Compression of unitary rank-structured matrices to CMV-like shape with an application to polynomial rootfinding

This paper is concerned with the reduction of a unitary matrix U to CMV-like shape. A Lanczos–type algorithm is presented which carries out the reduction by computing the block tridiagonal form of the Hermitian part of U , i.e., of the matrix U +UH . By elaborating on the Lanczos approach we also propose an alternative algorithm using elementary matrices which is numerically stable. If U is rank–structured then the same property holds for its Hermitian part and, therefore, the block tridiagonalization process can be performed using the rank–structured matrix technology with reduced complexity. Our interest in the CMV-like reduction is motivated by the unitary and almost unitary eigenvalue problem. In this respect, finally, we discuss the application of the CMV-like reduction for the design of fast companion eigensolvers based on the customary QR iteration.