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.

Part 5. The eigenvalue structure of order one quasiseparable matrices.- 21. Quasiseparable of order one matrices. Characteristic polynomials.- 22. Eigenvalues with geometric multiplicity one.- 23. Kernels of quasiseparable of order one matrices.- 24. Multiple eigenvalues.- Part 6. Divide and conquer method for eigenproblems.- 25. Divide step.- 26. Conquer step and rational matrix functions eigenproblem.- 27. Complete algorithm for Hermitian matrices.- 28. Complete algorithm for unitary Hessenberg matrices.- Part 7. Algorithms for qr iterations and for reduction to Hessenberg form.- 29. The QR iteration method for eigenvalues.- 30. The reduction to Hessenberg form.- 31. The implicit QR iteration method for eigenvalues of upper Hessenberg matrices.- Part 8. QR iterations for companion matrices.- 32. Companion and unitary matrices.- 33. Explicit methods.- 34. Implicit methods with compression.- 35. The factorization based implicit method.- 36. Implicit algorithms based on the QR representation.- Bibliography.