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

Abstract

This paper addresses the problem of computing an exact Greatest Common Right Divisor (GCRD) of several univariate polynomial matrices $B_1\left(s\right),\dots ,B_t\left(s\right)$. We construct a polynomial matrix $P\left(s\right)$ by stacking $B_1\left(s\right),\dots ,B_t\left(s\right)$, one below the other. This results in $P\left(s\right)$ being wide, square or tall, each examined individually. We further prove the equivalence of rank deficiency of a particular generalized Sylvester matrix associated with $P\left(s\right)$ to the degree of the determinant of a GCRD of $B_1\left(s\right),\dots ,B_t\left(s\right)$ when $P\left(s\right)$ 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.