Matrix Computations and Semiseparable Matrices : Linear Systems.

Format:

Book

Author/Creator:

Vandebril, Raf.

Contributor:

Van Barel, Marc.

Mastronardi, Nicola.

Language:

English

Physical Description:

1 online resource (594 pages)

Edition:

1st ed.

Place of Publication:

Baltimore : Johns Hopkins University Press, 2008.

Summary:

Many of the routines featured are implemented in Matlab and can be downloaded from the Web for further exploration.

Contents:

Intro

Contents

Preface

Notation

I: Introduction to semiseparable and related matrices

1 Semiseparable and related matrices: definitions and properties

1.1 Symmetric semiseparable and related matrices

1.2 Relations between the different symmetric definitions

1.3 Unsymmetric semiseparable and related matrices

1.4 Relations between the different "unsymmetric" definitions

1.5 Relations under inversion

1.6 Conclusions

2 The representation of semiseparable and related matrices

2.1 Representations

2.2 The symmetric generator representation

2.3 The symmetric diagonal-subdiagonal representation

2.4 The symmetric Givens-vector representation

2.5 The symmetric quasiseparable representation

2.6 Some examples

2.7 The unsymmetric generator representation

2.8 The unsymmetric Givens-vector representation

2.9 The unsymmetric quasiseparable representation

2.10 The decoupled representation for semiseparable matrices

2.11 Summary of the representations

2.12 Are there more representations?

2.13 Some algorithms related to representations

2.14 Conclusions

3 Historical applications and other topics

3.1 Oscillation matrices

3.2 Semiseparable matrices as covariance matrices

3.3 Discretization of integral equations

3.4 Orthogonal rational functions

3.5 Some comments

3.6 Conclusions

II: Linear systems with semiseparable and related matrices

4 Gaussian elimination

4.1 About Gaussian elimination and the LU-factorization

4.2 Backward substitution

4.3 Inversion of triangular semiseparable matrices

4.4 Theoretical considerations of the LU-decomposition

4.5 The LU-decomposition for semiseparable matrices

4.6 The LU-decomposition for quasiseparable matrices

4.7 Some comments

4.8 Conclusions

5 The QR-factorization

5.1 About the QR-decomposition.

5.2 Theoretical considerations of the QR-decomposition

5.3 A QR-factorization of semiseparable matrices

5.4 A QR-factorization of quasiseparable matrices

5.5 Implementing the QR-factorization

5.6 Other decompositions

5.7 Conclusions

6 A Levinson-like and Schur-like solver

6.1 About the Levinson algorithm

6.2 Generator representable semiseparable plus diagonal matrices

6.3 A Levinson framework

6.4 Examples

6.5 The Schur algorithm

6.6 Conclusions

7 Inverting semiseparable and related matrices

7.1 Known factorizations

7.2 Direct inversion methods

7.3 General formulas for inversion

7.4 Scaling of symmetric positive definite semiseparable matrices

7.5 Decay rates for the inverses of tridiagonal matrices

7.6 Conclusions

III: Structured rank matrices

8 Definitions of higher order semiseparable matrices

8.1 Structured rank matrices

8.2 Definition of higher order semiseparable and related matrices

8.3 Inverses of structured rank matrices

8.4 Generator representable semiseparable matrices

8.5 Representations

8.6 Conclusions

9 A QR-factorization for structured rank matrices

9.1 A sequence of Givens transformations from bottom to top

9.2 Making the structured rank matrix upper triangular

9.3 Different patterns of annihilation

9.4 Rank-expanding sequences of Givens transformations

9.5 QR-factorization for the Givens-vector representation

9.6 Extra material

9.7 Multiplication between structured rank matrices

9.8 Conclusions

10 A Gauss solver for higher order structured rank systems

10.1 A sequence of Gauss transformation matrices without pivoting

10.2 Effect of pivoting on rank structures

10.3 More on sequences of Gauss transforms

10.4 Solving systems with Gauss transforms

10.5 Different patterns of annihilation

10.6 Conclusions.

11 A Levinson-like solver for structured rank matrices

11.1 Higher order generator representable semiseparable matrices

11.2 General quasiseparable matrices

11.3 Band matrices

11.4 Unsymmetric structures

11.5 Summations of Levinson-conform matrices

11.6 Conclusions

12 Block quasiseparable matrices

12.1 Definition

12.2 Factorization of the block lower/upper triangular part

12.3 Connection to structured rank matrices

12.4 Special cases

12.5 Multiplication of a block quasiseparable matrix by a vector

12.6 Solver for block quasiseparable systems

12.7 Block quasiseparable matrices and descriptor systems

12.8 Conclusions

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

13.1 H-matrices or hierarchical matrices

13.2 H[sup(2)]-matrices

13.3 Hierarchically semiseparable matrices

13.4 Other classes of structured rank matrices

13.5 Conclusions

14 Inversion of structured rank matrices

14.1 Banded Toeplitz matrices

14.2 Inversion of (generalized) Hessenberg matrices

14.3 Inversion of higher order semiseparable and band matrices

14.4 Block matrices

14.5 Quasiseparable matrices

14.6 Generalized inverses

14.7 Conclusions

15 Concluding remarks &amp

software

15.1 Software

15.2 Conclusions

Bibliography

Author/Editor Index

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

R

S

T

U

V

W

Y

Z

Subject Index

Q

X

Z.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Vandebril, Raf Matrix Computations and Semiseparable Matrices

ISBN:

9780801896798

OCLC:

923194004