Treffer: Inversion of Displacement Operators: Inversion of displacement operators

It is known that an \(n\times n\) structured matrix \(M\) can be associated with an appropriate displacement operator \(L\) such that the rank \(r\) of \(L(M)\) satisfies \(r\ll n\) and the \(n^2\) entries of \(L(M)\) can be represented via only \(2rn\) parameters. The authors present a general method to express \(M\) via \(L(M)\) under very mild nonsingularity assumptions. The method unifies the derivation of known formulae and gives new formulae, in particular for the tangential Nevalinna-Pick problems. It accelerates known solution algorithms. The authors obtain general new matrix representations in the confluent case and substantially improve the known estimates for the norm \(\| L^{-1}\| \) which is critical in computations based on the displacement approach.