A High-Precision Chebyshev Collocation Framework for Sturm–Liouville Eigenproblems

Abstract

A high-precision spectral implementation framework is presented for computing eigenvalues of second- and fourth-order Sturm–Liouville problems on finite intervals. The framework employs the trigonometric substitution  to assemble all derivative operations on Chebyshev polynomials entry-by-entry from closed-form trigonometric identities, bypassing the repeated matrix multiplication that causes the standard Chebyshev differentiation matrix (CDM) approach to accumulate roundoff at large , high derivative order, and in clustered spectral regimes. By reformulating the differential problem as a generalised eigenvalue system solved via the QZ algorithm, the framework combines numerical reliability with high-order geometric convergence. Multiple boundary conditions at the same endpoint are incorporated through a node-reduction procedure; an empirical condition-number study is included to characterise its stability. All numerical experiments are conducted in MATLAB at 34-digit precision, enabling eigenvalue errors at the level of  and providing self-convergence verification independent of double-precision saturation. The framework is benchmarked across six problems including the Coffey–Evans equation and free vibration of exponentially functionally graded beams, demonstrating good accuracy beyond double-precision references.