Separable Type Representations of Matrices and Fast Algorithms : Volume 2 Eigenvalue Method 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. This two volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The primary focus is on fast algorithms (many of linear complexity) for matrices in semiseparable quasiseparable band and companion form. The work examines algorithms of multiplication inversion and description of eigenstructure and includes a wealth 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 for computing 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 also being derived for matrices with quasiseparable representation of order one. The QR iteration method for some classes of matrices with quasiseparable representations of any order is studied in the third part. This method is then used in the last part in order to provide 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 accessible style the text will be a valuable resource for engineers scientists numerical analysts computer scientists and mathematicians alike