Computation of the nearest non-prime polynomial matrix: Structured low-rank approximation approach

In this paper we consider the problem of computing the nearest non-prime polynomial matrix to a given left prime polynomial matrix. This problem is a generalization of the problem of computing the nearest non-coprime polynomials to several given coprime polynomials. In order to solve the problem, we first prove the equivalence of the left primeness property of a polynomial matrix with rank deficiency of a certain block Toeplitz matrix constructed from the given polynomial matrix. Using this equivalence, we reformulate the problem as the problem of computing the nearest Structured Low-Rank Approximation (SLRA) of the associated block Toeplitz matrix. This reformulation has been carried out when the leading coefficient matrix of the left prime polynomial matrix is of full row rank. We also prove that a non-prime polynomial matrix can be obtained arbitrarily close to a left prime polynomial matrix when the leading coefficient matrix is rank deficient. We implement numerical techniques such as STLN and regularized factorization approach, available in literature, to solve the SLRA problem. Furthermore, we demonstrate our proposed algorithm with several numerical examples along with the comparison with the results in literature wherever available.