Two algorithms for the stress update (i.e., time-integration of the constitutive equation) in large strain solid mechanics are discussed, with particular emphasis on two issues: the incremental objectivity and the implementation aspects. It is shown that both algorithms are incrementally objective (i.e., they treat rigid rotations properly) and that they can be employed to add large strain capabilities to a small large deformation paths (rigid rotation, simple shear, extension, extension and compression, dilatation, extension and rotation), have been used to test and compare the two algorithms, both for elastic and plastic analysis. These test evidence different time integration accuracy for each algorithm. However, it is demonstrated that the, in general, less accurate algorithm gives exact results for shear-free deformation paths.