A multiple shift QR-step for structured rank matrices

Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, QR-algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored. A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations. This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift QR-step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices. Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach.