  In this paper, we study the pseudo-Wigner solution of the quark gap equation
with a recently proposed algorithm in the framework of the (2+1)-flavor
Nambu-Jona-Lasinio (NJL) model. We find that for the current quark mass $m_{\rm
u,d}=5.5$ MeV and chemical potential $\mu<\mu_{\rm TCP}=272.5$ MeV, the Nambu
solution and the positive pseudo-Wigner solution obtained via this algorithm is
consistent with the physical solution obtained with the iterative method.
Furthermore, the algorithm we used can help to illustrate the evolution of the
solutions of the gap equation from the chiral limit to non-chiral limit and
gives a prediction where the crossover line is located in the phase diagram for
$\mu<272.5$ MeV. In addition, we also study the chiral susceptibilities as well
as the loss of solutions for different chemical potentials.
