l1Magic is a collection of customized interior-point methods (IPM) for solving problems related to compressed sensing, such as basis pursuit and variants, and TV denoising. The IPM are primal-dual methods based off the algorithms in Convex Optimization by Boyd and Vandenberghe. It uses conjugate gradients (CG) to solve the Newton system.
Performance of l1Magic depends on a few parameter such as the initial guess and the parameters used for the CG iteration. Because it is a second order method, it is highly accurate, but it may be slower than first order methods.