Breakdowns in the implementation of the Lánczos method for solving linear systems

AbstractThe Lánczos method for solving systems of linear equations is based on formal orthogonal polynomials. Its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials. A division by zero can occur in such recurrence relations, thus causing a breakdown in the algorithm which has to be stopped. In this paper, two types of breakdowns are discussed. The true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used. Among all the recurrence relationships which can be used and all the algorithms for implementing the Lánczos method which came out from them, the only reliable algorithm is Lánczos/Orthodir which can only suffer from true breakdowns. It is shown how to avoid true breakdowns in this algorithm. Other algorithms are also discussed and the case of near-breakdown is treated. The same treatment applies to other methods related to Lánczos'

Closer to the solutions: iterative linear solvers

The solution of dense linear systems received much attention after the second world war, and by the end of the sixties, most of the problems associated with it had been solved. For a long time, Wilkinson's \The Algebraic Eigenvalue Problem" [107], other than the title suggests, became also the standard textbook for the solution of linear systems. When it became clear that partial dierential equations could be solved numerically, to a level of accuracy that was of interest for application areas (such as reservoir engineering, and reactor diusion modeling), there was a strong need for the fast solution of the discretized systems, and iterative methods became popular for these problems

Lanczos-type solvers for nonsymmetric linear systems of equations

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 discusse

Iterative solvers for generalized finite element solution of boundary-value problems

Most of generalized finite element methods use dense direct solvers for the resulting linear systems. This is mainly the case due to the ill‐conditioned linear systems that are associated with these methods. In this study, we investigate the performance of a class of iterative solvers for the generalized finite element solution of time‐dependent boundary‐value problems. A fully implicit time‐stepping scheme is used for the time integration in the finite element framework. As enrichment, we consider a combination of exponential functions based on an approximation of the internal boundary layer in the problem under study. As iterative solvers, we consider the changing minimal residual method based on the Hessenberg reduction and the generalized minimal residual method. Compared with dense direct solvers, the iterative solvers achieve high accuracy and efficiency at low computational cost and less storage as only matrix–vector products are involved in their implementation. Two test examples for boundary‐value problems in two space dimensions are used to assess the performance of the iterative solvers. Comparison to dense direct solvers widely used in the framework of generalized finite element methods is also presented. The obtained results demonstrate the ability of the considered iterative solvers to capture the main solution features. It is also illustrated for the first time that this class of iterative solvers can be efficient in solving the ill‐conditioned linear systems resulting from the generalized finite element methods for time domain problems