Computing a compact local Smith–McMillan form (Q6665962)

The authors describe an algorithm to compute the compact local Smith-McMillan form of a rational complex matrix \(R(\lambda)\) as follows:\N\[\NR(\lambda) N_r(\lambda) = M_r(\lambda) \ \operatorname{diag} \left( (\lambda-\lambda_0)^{\sigma_1}, \ldots,(\lambda-\lambda_0)^{\sigma_r} \right),\N\]\Nwhere \(\sigma_1, \ldots, \sigma_r\) are the structural indices of \(R(\lambda)\) at \(\lambda_0\), \(N_r(\lambda)\) has a polynomial left inverse and \(M_r(\lambda)\) is left invertible at \(\lambda_0\), and\N\[\NM_l(\lambda) R(\lambda) = \ \operatorname{diag} \left( (\lambda-\lambda_0)^{\sigma_1}, \ldots,(\lambda-\lambda_0)^{\sigma_r} \right) N_l(\lambda),\N\]\N\(M_l(\lambda)\) has a polynomial right inverse and \(N_l(\lambda)\)is right invertible at \(\lambda_0\).\N\NThe authors give numerical results for the computation of the compact local Smith.