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A computer program, based on a fast direct solver, to solve for the dominant mode in a waveguide of arbitrary refractive index profile within a rectangular box is discussed. The two-dimensional field problem is reduced to a conventional eigenvalue problem in terms of the transverse magnetic field components. The two coupled scalar equations are solved using the FISHPAK software package, which solves a scalar Helmholtz problem using finite differences with an M by N mesh in time proportional to MNlog2N. This fast solver is within two iteration loops. The outer is a Rayleigh quotient iteration, and the inner solves the two coupled equations. Results emerge typically in between 2 and 30 total calls of FISHPAK.