Accurate spectral computations and error analysis for r-geometric Min and Max matrices

In this work, we introduce a new two-parameter family of structured matrices, termed r -geometric Min and Max matrices, which generalize both the r -Min/ r -Max and geometric Min/Max matrices. We derive explicit bidiagonal factorizations for these matrices using Neville elimination and establish necessary and sufficient conditions under which they are totally positive. Under these conditions, we develop algorithms capable of computing their eigenvalues and singular values to high relative accuracy, as well as closed-form expressions for their determinants. We also apply perturbation theory to analyze the sensitivity of these problems to input data, deriving structured condition numbers that quantify the impact of data perturbations. Numerical experiments confirm the theoretical results and demonstrate the reliability and efficiency of the proposed algorithms across both the general r -geometric case and key special instances.