Separable type representations of matrices and fast algorithms. Volume 1. Basics. Completion problems. Multiplication and inversion algorithms
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)].
scientific article; zbMATH DE number 6151541