A constructive proof of the Gohberg-Semencul formula

AbstractGohberg and Semencul gave some elegant formulas for the inverse of a Toeplitz matrix as a difference of products of lower and upper triangular Toeplitz matrices. There are several algebraic and analytic proof of these formulas. Here we give a “constructive” proof for two of the Gohberg-Semencul formulas, under the assumption that the matrices are strongly nonsingular, i.e., all leading minors are nonzero. This assumption is stronger than necessary, but it enables fast O(n2) constructions for the entries in the Gohberg-Semencul formulas. Our method also gives a new proof of the relation between the reflection coefficients of a Toeplitz matrix and its inverse