Computation of an exact GCRD of several polynomial matrices: QR decomposition approach

• Published in: Linear algebra and its applications Vol. 710; pp. 471 - 506
• Main Authors: Beniwal, Anjali, Saha, Tanay, Khare, Swanand R.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.04.2025
• Subjects: Effectively eliminating QR decomposition; GCRD of polynomial matrices; Generalized Sylvester matrices; Index Sum theorem; Right prime polynomial matrix; 15A23; 15A54; 15B05; 65F25; Index Sum theorem; Right prime polynomial matrix; Effectively eliminating QR decomposition; Generalized Sylvester matrices; GCRD of polynomial matrices
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2025.02.001

This paper addresses the problem of computing an exact Greatest Common Right Divisor (GCRD) of several univariate polynomial matrices B1(s),…,Bt(s). We construct a polynomial matrix P(s) by stacking B1(s),…,Bt(s), one below the other. This results in P(s) being wide, square or tall, each examined individually. We further prove the equivalence of rank deficiency of a particular generalized Sylvester matrix associated with P(s) to the degree of the determinant of a GCRD of B1(s),…,Bt(s) when P(s) is a tall matrix with full normal rank. This equivalence enables us to propose a method to extract a GCRD based on the ‘effectively eliminating’ QR (EEQR) decomposition of that generalized Sylvester matrix. We also propose a computationally efficient algorithm to extract the exact GCRD. To validate the theoretical findings, we provide several numerical examples.