We derive several closed-form expressions that generalize co-prime array system model and study a non negative gridless compressive sensing formulation of the problem of estimating direction-of-arrival (DOA) based on the derived model. To solve the problem, two computationally efficient cyclic block coordinate minimization algorithms are proposed; the algorithms perform atomic norm minimization of an objective function through a sequence of computationally efficient atom merging and atom activation steps conducted in subdomains of a continuous atom search space. The convergence properties of the developed algorithms are analyzed. Numerical simulations demonstrate that the proposed techniques outperform the joint sparsity reconstruction method (JLASSO) and the ESPRIT method with spatial smoothing (SS ESPRIT) in terms of various criteria. It is also demonstrated that our methods are significantly faster and yield competitive performance in terms of root mean square error (RMSE), detection probability, and false alarms when compared to the recent convex optimization based methods, i.e. the gridless SPICE with ESPRIT (GLS-ESPRIT), the atomic norm minimization with dimension reduction and ESPRIT (ANM-ESPRIT), and the nuclear norm minimization with ESPRIT (NNM-ESPRIT).