  We extend the matrix-resolvent method of computing logarithmic derivatives of
tau-functions to the nonlinear Schr\"odinger (NLS) hierarchy. Based on this
method we give a detailed proof of a theorem of Carlet, Dubrovin and Zhang
regarding the relationship between the Toda lattice hierarchy and the NLS
hierarchy. As an application, we give an improvement of an algorithm of
computing correlators in hermitian matrix models.
