  We provide a systematic and self-consistent method to calculate the
generalized Brillouin Zone (GBZ) analytically in one dimensional non-Hermitian
systems, which helps us to understand the non-Hermitian bulk-boundary
correspondence. In general, a n-band non-Hermitian Hamiltonian is constituted
by n distinct sub-GBZs, each of which is a piecewise analytic closed loop.
Based on the concept of resultant, we can show that all the analytic properties
of the GBZ can be characterized by an algebraic equation, the solution of which
in the complex plane is dubbed as auxiliary GBZ (aGBZ). We also provide a
systematic method to obtain the GBZ from aGBZ. Two physical applications are
also discussed. Our method provides an analytic approach to the spectral
problem of open boundary non-Hermitian systems in the thermodynamic limit.
