Direct methods for sparse matrices (Q2827272)

This is a new, updated edition of the original from 1986 [\textit{I. S. Duff} et al., Direct methods for sparse matrices. Oxford: Clarendon Press (1986; Zbl 0604.65011)]. While the literature about iterative methods for large sparse systems is overwhelming, there is much less available about direct methods, and yet, also this area has evolved since 1986. After 2000 essentially only the SIAM book by \textit{T. A. Davis} [Direct methods for sparse linear systems. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (2006; Zbl 1119.65021)] was published but that is not as general and puts emphasis on the matlab/C-code for the package CSparse.NEWLINENEWLINEThis new edition is faithful to the original version in that it is mainly discussing storage techniques and how this influences the implementation of LU factorizations such as the Gaussian elimination and the subsequent solution of the triangular system, either in scalar or block version. Of course pivoting to maintain numerical stability has to be considered carefully to avoid too much of fill-in. Therefore, a preprocessing step reordering the matrix is important so that Gaussian elimination and pivoting can operate locally. The use of graph theory and static or dynamic data structures and how this interferes with parallelism of the implementation are discussed. The most drastic rewriting is in the last chapters where new evolutions are presented. There is a new chapter on Gaussian elimination using trees and one using more general graphs. Also, the last chapter has new contributions dealing with backward error analysis, nonlinear problems, orthogonal transformations, and hybrid methods mixing direct and iterative methods. The storage techniques and the code in the original book relied on FORTRAN, but that is given up here, using a more modern pseudocode. The exercises following the chapters and the solutions and hints at the end of the book are adapted and new tasks, called research exercises are added without solutions. Obviously, the reference list is extended with many more recent publications.