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Matrix computations and semiseparable matrices. Vol. 2: Eigenvalue and singular value methods. (English) Zbl 1175.65045
Baltimore, MD: Johns Hopkins University Press (ISBN 978-0-8018-9052-9/hbk). xvi, 498 p. (2008).
The general properties of semiseparable matrices and similar structured rank matrices were presented in Volume 1 of this doublet of books [for Vol. 1 (2008) see Zbl 1141.65019]. A matrix is called semiseparable if all rectangular submatrices taken out of the lower or the upper triangular part of the matrix have a rank of at most 1. These matrices are related to inverses of triangular matrices.
The present volume is concerned with eigenvalue problems, singular value decompositions, and the reduction procedures that are used in this connection. In addition to the reduction of semiseparable matrices to Hessenberg form, tridiagonal or diagonal matrices, we find now also the reduction of general matrices to semiseparable matrices. A generalization of the Hessenberg form is introduced in this context.
The first part of the book is concerned with different reduction procedures. The second part focuses on implicit QR-algorithms for semiseparable matrices. Miscellaneous topics that fit into the context of eigenvalue problems of rank-structured matrices make part three. The coefficients of recurrence relations for orthogonal polynomials can be stored in matrices of Hessenberg form. This is the starting point of part 4 on orthogonal functions and inverse eigenvalue problems.

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65F18 Numerical solutions to inverse eigenvalue problems
65F20 Numerical solutions to overdetermined systems, pseudoinverses