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Lanczos-type solvers for nonsymmetric linear systems of equations. (English) Zbl 0888.65030
Iserles, A. (ed.), Acta Numerica Vol. 6, 1997. Cambridge: Cambridge University Press. 271-397 (1997).
The paper is a detailed and well written review of various Lanczos-type solvers for large linear systems with sparse nonsymmetric matrices. A list of about 150 references is given. Special attention is paid to possible breakdowns of the algorithms and ways to cure them. The important role of preconditioning is underlined and several examples of good preconditioners are given. Reviewer’s remarks: One of the most effective approaches to grid systems $Ax= b$ should be mentioned. It deals with the symmetrization $B^{-1}A^* B^{-1}(Ax-b)= 0$, where $B= B^*>0$ is a good model operator for $A$ [see {\it E. G. D’yakonov}, Optimization in solving elliptic problems (1996; Zbl 0852.65087)]. It enables one to use modified Richardson iterations with proper parameters and their adaption, as well. For the entire collection see [Zbl 0868.00024].

65F10Iterative methods for linear systems
65F50Sparse matrices (numerical linear algebra)
65F35Matrix norms, conditioning, scaling (numerical linear algebra)