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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.