Efficient computation of dispersion curves in damped and anisotropic waveguides often requires solving multivariate transcendental equations in a complex-domain parameter space, where the physically admissible solution set occupies only a small fraction of the search volume. This paper presents a multivariate complex-domain root-finding algorithm that combines Shift-Differencing Screening (SDS), occupancy-driven mesh refinement, and localized Modulus-Ratio Convergence Method (MRCM) confirmation. SDS is introduced as a necessary-stage screening paradigm that detects candidate root neighborhoods through shift-based comparisons on precomputed determinant values, thereby reducing dependence on fixed-axis slicing and suppressing pseudo-candidates generated by cellwise scans. The same screening mechanism drives adaptive mesh refinement so that determinant evaluations are concentrated in the small occupied fraction of the domain rather than the full Cartesian grid. MRCM is retained as a sufficient local confirmation stage and applied only to the refined candidates, preserving direct root verification while decoupling it from expensive domain-wide scanning. Relative to conventional UCS--MRCM implementations, the resulting framework reduces determinant evaluations by one to two orders of magnitude while preserving finest-grid-consistent dispersion branches. Numerical results for non-dimensional Lamb waves and a fully triclinic hysteretic plate show improved candidate localization, suppression of valley-type and plateau-type pseudo-roots, and substantial runtime reductions. The formulation is broadly applicable to multivariate complex-domain root-finding problems whenever an analytical or semi-analytical residual is available.
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