The second volume, consisting of four parts, addresses the eigenvalue problem for matrices with quasiseparable structure and applications to the polynomial root finding problem. In the first part the properties of the characteristic polynomials of principal leading submatrices, the structure of eigenspaces and the basic methods to compute eigenvalues are studied in detail for matrices with quasiseparable representation of the first order. The second part is devoted to the divide and conquer method, with the main algorithms being derived also for matrices with quasiseparable representation of order one. The QR iteration method for some classes of matrices with quasiseparable of any order representations is studied in the third part. This method is then used in the last part in order to get a fast solver for the polynomial root finding problem. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.
The book begins with his biography and list of publications. It contains a selection of research papers, most of which are devoted to spectral analysis of operators or operator pencils with applications to ordinary and partial differential equations. Other papers deal with time-varying systems, interpolation and factorization problems, and topics from mathematical physics. About half of the papers contain further developments in the theory of operators in spaces with an indefinite metric and treat new applications. The book is of interest to a wide audience of pure and applied mathematicians.