High relative accuracy with some special matrices related to Γ and β functions

Abstract

For some families of totally positive matrices using Γ$$ \Gamma $$ and β$$ \beta $$ functions, we provide their bidiagonal factorization. Moreover, when these functions are defined over integers, we prove that the bidiagonal factorization can be computed with high relative accuracy and so we can compute with high relative accuracy their eigenvalues, singular values, inverses and the solutions of some associated linear systems. We provide numerical examples illustrating this high relative accuracy.