Inversion of Displacement Operators

We recall briefly the displacement rank approach to the computations with structured matrices, which we trace back to the seminal paper by Kailath, Kung, and Morf [J. Math. Anal. Appl., 68 (1979), pp. 395–407]. The concluding stage of the computations is the recovery of the output from its compressed representation via the associated displacement operator L. The recovery amounts to the inversion of the operator. That is, one must express a structured matrix M via its image L(M). We show a general method for obtaining such expressions that works for all displacement operators (under only the mildest nonsingularity assumptions) and thus provides the foundation for the displacement rank approach to practical computations with structured matrices. We also apply our techniques to specify the expressions for various important classes of matrices. Besides unified derivation of several known formulae, we obtain some new ones, in particular, for the matrices associated with the tangential Nevanlinna–Pick problems. This enables acceleration of the known solution algorithms. We show several new matrix representations of the problem in the important confluent case. Finally, we substantially improve the known estimates for the norms of the inverse displacement operators, which are critical numerical parameters for computations based on the displacement approach.