2023/03/18 by Vanni Noferini, Paul Van Dooren, Noferini, Vanni +1
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2303.10403
openalex publication_date 2023/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define a compact local Smith-McMillan form of a rational matrix R(λ) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of R(λ). We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of R(λ) around an arbitrary complex point λ0, allows us to compute a compact local Smith-McMillan form of that rational matrix R(λ) at the point λ0, provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point λ0. Numerical tests illustrate the promising performance of the resulting algorithm.