
Matrix Computations and Semiseparable Matrices : Linear Systems.
Title:
Matrix Computations and Semiseparable Matrices : Linear Systems.
Author:
Vandebril, Raf.
ISBN:
9780801896798
Personal Author:
Physical Description:
1 online resource (594 pages)
Contents:
Contents -- Preface -- Notation -- I: Introduction to semiseparable and related matrices -- 1 Semiseparable and related matrices: definitions and properties -- 1.1 Symmetric semiseparable and related matrices -- 1.2 Relations between the different symmetric definitions -- 1.3 Unsymmetric semiseparable and related matrices -- 1.4 Relations between the different "unsymmetric" definitions -- 1.5 Relations under inversion -- 1.6 Conclusions -- 2 The representation of semiseparable and related matrices -- 2.1 Representations -- 2.2 The symmetric generator representation -- 2.3 The symmetric diagonal-subdiagonal representation -- 2.4 The symmetric Givens-vector representation -- 2.5 The symmetric quasiseparable representation -- 2.6 Some examples -- 2.7 The unsymmetric generator representation -- 2.8 The unsymmetric Givens-vector representation -- 2.9 The unsymmetric quasiseparable representation -- 2.10 The decoupled representation for semiseparable matrices -- 2.11 Summary of the representations -- 2.12 Are there more representations? -- 2.13 Some algorithms related to representations -- 2.14 Conclusions -- 3 Historical applications and other topics -- 3.1 Oscillation matrices -- 3.2 Semiseparable matrices as covariance matrices -- 3.3 Discretization of integral equations -- 3.4 Orthogonal rational functions -- 3.5 Some comments -- 3.6 Conclusions -- II: Linear systems with semiseparable and related matrices -- 4 Gaussian elimination -- 4.1 About Gaussian elimination and the LU-factorization -- 4.2 Backward substitution -- 4.3 Inversion of triangular semiseparable matrices -- 4.4 Theoretical considerations of the LU-decomposition -- 4.5 The LU-decomposition for semiseparable matrices -- 4.6 The LU-decomposition for quasiseparable matrices -- 4.7 Some comments -- 4.8 Conclusions -- 5 The QR-factorization -- 5.1 About the QR-decomposition.
5.2 Theoretical considerations of the QR-decomposition -- 5.3 A QR-factorization of semiseparable matrices -- 5.4 A QR-factorization of quasiseparable matrices -- 5.5 Implementing the QR-factorization -- 5.6 Other decompositions -- 5.7 Conclusions -- 6 A Levinson-like and Schur-like solver -- 6.1 About the Levinson algorithm -- 6.2 Generator representable semiseparable plus diagonal matrices -- 6.3 A Levinson framework -- 6.4 Examples -- 6.5 The Schur algorithm -- 6.6 Conclusions -- 7 Inverting semiseparable and related matrices -- 7.1 Known factorizations -- 7.2 Direct inversion methods -- 7.3 General formulas for inversion -- 7.4 Scaling of symmetric positive definite semiseparable matrices -- 7.5 Decay rates for the inverses of tridiagonal matrices -- 7.6 Conclusions -- III: Structured rank matrices -- 8 Definitions of higher order semiseparable matrices -- 8.1 Structured rank matrices -- 8.2 Definition of higher order semiseparable and related matrices -- 8.3 Inverses of structured rank matrices -- 8.4 Generator representable semiseparable matrices -- 8.5 Representations -- 8.6 Conclusions -- 9 A QR-factorization for structured rank matrices -- 9.1 A sequence of Givens transformations from bottom to top -- 9.2 Making the structured rank matrix upper triangular -- 9.3 Different patterns of annihilation -- 9.4 Rank-expanding sequences of Givens transformations -- 9.5 QR-factorization for the Givens-vector representation -- 9.6 Extra material -- 9.7 Multiplication between structured rank matrices -- 9.8 Conclusions -- 10 A Gauss solver for higher order structured rank systems -- 10.1 A sequence of Gauss transformation matrices without pivoting -- 10.2 Effect of pivoting on rank structures -- 10.3 More on sequences of Gauss transforms -- 10.4 Solving systems with Gauss transforms -- 10.5 Different patterns of annihilation -- 10.6 Conclusions.
11 A Levinson-like solver for structured rank matrices -- 11.1 Higher order generator representable semiseparable matrices -- 11.2 General quasiseparable matrices -- 11.3 Band matrices -- 11.4 Unsymmetric structures -- 11.5 Summations of Levinson-conform matrices -- 11.6 Conclusions -- 12 Block quasiseparable matrices -- 12.1 Definition -- 12.2 Factorization of the block lower/upper triangular part -- 12.3 Connection to structured rank matrices -- 12.4 Special cases -- 12.5 Multiplication of a block quasiseparable matrix by a vector -- 12.6 Solver for block quasiseparable systems -- 12.7 Block quasiseparable matrices and descriptor systems -- 12.8 Conclusions -- 13 H, H[sup(2)] and hierarchically semiseparable matrices -- 13.1 H-matrices or hierarchical matrices -- 13.2 H[sup(2)]-matrices -- 13.3 Hierarchically semiseparable matrices -- 13.4 Other classes of structured rank matrices -- 13.5 Conclusions -- 14 Inversion of structured rank matrices -- 14.1 Banded Toeplitz matrices -- 14.2 Inversion of (generalized) Hessenberg matrices -- 14.3 Inversion of higher order semiseparable and band matrices -- 14.4 Block matrices -- 14.5 Quasiseparable matrices -- 14.6 Generalized inverses -- 14.7 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 -- R -- S -- T -- U -- V -- W -- Y -- Z -- Subject Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Y -- Z.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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