دانلود کتاب Separable Type Representations of Matrices and Fast Algorithms

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Contents
Preface
Introduction
Part I Basics on Separable, Semiseparable and Quasiseparable Representations of Matrices
Introduction to Part I
Chapter 1 Matrices with Separable
Representation and Low
Complexity Algorithms
§1.1 Rank and related factorizations
§1.2 Definitions and first examples
§1.3 The algorithm of multiplication by a vector
§1.3.1 Forward and backward computation of y
§1.3.2 Forward-backward computation of y
§1.4 Systems with homogeneous boundary conditions associated with matrices in diagonal plus separable form
§1.4.1 Forward and backward systems
§1.4.2 Forward-backward descriptor systems
§1.5 Multiplication of matrices
§1.5.1 Product of matrices with separable representations
§1.5.2 Product of matrices with diagonal plus separable representations
§1.6 Schur factorization and inversion of block matrices
§1.7 A general inversion formula
§1.8 Inversion of matrices with diagonal plus separable representation
§1.9 LDU factorization of matrices with diagonal plus separable representation
§1.10 Solution of linear systems in the presence of the LDU factorization of the matrixof the system in diagonal plus separable form
§1.11 Comments
Chapter 2 The Minimal Rank
Completion Problem
§2.1 The definition. The case of a 2 × 2 block matrix
§2.2 Solution of the general minimal rank completion problem. Examples
§2.3 Uniqueness of the minimal rank completion
§2.4 Comments
Chapter 3 Matrices in Diagonal Plus
Semiseparable Form
§3.1 Definitions
§3.2 Semiseparable order and minimal semiseparable generators
§3.3 Comments
Chapter 4 Quasiseparable Representations:
The Basics
§4.1 The rank numbers and quasiseparable order. Examples
§4.1.1 The definitions
§4.1.2 The companion matrix
§4.1.3 The block companion matrix
§4.1.4 Tridiagonal matrices and band matrices
§4.1.5 Matrices with diagonal plus semiseparable representations
§4.2 Quasiseparable generators
§4.3 Minimal completion rank, rank numbers, and quasiseparable order
§4.4 The quasiseparable and semiseparable generators
§4.5 Comments
Chapter 5 Quasiseparable Generators
§5.1 Auxiliary matrices and relations
§5.2 Existence and minimality of quasiseparable generators
§5.3 Examples
§5.4 Quasiseparable generators of block companion matrices viewed as scalar matrices
§5.5 Minimality conditions
§5.6 Sets of generators. Minimality and similarity
§5.7 Reduction to minimal quasiseparable generators
§5.8 Normal quasiseparable generators
§5.9 Approximation by matrices with quasiseparable representation
§5.10 Comments
Chapter 6 Rank Numbers of Pairs of Mutually
Inverse Matrices, Asplund Theorems
§6.1 Rank numbers of pairs of inverse matrices
§6.2 Rank numbers relative to the main diagonal. Quasiseparable orders
§6.3 Green and band matrices
§6.4 The inverses of diagonal plus Green of order one matrices
§6.5 Minimal completion ranks of pairs of mutually inverse matrices. The inverse of an irreducible band matrix
§6.6 Linear-fractional transformations of matrices
§6.6.1 The definition and the basic property
§6.6.2 Linear-fractional transformations of Green and band matrices
§6.6.3 Unitary Hessenberg and Hermitian matrices
§6.6.4 Linear-fractional transformations of diagonal plus Green of order one matrices
§6.7 Comments
Chapter 7 Unitary Matrices with
Quasiseparable Representations
§7.1 QR and related factorizations of matrices
§7.2 The rank numbers and quasiseparable generators
§7.3 Factorization representations
§7.3.1 Block triangular matrices
§7.3.2 Factorization of general unitary matrices and compression of generators
§7.3.3 Generators via factorization
§7.4 Unitary Hessenberg matrices
§7.5 Efficient generators
§7.6 Comments
Part II Completion of Matrices with Specified Bands
Introduction to Part II
Chapter 8 Completion to Green Matrices
§8.1 Auxiliary relations
§8.2 Completion formulas
§8.3 Comments
Chapter 9 Completion to Matrices with Band
Inverses and with Minimal Ranks
§9.1 Completion to invertible matrices
§9.2 The LDU factorization
§9.3 The Permanence Principle
§9.4 The inversion formula
§9.5 Completion to matrices of minimal ranks
§9.6 Comments
Chapter 10 Completion of Special Types
of Matrices
§10.1 The positive case
§10.2 The Toeplitz case
§10.3 Completion of specified tridiagonal parts with identities on the main diagonal
§10.3.1 The general case
§10.3.2 The Toeplitz case
§10.4 Completion of special 2 × 2 block matrices
§10.4.1 Completion formulas
§10.4.2 Completion to invertible and positive matrices
§10.4.3 Completion to matrices of minimal ranks
§10.5 Comments
Chapter 11 Completion of Mutually
Inverse Matrices
§11.1 The statement and preliminaries
§11.2 The basic theorem
§11.3 The direct method
§11.4 The factorization
§11.5 Comments
Chapter 12 Completion to Unitary Matrices
§12.1 Auxiliary relations
§12.2 An existence and uniqueness theorem
§12.3 Unitary completion via quasiseparable representation
§12.3.1 Existence theorem
§12.3.2 Diagonal correction for scalar matrices
§12.4 Comments
Part III Quasiseparable Representations of Matrices, Descriptor Systems with Boundary Conditions and First Applications
Introduction to Part III
Chapter 13 Quasiseparable Representations
and Descriptor Systems
with Boundary Conditions
§13.1 The algorithm of multiplication by a vector
§13.2 Descriptor systems with homogeneous boundary conditions
§13.3 Examples
§13.4 Inversion of triangular matrices
§13.5 Comments
Chapter 14 The First Inversion Algorithms
§14.1 Inversion of matrices in quasiseparable representation with invertible diagonal elements
§14.2 The extension method for matrices with quasiseparable/semiseparable representations
§14.2.1 The inversion formula
§14.2.2 The orthogonalization procedure
§14.3 Comments
Chapter 15 Inversion of Matrices in Diagonal
Plus Semiseparable Form
§15.1 The modified inversion formula
§15.2 Scalar matrices with diagonal plus semiseparable representation
§15.3 Comments
Chapter 16 Quasiseparable/Semiseparable
Representations and
One-direction Systems
§16.1 Systems with diagonal main coefficients and homogeneous boundary conditions
§16.2 The general one-direction systems
§16.3 Inversion of matrices with quasiseparable/ semiseparable representations via one-direction systems
§16.4 Comments
Chapter 17 Multiplication of Matrices
§17.1 The rank numbers of the product
§17.2 Multiplication of triangular matrices
§17.3 The general case
§17.4 Multiplication by triangular matrices
§17.5 Complexity analysis
§17.6 Product of matrices with semiseparable representations
§17.7 Comments
Part IV Factorization and Inversion
Introduction to Part IV
Chapter 18 The LDU Factorization and Inversion
§18.1 Rank numbers and minimal completion ranks
§18.2 The factorization algorithm
§18.3 Solution of linear systems and analog of Levinson algorithm
§18.4 The inversion formula
§18.5 The case of a diagonal plus semiseparable representation
§18.6 Comments
Chapter 19 Scalar Matrices with
Quasiseparable Order One
§19.1 Inversion formula
§19.2 Examples
§19.3 Inversion algorithm with scaling
§19.4 The case of diagonal plus semiseparable representation
§19.5 The case of a tridiagonal matrix
§19.6 Comments
Chapter 20 The QR-Factorization Based Method
§20.1 Factorization of triangular matrices
§20.2 The first factorization theorem
§20.3 The second factorization theorem
§20.4 Solution of linear systems
§20.5 Complexity
§20.6 The case of scalar matrices
§20.7 Comments
Bibliography
Index