Separable Type Representations of Matrices and Fast Algorithms

1. Front Matter

   Pages i-xv

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2. Basics on Separable, Semi-separable and Quasiseparable Representations of Matrices

   1. Front Matter

      Pages 1-2

      PDF

   2. Matrices with Separable Representation and Low Complexity Algorithms

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 3-44

   3. The Minimal Rank Completion Problem

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 45-66

   4. Matrices in Diagonal Plus Semiseparable Form

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 67-73

   5. Quasiseparable Representations: The Basics

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 75-84

   6. Quasiseparable Generators

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 85-117

   7. Rank Numbers of Pairs of Mutually Inverse Matrices, Asplund Theorems

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 119-137

   8. Unitary Matrices with Quasiseparable Representations

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 139-161

3. Completion of Matrices with Specified Bands

   1. Front Matter

      Pages 163-164

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   2. Completion to Green Matrices

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 165-177

   3. Completion to Matrices with Band Inverses and with Minimal Ranks

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 179-199

   4. Completion of Special Types of Matrices

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 201-217

   5. Completion of Mutually Inverse Matrices

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 219-228

   6. Completion to Unitary Matrices

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 229-244

4. Quasiseparable Representations of Matrices, Descriptor Systems with Boundary Conditions and First Applications

   1. Front Matter

      Pages 245-246

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   2. Quasiseparable Representations and Descriptor Systems with Boundary Conditions

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 247-260

   3. The First Inversion Algorithms

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 261-278

   4. Inversion of Matrices in Diagonal Plus Semiseparable Form

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 279-293

   5. Quasiseparable/Semiseparable Representations and One-direction Systems

      • Yuli Eidelman, Israel Gohberg, Iulian Haimovici

      Pages 295-307

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.​

• Completion problems
• Factorization and inversion
• Fast algorithms
• Quasiseparable representations
• Structured matrices
• matrix theory

• School of Mathematical Sciences, Tel Aviv University Raymond & Beverly Sackler Faculty of Exa, Tel Aviv, Israel

  Yuli Eidelman, Israel Gohberg, Iulian Haimovici

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