Matrix Computations and Semiseparable Matrices
Publication Year: 2007
Published by: The Johns Hopkins University Press
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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 . . .
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I: Introduction to semiseparable and related matrices
1 Semiseparable and related matrices: definitions and properties
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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
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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
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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 . . .
II: Linear systems with semiseparable and related matrices
4 Gaussian elimination
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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
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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
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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
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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 . . .
III: Structured rank matrices
8 Definitions of higher order semiseparable matrices
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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
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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
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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
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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
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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
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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
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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
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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: . . .
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Page Count: 584
Illustrations: 7 halftones, 75 line drawings
Publication Year: 2007