Accurate algebra and error analysis for geometric r-Frank matrices

Khiar, Yasmina (Universidad de Zaragoza) ; Mainar, Esmeralda (Universidad de Zaragoza) ; Royo-Amondarain, Eduardo (Universidad de Zaragoza)

Resumen: Computational methods that guarantee accurate solutions to linear algebra problems are of great interest in many applied contexts. These scenarios often involve particular matrix families that can benefit from a tailored analysis. In this work, we study a recently introduced class of structured matrices termed geometric r-Frank matrices, which are a one-parameter generalization of their classical version. Explicit bidiagonal factorizations for these matrices are derived, providing necessary and sufficient conditions for their total positivity. As a consequence, all eigenvalues and singular values can be determined with excellent relative accuracy under mild assumptions. Furthermore, we carry out a perturbation analysis for the bidiagonal factors and the determinants, establishing structured condition numbers that depend on the relative gaps of the underlying data. In addition, we develop efficient algorithms to compute the determinant of geometric r-Frank matrices together with running absolute and relative error bounds. Numerical experiments demonstrate the effectiveness and reliability of the proposed methods, even under challenging conditions.

Idioma: Inglés

DOI: 10.1080/03081087.2026.2646939

Año: 2026

Publicado en: Linear and Multilinear Algebra 74, 6 (2026), 757-778
ISSN: 0308-1087

Tipo y forma: Article (PostPrint)

Área (Departamento): Área Matemática Aplicada (Dpto. Matemática Aplicada)