A Bernstein Roots–Based Numerical Framework for Solving Second-Order Eigenvalue Problems with Enhanced Stability

Abstract

Second-order eigenvalue problems are the foundation of engineering and applied sciences. In this paper, we present a Bernstein root–based method that combines Bernstein polynomial expansion, operational derivative matrices, and root-based collocation to produce a well-conditioned generalized eigenvalue problem. Analytical matrix formulations of the stiffness and mass matrices are derived, leading to enhanced stability and accuracy (errors ≈ ). The approach is a fine compromise between simplicity, efficiency, and robustness and is a competitive alternative to conventional solvers.