Computing a compact local Smith–McMillan form

We define a compact local Smith–McMillan form of a rational matrix (Formula presented.) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of (Formula presented.). We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of (Formula presented.) around an arbitrary complex point (Formula presented.), allows us to compute a compact local Smith-McMillan form of that rational matrix (Formula presented.) at the point (Formula presented.), 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 (Formula presented.). Numerical tests illustrate the promising performance of the resulting algorithm.