Separable type representations of matrices and fast algorithms. Volume 1. Basics. Completion problems. Multiplication and inversion algorithms
type: Item
instance of: scholarly article
title: Separable type representations of matrices and fast algorithms. Volume 1. Basics. Completion problems. Multiplication and inversion algorithms
zbMATH Open document ID: 1280.65027
DOI: 10.1007/978-3-0348-0606-0
published in: Operator Theory: Advances and Applications
publication date: 2013-04-08T00:00:00Z
review text: The volume 1 of the book is concerned with the theoretical and computational analysis of some classes of separable matrices and their generalizations. It is organized in four parts, including 20 chapters. The first part (Chapters 1--7) has an introductory aim. It describes matrices with separable representation, minimal rank completion, quasiseparable representations and generators, rank numbers, mutually inverse matrices and unitary ones. The authors derive results related to quasiseparable and semiseparable structure, which will be used in the rest of the paper. The second part (Chapters 8--12) is essentially related to the completion of different types of matrices: Green, with band inverses, with minimal rank, mutually inverse, unitary. The third part (Chapters 13--17) deals with the description and analysis of several algorithms for quasiseparable matrices: multiplication by vectors, multiplication of matrices, inversion methods. The authors emphasize the important role of `the interplay between the quasiseparable structure and discrete-time varying linear systems with boundary conditions'. Part Four (Chapters 18--20) refers to factorization and inversion algorithms for quasiseparable and semiseparable structured matrices. In this respect the authors present the LDU factorization and inversion for strongly regular matrices and QR factorization of more general ones. The book is a self-contained graduate text. It is presented following the lines and principles of a classical linear algebra course. It addresses to engineers, scientists, numerical analists, computer scientists and mathematicians alike. For Volume 2 see [ibid. 235 (2014; Zbl 1280.65034)].
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65-01
Mathematics Subject Classification ID: 15A83
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 6151541
zbMATH Keywords: textbook
zbMATH Keywords: quasiseparable matrices
zbMATH Keywords: minimal rank
zbMATH Keywords: quasiseparable representations
zbMATH Keywords: inverse matrices
zbMATH Keywords: unitary matrices
zbMATH Keywords: completion
zbMATH Keywords: multiplication
zbMATH Keywords: LDU factorization
zbMATH Keywords: QR-factorization
zbMATH Keywords: algorithm
author: Y. Eidelman
author: I. Gohberg
author: I. Haimovici
reviewed by: Constantin Popa
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-0348-0606-0
OpenAlex ID: W2503785863
description: scientific article; zbMATH DE number 6151541