficient solutions to the secrecy
capacity problem is challenging since the secrecy rate is expressed as a
difference of convex functions (DC) of the transmit covariance matrix, for
which its convexity is only known for some special cases. In this paper we
propose two low-complexity methods to compute the secrecy capacity along with a
convex reformulation for degraded channels. In the first method we capitalize
on the accelerated DC algorithm which requires solving a sequence of convex
subproblems, for which we propose an efficient iterative algorithm where each
iteration admits a closed-form solution. In the second method, we rely on the
concave-convex equivalent reformulation of the secrecy capacity problem which
allows us to derive the so-called partial best response algorithm to obtain an
optimal solution. Notably, each iteration of the second method can also be done
in closed form. The simulation results demonstrate a faster convergence rate of
our methods compared to other known solutions. We carry out extensive numerical
experiments to evaluate the impact of various parameters on the achieved
secrecy capacity.
