Computing a compact local Smith McMillan form

We define a compact local Smith-McMillan form of a rational matrix $R(\lambda)$ as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of $R(\lambda)$. We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of $R(\lambda)$ around an arbitrary complex point $\lambda_0$, allows us to compute a compact local Smith-McMillan form of that rational matrix $R(\lambda)$ at the point $\lambda_0$, 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 $\lambda_0$. Numerical tests illustrate the promising performance of the resulting algorithm.