Computing a compact local Smith McMillan form

03/18/2023

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by   Vanni Noferini, et al.

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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 λ_0, allows us to compute a compact local Smith-McMillan form of that rational matrix R(λ) at the point λ_0, 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 λ_0. Numerical tests illustrate the promising performance of the resulting algorithm.