Matrix Computations and Semiseparable Matrices

Transcription

1 Matrix Computations and Semiseparable Matrices Volume I: Linear Systems Raf Vandebril Department of Computer Science Catholic University of Louvain Marc Van Barel Department of Computer Science Catholic University of Louvain Nicola Mastronardi M. Picone Institute for Applied Mathematics, Bari The Johns Hopkins University Press Baltimore

2 Preface Notation xiii xvii I Introduction to semiseparable and related matrices 1 1 Semiseparable and related matrices: definitions and properties Symmetric semiseparable and related matrices Relations between the different symmetric definitions Known relations Common misunderstandings about generator representable semiseparable matrices A theoretical problem with numerical consequences Generator representable semiseparable and semiseparable matrices Semiseparable plus diagonal and quasiseparable matrices Summary of the relations More on the pointwise convergence Unsymmetric semiseparable and related matrices Relations between the different "unsymmetric" definitions Extension of known symmetric relations Quasiseparable matrices having a symmetric rank structure Summary of the relations Relations under inversion The nullity theorem The inverse of semiseparable and tridiagonal matrices The inverse of a quasiseparable matrix The inverse of a semiseparable plus diagonal matrix Summary of the relations Conclusions 50

3 The representation of semiseparable and related matrices Representations The definition of a representation A representation is "just" a representation Covered representations The symmetric generator representation The representation for symmetric semiseparables Application to other classes of matrices The symmetric diagonal-subdiagonal representation The representation for symmetric semiseparables The representation for symmetric quasiseparables The symmetric Givens-vector representation The representation for symmetric semiseparables Examples Retrieving the Givens-vector representation Swapping the representation Application to other classes of matrices The symmetric quasiseparable representation The representation for symmetric semiseparables Application to other classes of matrices Some examples The unsymmetric generator representation The representation for semiseparables Application to other classes of matrices The unsymmetric Givens-vector representation The representation for semiseparables Application to other classes of matrices The unsymmetric quasiseparable representation The representation for semiseparables Application to other classes of matrices The decoupled representation for semiseparable matrices The decoupled generator representation The decoupled Givens-vector representation Summary of the representations Are there more representations? Some algorithms related to representations A fast matrix vector multiplication Changing representations Computing the determinant of a semiseparable matrix in O{n) flops Conclusions 106 Historical applications and other topics Oscillation matrices Introduction Ill ' Definition and examples 113

4 3.1.3 The inverse of a one-pair matrix The example of the one-pair matrix Some other interesting applications The connection with eigenvalues and eigenvectors Semiseparable matrices as covariance matrices Covariance calculations The multinomial distribution Some other matrices Discretization of integral equations Orthogonal rational functions Some comments The name "semiseparable" matrix Eigenvalue problems References to applications Conclusions 141 II Linear systems with semiseparable and related matrices Gaussian elimination About Gaussian elimination and the LU-factorization Backward substitution Inversion of triangular semiseparable matrices The inverse of a bidiagonal The inverse of lower semiseparable matrices Examples Theoretical considerations of the LL/-decomposition The -L[/-decomposition for semiseparable matrices Strongly nonsingular matrices General semiseparable matrices The Lt/-decomposition for quasiseparable matrices A first naive factorization scheme The strongly nonsingular case, without pivoting The strongly nonsingular case Some comments Numerical stability A representation? Conclusions The QiMactorization About the Q-R-decomposition Theoretical considerations of the Q-R-decomposition A Qi?-factorization of semiseparable matrices The factorization using Givens transformations The generators of the factors The Givens-vector representation 191

5 viii Contents Solving systems of equations A Qi?-factorization of quasiseparable matrices Implementing the QR-factorization Other decompositions The [/^-decomposition Some other orthogonal decompositions Conclusions A Levinson-like and Schur-like solver About the Levinson algorithm The Yule-Walker problem and the Durbin algorithm The Levinson algorithm An upper triangular factorization of the inverse Generator representable semiseparable plus diagonal matrices The class of matrices A Yule-Walker-like problem A Levinson-type algorithm An upper triangular factorization of the inverse Some general remarks A Levinson framework The matrix block decomposition Simple {pi,p2}-levinson conform matrices An upper triangular factorization The look-ahead procedure Examples Givens-vector-represented semiseparable matrices Quasiseparable matrices Tridiagonal matrices Arrowhead matrices Unsymmetric structures Upper triangular matrices Dense matrices Summations of Levinson-conform matrices Matrices with errors in structures Companion matrices Comrade matrix Fellow matrices The Schur algorithm Basic concepts The Schur reduction The Schur complement of quasiseparable matrices A Schur-like algorithm for quasiseparable matrices A more general framework for the Schur reduction Conclusions Inverting semiseparable and related matrices 257

6 7.1 Known factorizations Inversion via the QR-factorization Inversion via the L /-factorization Inversion via the Levinson algorithm Direct inversion methods The inverse of a symmetric tridiagonal matrix The inverse of a symmetric semiseparable matrix The inverse of a tridiagonal matrix The inverse of a specific semiseparable matrix General formulas for inversion Scaling of symmetric positive definite semiseparable matrices Decay rates for the inverses of tridiagonal matrices M-matrices Decay rates for the inverse of diagonally dominant tridiagonal M-matrices Decay rates for the inverse of diagonally dominant tridiagonal matrices Conclusions 285 III Structured rank matrices Definitions of higher order semiseparable matrices Structured rank matrices Definition of higher order semiseparable and related matrices Some inner structured rank relations Semiseparable matrices Quasiseparable matrices Generator representable semiseparable matrices Extended semiseparable matrices Hessenberg-like matrices Sparse matrices What is in a name Inverses of structured rank matrices Inverse of structured rank matrices Some particular inverses Generator representable semiseparable matrices Generator representation When is a matrix generator representable? Decomposition of structured rank matrices Decomposition of semiseparable and related matrices Representations Givens-vector representation Quasiseparable representation Split representations Conclusions 345

7 A Q-R-factorization for structured rank matrices A sequence of Givens transformations from bottom to top Annihilating Givens transformations on lower rank 1 structures Arbitrary Givens transformations on lower rank 1 structures Givens transformations on lower rank structures Can the value of p in E p) be larger than 1? Givens transformations on upper rank 1 structures Givens transformations on upper rank structures Givens transformations on rank structures Other directions of sequences of Givens transformations Examples Summary Making the structured rank matrix upper triangular Annihilating completely the rank structure Expanding the zero rank structure Combination of ascending and descending sequences Examples Solving systems of equations Other decompositions Different patterns of annihilation The leaf form for removing the rank structure The pyramid form for removing the rank structure The leaf form for creating zeros The diamond form for creating zeros Theorems connected to Givens transformations The A-pattern The X-pattern The V-pattern Givens transformations in the V-pattern Rank-expanding sequences of Givens transformations The Givens transformation Rank-expanding Givens transformations on upper rank 1 structures Existence and the effect on upper rank structures Global theorem for sequences from bottom to top Q-R-factorization for the Givens-vector representation The Givens-vector representation Applying transformations on the Givens-vector representation Computing the rank-expanding Givens transformations Extra material A rank-expanding QR-factorization 425

8 xi Parallel factorization QZ-factorization of an unstructured matrix Multiplication between structured rank matrices Products of structured rank matrices Examples Conclusions A Gauss solver for higher order structured rank systems A sequence of Gauss transformation matrices without pivoting Annihilating Gauss transforms on lower rank 1 structures Arbitrary Gauss transforms on lower rank structures Gauss transforms on upper rank structures A sequence of Gauss transforms from bottom to top Ascending and descending Gauss transforms Zero-creating Gauss transforms Rank-expanding Gauss transforms A sequence of Gauss transforms from top to bottom Effect of pivoting on rank structures The effect of pivoting on the rank structure Combining Gauss transforms with pivoting Sequences of transformations involving pivoting Numerical stability More on sequences of Gauss transforms Upper triangular Gauss transforms Transformations on the right of the matrix Solving systems with Gauss transforms Making the structured rank matrix upper triangular Gauss solver and the Lf/-factorization Other decompositions Different patterns of annihilation The graphical representation Some standard patterns Some theorems connected to Gauss transforms Conclusions A Levinson-like solver for structured rank matrices Higher order generator representable semiseparable matrices General quasiseparable matrices Band matrices Unsymmetric structures Summations of Levinson-conform matrices Conclusions Block quasiseparable matrices Definition 481

9 xii Contents 12.2 Factorization of the block lower/upper triangular part Connection to structured rank matrices Special cases 484, 12.5 Multiplication of a block quasiseparable matrix by a vector Solver for block quasiseparable systems Block quasiseparable matrices and descriptor systems Conclusions H, H 2 and hierarchically semiseparable matrices "H-matrices or hierarchical matrices W 2 -matrices Hierarchically semiseparable matrices Other classes of structured rank matrices Conclusions ' Inversion of structured rank matrices Banded Toeplitz matrices Inversion of (generalized) Hessenberg matrices Inversion of higher order semiseparable and band matrices Strict band and generator representable semiseparable matrices Band and semiseparable matrices Block matrices Quasiseparable matrices Generalized inverses Conclusions Concluding remarks & software Software Conclusions 532 Bibliography 533 Author/Editor Index 557 Subject Index 565