000170411 001__ 170411 000170411 005__ 20260420103355.0 000170411 0247_ $$2doi$$a10.1080/03081087.2026.2646939 000170411 0248_ $$2sideral$$a148924 000170411 037__ $$aART-2026-148924 000170411 041__ $$aeng 000170411 100__ $$0(orcid)0000-0002-6497-7158$$aKhiar, Yasmina$$uUniversidad de Zaragoza 000170411 245__ $$aAccurate algebra and error analysis for geometric r-Frank matrices 000170411 260__ $$c2026 000170411 5060_ $$aAccess copy available to the general public$$fUnrestricted 000170411 5203_ $$aComputational 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. 000170411 540__ $$9info:eu-repo/semantics/embargoedAccess$$aby-nc-nd$$uhttps://creativecommons.org/licenses/by-nc-nd/4.0/deed.es 000170411 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/acceptedVersion 000170411 700__ $$0(orcid)0000-0002-1101-6230$$aMainar, Esmeralda$$uUniversidad de Zaragoza 000170411 700__ $$0(orcid)0000-0003-1550-8168$$aRoyo-Amondarain, Eduardo$$uUniversidad de Zaragoza 000170411 7102_ $$12005$$2595$$aUniversidad de Zaragoza$$bDpto. Matemática Aplicada$$cÁrea Matemática Aplicada 000170411 773__ $$g74, 6 (2026), 757-778$$pLinear multilinear algebra$$tLinear and Multilinear Algebra$$x0308-1087 000170411 8564_ $$s294185$$uhttps://zaguan.unizar.es/record/170411/files/texto_completo.pdf$$yPostprint$$zinfo:eu-repo/date/embargoEnd/2027-03-22 000170411 8564_ $$s1214606$$uhttps://zaguan.unizar.es/record/170411/files/texto_completo.jpg?subformat=icon$$xicon$$yPostprint$$zinfo:eu-repo/date/embargoEnd/2027-03-22 000170411 909CO $$ooai:zaguan.unizar.es:170411$$particulos$$pdriver 000170411 951__ $$a2026-04-18-10:49:04 000170411 980__ $$aARTICLE