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# Matrix Computations and Semiseparable Matrices

Linear Systems

Publication Year: 2007

In recent years several new classes of matrices have been discovered and their structure exploited to design fast and accurate algorithms. In this new reference work, Raf Vandebril, Marc Van Barel, and Nicola Mastronardi present the first comprehensive overview of the mathematical and numerical properties of the family's newest member: semiseparable matrices. The text is divided into three parts. The first provides some historical background and introduces concepts and definitions concerning structured rank matrices. The second offers some traditional methods for solving systems of equations involving the basic subclasses of these matrices. The third section discusses structured rank matrices in a broader context, presents algorithms for solving higher-order structured rank matrices, and examines hybrid variants such as block quasiseparable matrices. An accessible case study clearly demonstrates the general topic of each new concept discussed. Many of the routines featured are implemented in Matlab and can be downloaded from the Web for further exploration.

Published by: The Johns Hopkins University Press

#### Preface

pp. xiii-xvi

In this book we study the class of semiseparable matrices, which is nowadays a “hot topic”. Matrices arising in a wide variety of applications, e.g., quasiseparable, hierarchically semiseparable, H-matrices, structured rank . . .

pp. xvii-xviii

#### 1 Semiseparable and related matrices: definitions and properties

pp. 5-51

In this first chapter of the book we will pay special attention to the definition of semiseparable matrices and closely related classes such as ‘generator representable’ semiseparable, quasiseparable and semiseparable plus diagonal matrices. . . .

#### 2 The representation of semiseparable and related matrices

pp. 53-107

In the previous chapter it was shown that, when one wants to solve the eigenvalue problem by means of the QR-algorithm, the definition of semiseparable matrices with generators has some disadvantages. Therefore we proposed the more elaborate . . .

#### 3 Historical applications and other topics

pp. 109-141

In this chapter some historical applications and early appearances of semiseparable and related matrices are investigated. Some links to other closely related topics not covered in this book are also presented, e.g., the eigenvalue problem via . . .

#### 4 Gaussian elimination

pp. 149-180

In this chapter we will investigate an efficient Gaussian elimination pattern for some classes of structured rank matrices. Gaussian elimination with partial row pivoting computes a factorization of the matrix . . .

#### 5 The QR-factorization

pp. 181-204

In this chapter we will consider two fast algorithms for solving linear systems involving semiseparable, quasiseparable and semiseparable plus diagonal matrices. Exploiting the structure of such matrices, algorithms . . .

#### 6 A Levinson-like and Schur-like solver

pp. 205-256

Different algorithms for solving systems of equations with semiseparable plus diagonal coefficient matrices have been proposed in the previous chapters. The method proposed in this chapter is based on the underlying idea of the Durbin and the . . .

#### 7 Inverting semiseparable and related matrices

pp. 257-285

The inverses of semiseparable and semiseparable plus diagonal matrices have often been used in statistical applications; hence, the extensive list of references related to this . . .

#### 8 Definitions of higher order semiseparable matrices

pp. 293-345

In this chapter we will discuss some new classes of structured rank matrices. In fact we will extend in a natural way the classes as defined in Chapter 1 of this book towards their higher order generalizations. The definitions, however, will be . . .

#### 9 A QR-factorization for structured rank matrices

pp. 347-433

In the beginning of the book we discussed the QR-factorization for the easiest classes of structured rank matrices, e.g., semiseparable, semiseparable plus diagonal and quasiseparable matrices. For quasiseparable matrices, we investigated the . .

#### 10 A Gauss solver for higher order structured rank systems

pp. 435-471

In the previous chapter we studied thoroughly the QR-factorization of structured rank matrices. Special attention was paid to the rank structure of all involved matrices. The matrix Q was factored as a product of Givens transformations . . .

#### 11 A Levinson-like solver for structured rank matrices

pp. 473-479

The second part of this book was dedicated to solving systems of equations for the easy classes of structured rank matrices, such as semiseparable, quasiseparable, semiseparable plus diagonal and so forth. The chapter on the . . .

#### 12 Block quasiseparable matrices

pp. 481-490

In this chapter we will give an introduction on the class of block quasiseparable matrices. This is again a generalization of the class of higher order quasiseparable matrices. Quasiseparable matrices are also known under the name . . .

#### 13 H, H[sup(2)] and hierarchically semiseparable matrices

pp. 491-497

In this chapter we will give a brief overview of some other classes of structured rank matrices. In Section 13.1 the class of H-matrices or hierarchical matrices is defined. It is shown how these matrices can be used to solve integral equations. . . .

#### 14 Inversion of structured rank matrices

pp. 499-529

In this chapter we will discuss some references related to the inversion of structured rank matrices. This chapter focuses attention on higher order structured rank matrices such as semiseparable, quasiseparable, generalized Hessenberg, Hessenberg-like . . .

#### 15 Concluding remarks & software

pp. 531-532

As already mentioned, several of the proposed methods were implemented by the authors in Matlab and are freely available for download at the following site: . . .

pp. 533-556

pp. 557-563

#### Subject Index

pp. 565-575

E-ISBN-13: 9780801896798
E-ISBN-10: 0801896797
Print-ISBN-13: 9780801887147
Print-ISBN-10: 0801887143

Page Count: 584
Illustrations: 7 halftones, 75 line drawings
Publication Year: 2007