A two-way coupled-mode model has been developed to solve propagation problems in fluid-layered, range-dependent ocean environments. The model incorporates piecewise-linear depth-dependent, range-varying sound-speed profiles (SVPs). A fast and efficient normal-mode model which is based on a propagator matrix method is used to obtain the eigenvalues and eigenfunctions. The coupling coefficients are expressed in terms of depth integrals of products of Airy functions and are solved analytically. The coupled equations are solved exactly by an approach referred to as the U--K method, which incorporates the Lanczos method, previously used in solving similar coupled-channel equations in nuclear theory and successfully applied to nuclear response problems in heavy nuclei [Knobles, ``Solutions of coupled-mode equations with a large dimension in underwater acoustics,'' J. Acoust. Soc. Am. 96, 1741--1747 (1994)]. Highlights of the formalism will be presented which expand on the overview given previously [Stotts et al., ``Solutions of coupled-mode equations with a large dimension in underwater acoustics,'' J. Acoust. Soc. Am. 95, 2909(A) (1994)]. Several benchmark comparisons will be presented including propagation over a hill, upslope wedge propagation, and an analytic depth and range-dependent SVP in a flat waveguide.