  The nuclear matrix element (NME) of the neutrinoless double-$\beta$
($0\nu\beta\beta$) decay is an essential input for determining the neutrino
effective mass, if the half-life of this decay is measured. The reliable
calculation of this NME has been a long-standing problem because of the
diversity of the predicted values of the NME depending on the calculation
method. In this paper, we focus on the shell model and the QRPA. The shell
model have a rich amount of the many-particle many-hole correlations, and the
QRPA can obtain the convergence of the result of calculation with respect to
the extension of the single-particle space. It is difficult for the shell model
to obtain the convergence of the $0\nu\beta\beta$ NME with respect to the
valence single-particle space. The many-body correlations of the QRPA are
insufficient depending on nuclei. We propose a new method to modify
phenomenologically the results of the shell model and the QRPA compensating the
insufficient point of each method by using the information of other method
complementarily. Extrapolations of the components of the $0\nu\beta\beta$ NME
of the shell model are made toward a very large valence single-particle space.
We introduce a modification factor to the components of the $0\nu\beta\beta$
NME of the QRPA. Our modification method gives similar values of the
$0\nu\beta\beta$ NME of the two methods for $^{48}$Ca. The discrepancy of the
shell model and the QRPA for the NME of the two-neutrino double-$\beta$ decay
is much smaller than that for the $0\nu\beta\beta$ NME without major
modifications.
