PC-PK1 gives good description for the saturation properties of nuclear matter. In particular, the predicted values for binding energy per nucleon and density at the saturation point are − 16 . 12 MeV and 0 . 154 fm − $^{3}$, which agree well with the empirical values − 16 ± 1 MeV and 0 . 166 ± 0 . 018 fm − 3 [34], respectively. Moreover, the incompressibility given by PC-PK1 is 238 MeV. For the effective masses, all the effective interactions give reasonable values between 0 . 55 and 0 . 60 for the Dirac mass M ∗ $_{D}$/M [35] as required by the spin-orbit splitting data in finite nuclei, but smaller Landau masses M ∗ $_{L}$/M compared with the empirical constraint 0 . 8 ± 0 . 1 [36], implying that they would give a small single-particle level density at the Fermi energy in finite nuclei as compared with data. The symmetry energies in the calculations with the nonlinear effective interactions PC- F1, PC-LA, NL3* and PK1 are always larger than the empirical value (around 32 MeV) by around 16-18%, which is reduced to 11% for PC-PK1. Therefore all the interactions would predict large neutron skin thicknesses in finite nuclei except DD-PC1, which is adjusted by fixing E$_{sym}$ = 33 MeV. Moreover, the empirical L (88 ± 25 MeV) [33] and K$_{asy}$ ( − 550 ± 100 MeV) [37] have been reproduced quite well by PC-PK1. In Fig. 1, the binding energy per nucleon E/A for nuclear matter as a function of the baryon density ρ$_{B}$ given by PC-PK1 is shown in comparison with those by DD-PC1, PC-F1, PC-LA, NL3*, and PK1. All the effective interactions predict the similar E/A behavior with density below ρ$_{B}$ = 0 . 20 fm − 3 due to the constraints from the properties of the finite nuclei. Divergence appears at supra-saturation densities, especially for the results given by PC-LA. It implies that the properties of finite nuclei are not sufficient in the determination of EDF to describe the EOS at supra-saturation densities that are directly related to the maximal mass of neutron star. The prediction by PC-PK1 is consistent with those by PK1 and PC-F1, while softer than those by NL3*. The ab-initio variational calculation for the symmetric nuclear matter [38] is also given for comparison, which coincides with the relativistic EOS with density below ρ$_{B}$ = 0 . 20 fm − 3 but predicts softer EOS behavior at supra-saturation densities than the relativistic ones, except those given by PC-LA and DD-PC1. One should note that DD-PC1 has been adjusted to the EOS given by ab-initio B. Equation of state 11