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Matrix computations and semiseparable matrices. Vol. 1: Linear systems. (English) Zbl 1141.65019
Baltimore, MD: Johns Hopkins University Press (ISBN 978-0-8018-8714-7/hbk). xviii, 575 p. (2008).
The book is concerned with the class of semiseparable matrices and some generalizations. A matrix is called semiseparable if all submatrices taken out of the lower and upper triangular part of the matrix are of rank 1. The role of these matrices becomes clearer from another characterization. A nonsingular matrix is semiseparable iff it is the inverse of a tridiagonal matrix. Therefore, there are connections with oscillation matrices that are encountered with the discretization of ordinary differential equations or with matrices that arise with integral equations or with covariance matrices of multinomial distributions. These applications, basic properties, and some representations (for these and related matrices) form the first part of the book.
The second part of the book is devoted to numerical linear algebra with semiseparable and related matrices, e.g., Gaussian elimination, the QR-factorization, Levinson-like algorithms and Schur-like solvers. Of course, the direct computation of the inverse of a semiseparable function is cheaper than its LU-decomposition. Moreover, one may prefer to perform (also other) computations with the inverse matrices since they are tridiagonal. A motivation for the algorithms without the reference to the tridiagonal inverses is the generalization to the larger class of quasiseparable matrices.
The third part deals with structured rank matrices, i.e., submatrices taken out of the triangular parts (as above) may have a rank greater than one. Such matrices occur, e.g., as inverses of band matrices with broader bands. The topics of part one and two are generalized to matrices with the more general structure.
The style of the book can be described by the fact that properties of the matrices and the algorithms are elaborated with many details. Symmetric semiseparable matrices and unsymmetric semiseparable matrices are independently defined. – Another example. The solution of a linear system with a semiseparable matrix is obtained by a multiplication with a tridiagonal matrix that is obtained by a recursive procedure. If the given matrix is a semiseparable matrix plus a diagonal matrix, then an additional linear system with a modified tridiagonal matrix has to be solved. Here, the book does not restrict the discussion to the associated reduction of the given problem, but contains the complete algorithms and provides variants which are related to big names.
In particular, the relation with algorithms of historical interest will make the book interesting for its readers.

65Fxx Numerical linear algebra
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra