For　a　class　of　nonlinear　least　squares　problems,　it　is　usually　very　beneficial　to　separate　the　variables　into　a　linear　and　a　nonlinear　part　and　take　full　advantage　of　reliable　linear　least　squares　techniques.　Consequently,　the　original　problem　is　turned　into　a　reduced　problem　which　involves　only　nonlinear　parameters.　We　consider　in　this　paper　four　separated　algorithms　for　such　problems.　The　first　one　is　the　variable　projection　(VP)　algorithm　with　full　Jacobian　matrix　of　Golub　and　Pereyra.　The　second　and　third　ones　are　VP　algorithms　with　simplified　Jacobian　matrices　proposed　by　Kaufman　and　Ruano　et　al.　respectively.　The　fourth　one　only　uses　the　gradient　of　the　reduced　problem.　Monte　Carlo　experiments　are　conducted　to　compare　the　performance　of　these　four　algorithms.　From　the　results　of　the　experiments,　we　find　that:　1)　the　simplified　Jacobian　proposed　by　Ruano　et　al.　is　not　a　good　choice　for　the　VP　algorithm;　moreover,　it　may　render　the　algorithm　hard　to　converge;　2)　the　fourth　algorithm　perform　moderately　among　these　four　algorithms;　3)　the　VP　algorithm　with　the　full　Jacobian　matrix　perform　more　stable　than　that　of　the　VP　algorithm　with　Kuafman＇s　simplified　one;　and　4)　the　combination　of　VP　algorithm　and　Levenberg-Marquardt　method　is　more　effective　than　the　combination　of　VP　algorithm　and　Gauss-Newton　method.