Contents -- Preface -- Notation -- 1 Introduction to semiseparable matrices -- 1.1 Definition of semiseparable matrices -- 1.2 Some properties -- 1.2.1 Relations under inversion -- 1.2.2 Generator representable semiseparable matrices -- 1.3 The representations -- 1.3.1 The generator representation -- 1.3.2 The Givens-vector representation -- 1.4 Conclusions -- I: The reduction of matrices -- 2 Algorithms for reducing matrices -- 2.1 Introduction -- 2.2 Orthogonal similarity transformations of symmetric matrices -- 2.3 Orthogonal similarity transformation of (unsymmetric) matrices -- 2.4 Orthogonal transformations of matrices -- 2.5 Transformations from sparse to structured rank form -- 2.6 From structured rank to sparse form -- 2.7 Conclusions -- 3 Convergence properties of the reduction algorithms -- 3.1 The Arnoldi(Lanczos)-Ritz values -- 3.2 Subspace iteration inside the reduction algorithms -- 3.3 Interaction of the convergence behaviors -- 3.4 Conclusions -- 4 Implementation of the algorithms and numerical experiments -- 4.1 Working with Givens transformations -- 4.2 Implementation details -- 4.3 Numerical experiments -- 4.4 Conclusions -- II: QR-algorithms (eigenvalue problems) -- 5 Introduction: traditional sparse QR-algorithms -- 5.1 On the QR-algorithm -- 5.2 A QR-algorithm for sparse matrices -- 5.3 An implicit QR-method for sparse matrices -- 5.4 On computing the singular values -- 5.5 Conclusions -- 6 Theoretical results for structured rank QR-algorithms -- 6.1 Preserving the structure under a QR-step -- 6.2 An implicit Q-theorem -- 6.3 On Hessenberg-like plus diagonal matrices -- 6.4 Conclusions -- 7 Implicit QR-methods for semiseparable matrices -- 7.1 An implicit QR-algorithm for symmetric semiseparable matrices -- 7.2 A QR-algorithm for semiseparable plus diagonal -- 7.3 An implicit QR-algorithm for Hessenberg-like matrices.

7.4 An implicit QR-algorithm for computing the singular values -- 7.5 Conclusions -- 8 Implementation and numerical experiments -- 8.1 Working with Givens transformations -- 8.2 Implementation of the QR-algorithm for semiseparable matrices -- 8.3 Computing the eigenvectors -- 8.4 Numerical experiments -- 8.5 Conclusions -- 9 More on QR-related algorithms -- 9.1 Complex arithmetic and Givens transformations -- 9.2 Variations of the QR-algorithm -- 9.3 The QR-method for quasiseparable matrices -- 9.4 The multishift QR-algorithm -- 9.5 A QH-algorithm -- 9.6 Computing zeros of polynomials -- 9.7 References to related subjects -- 9.8 Conclusions -- III: Some generalizations and miscellaneous topics -- 10 Divide-and-conquer algorithms for the eigendecomposition -- 10.1 Arrowhead and diagonal plus rank 1 matrices -- 10.2 Divide-and-conquer algorithms for tridiagonal matrices -- 10.3 Divide-and-conquer methods for quasiseparable matrices -- 10.4 Computational complexity and numerical experiments -- 10.5 Conclusions -- 11 A Lanczos-type algorithm and rank revealing -- 11.1 Lanczos semiseparabilization -- 11.2 Rank-revealing properties of the orthogonal similarity reduction -- 11.3 Conclusions -- IV: Orthogonal (rational) functions (Inverse eigenvalue problems) -- 12 Orthogonal polynomials and discrete least squares -- 12.1 Recurrence relation and Hessenberg matrix -- 12.2 Discrete inner product -- 12.3 Inverse eigenvalue problem -- 12.4 Polynomial least squares approximation -- 12.5 Updating algorithm -- 12.6 Special cases -- 12.7 Conclusions -- 13 Orthonormal polynomial vectors -- 13.1 Vector approximants -- 13.2 Equal degrees -- 13.3 Arbitrary degrees -- 13.4 The singular case -- 13.5 Conclusions -- 14 Orthogonal rational functions -- 14.1 The computation of orthonormal rational functions -- 14.2 Solving the inverse eigenvalue problem.

14.3 Special configurations of points zi -- 14.4 Conclusions -- 15 Concluding remarks & software -- 15.1 Software -- 15.2 Conclusions -- Bibliography -- Author/Editor Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- V -- W -- X -- Y -- Z -- Subject Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.