This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is of a mainly theoretical character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part contains factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structure. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.
Author(s): Yuli Eidelman, Israel Gohberg, Iulian Haimovici (auth.)
Series: Operator Theory: Advances and Applications 234
Edition: 1
Publisher: Birkhäuser Basel
Year: 2014
Language: English
Pages: 399
Tags: Linear and Multilinear Algebras, Matrix Theory; Numerical Analysis
Front Matter....Pages i-xv
Front Matter....Pages 1-2
Matrices with Separable Representation and Low Complexity Algorithms....Pages 3-44
The Minimal Rank Completion Problem....Pages 45-66
Matrices in Diagonal Plus Semiseparable Form....Pages 67-73
Quasiseparable Representations: The Basics....Pages 75-84
Quasiseparable Generators....Pages 85-117
Rank Numbers of Pairs of Mutually Inverse Matrices, Asplund Theorems....Pages 119-137
Unitary Matrices with Quasiseparable Representations....Pages 139-161
Front Matter....Pages 163-164
Completion to Green Matrices....Pages 165-177
Completion to Matrices with Band Inverses and with Minimal Ranks....Pages 179-199
Completion of Special Types of Matrices....Pages 201-217
Completion of Mutually Inverse Matrices....Pages 219-228
Completion to Unitary Matrices....Pages 229-244
Front Matter....Pages 245-246
Quasiseparable Representations and Descriptor Systems with Boundary Conditions....Pages 247-260
The First Inversion Algorithms....Pages 261-278
Inversion of Matrices in Diagonal Plus Semiseparable Form....Pages 279-293
Quasiseparable/Semiseparable Representations and One-direction Systems....Pages 295-307
Multiplication of Matrices....Pages 309-326
Front Matter....Pages 327-328
The LDU Factorization and Inversion....Pages 329-351
Scalar Matrices with Quasiseparable Order One....Pages 353-372
The QR-Factorization Based Method....Pages 373-391
Back Matter....Pages 393-399
