Zeros of para-orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para-orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle d mu and para-orthogonal polynomials associated with a modification of d mu by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.The authors thank the referees for their comments and suggestions. This work is partially supported by the CMUC, funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the Fundacao para a Ciência e a Tecnologia (FCT) under the project PEst-C/MAT/UI0324/2013. The research of the first author is supported by the Portuguese Government through the FCT under the grant SFRH/BPD/101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The second author also acknowledges the financial support by the Brazilian Government through the CAPES under the project 107/2012

Structured Eigenvalue Problems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices