  We compare three methods to calculate the nucleon-nucleon t-matrix based on
the three-dimensional formulation of J. Golak et al., Phys. Rev. C 81, 034006,
(2010). In the first place we solve a system of complex linear inhomogeneous
equations directly for the t-matrix. Our second method is based on iterations
and a variant of the Lanczos algorithm. In the third case we obtain the
t-matrix in two steps, solving a system of real linear equations for the
k-matrix expansion coefficients and then solving an on-shell equation, which
connects the scalar coefficients of the k- and t-matrices. A very good
agreement among the three methods is demonstrated for selected nucleon-nucleon
scattering observables using a chiral next-to-next-to-leading-order
neutron-proton potential. We also apply our three-dimensional framework to the
demanding problem of proton-proton scattering, using a corresponding version of
the nucleon-nucleon potential and supplementing it with the (screened) Coulomb
force, taken also in the three-dimensional form. We show converged results for
two different screening functions and find a very good agreement with other
methods dealing with proton-proton scattering.
