A note on the structured perturbation analysis for the inversion formula of Toeplitz matrices

The invertibility of a Toeplitz matrix can be assessed based on the solvability of two standard equations. The inverse of the nonsingular Toeplitz matrix can then be represented as the sum of products of circulant and skew-circulant (CS) matrices. In this note, we provide a new structured perturbation analysis for the CS representation of Toeplitz inversion and the new upper bound is just half as large as the existing upper bound proposed by Wu et al. (Numer Linear Algebra Appl 22(4):777–792, 2015) and Feng et al. (East Asian J Appl Math 5(2):160–175, 2015). Meanwhile, some practical issues and numerical experiments involving the numerical solutions of fractional partial differential equations are reported to support our theoretical findings.