The　aim　of　this　paper　is　to　present　a　general　algorithm　for　the　branching　solution　of　nonlinear　operator　equations　in　a　Hilbert　space,　namely　the　k-order　Taylor　expansion　algorithm,　k　＞=　1.　The　standard　Galerkin　method　can　be　viewed　as　the　1-order　Taylor　expansion　algorithm;　while　the　optimum　nonlinear　Galerkin　method　can　be　viewed　as　the　2-order　Taylor　expansion　algorithm.　The　general　algorithm　is　then　applied　to　the　study　of　the　numerical　approximations　for　the　steady　Navier-Stokes　equations.　Finally,　the　theoretical　analysis　and　numerical　experiments　show　that,　in　some　situations,　the　optimum　nonlinear　Galerkin　method　provides　higher　convergence　rate　than　the　standard　Galerkin　method　and　the　nonlinear　Galerkin　method.