  We put forward a scheme to study the anisotropic magnetic couplings in
Sr2IrO4 by mapping fully relativistic constrained noncollinear density
functional theory including an on-site Hubbard U correction onto a general spin
model Hamiltonian. This procedure allows for the simultaneous account and
direct control of the lattice, spin and orbital interactions within a fully ab
initio scheme. We compute the isotropic, single site anisotropy and
Dzyaloshinskii-Moriya (DM) coupling parameters, and clarify that the origin of
the canted magnetic state in Sr2IrO4 arises from the interplay between
structural distortions and the competition between isotropic exchange and DM
interactions. A complete magnetic phase diagram with respect to the tetragonal
distortion and the rotation of IrO6 octahedra is constructed, revealing the
presence of two types of canted to collinear magnetic transitions: a spin-flop
transition with increasing tetragonal distortion and a complete quenching of
the basal weak ferromagnetic moment below a critical octahedral rotation.
