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In this paper, a frequency-domain harmonic balance method is developed for solving the Maxwell-Bloch equations or simplified versions thereof for semiconductor lasers. The numerical algorithms are formulated to track stationary, periodic, and quasi-periodic steady-state behavior, including passive-mode-locking operation. The nonlinear set of equations with the Fourier coefficients and the fundamental angular frequency as unknowns is solved by a Newton method. Special focus is put on the derivation of a comprehensive bifurcation analysis tool including homotopy, continuation, stability analysis, bifurcation detection, and methods to handle branch switching. As an illustrative example of the method, nonlinear multimode characteristics of an edge-emitting laser are presented.