The nonsymmetric Lanczos method can be used to compute eigenvalues of large sparse non-Hermitian matrices or to solve large sparse non-Hermitian linear systems. However, the original Lanczos algorithm is susceptible to possible breakdowns and potential instabilities. An implementation of a look-ahead version of the Lanczos algorithm is presented that, except for the very special situation of an incurable breakdown, overcomes these problems by skipping over those steps in which a breakdown or near-breakdown would occur in the standard process. The proposed algorithm can handle look-ahead steps of any length and requires the same number of matrix-vector products and inner products as the standard Lanczos process without look-ahead.
Recently Van der Vorst [S/AM Y. Sci. Statist. Comput., 13 (1992), pp. 631--644] proposed for solving nonsymmetric linear systems Az b a biconjugate gradient (BICG)-based Krylov space method called BICGSTAB that, like the biconjugate gradient squared (BICGS) method ofSonneveld, does not require matrixvector multiplications with the transposed matrix AT, and that has typically a much smoother convergence behavior than BCG and BICGS. Its nth residual polynomial is the product of the one of BICG (i.e., the nth Lanczos polynomial) with a polynomial of the same degree with real zeros. Therefore, nonreal eigenvalues of A are not approximated well by the second polynomial factor. Here, the author presents for real nonsymmetric matrices a method BCGSTAB2 in which the second factor may have complex conjugate zeros. Moreover, versions suitable for complex matrices are given for both methods.
We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) "removes" certain parts from the operator making it singular, while augmentation adds a subspace to the Krylov subspace (often the one that is generated by the singular operator); in contrast, preconditioning changes the spectrum of the operator without making it singular. Deflation and augmentation have been used in a variety of methods and settings. Typically, deflation is combined with augmentation to compensate for the singularity of the operator, but both techniques can be applied separately.We introduce a framework of Krylov subspace methods that satisfy a Galerkin condition. It includes the families of orthogonal residual (OR) and minimal residual (MR) methods. We show that in this framework augmentation can be achieved either explicitly or, equivalently, implicitly by projecting the residuals appropriately and correcting the approximate solutions in a final step. We study conditions for a breakdown of the deflated methods, and we show several possibilities to avoid such breakdowns for the deflated MinRes method. Numerical experiments illustrate properties of different variants of deflated MinRes analyzed in this paper.
Among the iterative methods for solving large linear systems with a sparse (or, possibly, structured) nonsymmetric matrix, those that are based on the Lanczos process feature short recurrences for the generation of the Krylov space. This means low cost and low memory requirement. This review article introduces the reader not only to the basic forms of the Lanczos process and some of the related theory, but also describes in detail a number of solvers that are based on it, including those that are considered to be the most efficient ones. Possible breakdowns of the algorithms and ways to cure them by look-ahead are also discussed.
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