This two-volume work presents a systematic theoretical and computational
study of several types of generalizations of separable matrices. The
main attention is paid to fast algorithms (many of linear complexity)
for matrices in semiseparable, quasiseparable, band and companion form.
The work is focused on algorithms of multiplication, inversion and
description of eigenstructure and includes a large number of
illustrative examples throughout the different chapters.
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.
