Variations of Zhang's Lanczos-type product method
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
The Chebyshev iteration revisited
Parallel Computing - Parallel matrix algorithms and applications
Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems
Applied Numerical Mathematics
An Augmented Stability Result for the Lanczos Hermitian Matrix Tridiagonalization Process
SIAM Journal on Matrix Analysis and Applications
On the Numerical Analysis of Oblique Projectors
SIAM Journal on Matrix Analysis and Applications
Minimizing synchronizations in sparse iterative solvers for distributed supercomputers
Computers & Mathematics with Applications
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It has been widely observed that Krylov space solvers based on two three-term recurrences can give significantly less accurate residuals than mathematically equivalent solvers implemented with three two-term recurrences. In this paper we attempt to clarify and justify this difference theoretically by analyzing the gaps between recursively and explicitly computed residuals.It is shown that, in contrast with the two-term recurrences analyzed by Sleijpen, van der Vorst, and Fokkema [ Numer. Algorithms, 7 (1994), pp. 75--109] and Greenbaum [SIAM J. Matrix Anal. Appl., 18 (1997), pp. 535--551], in the two three-term recurrences the contributions of the local roundoff errors to the analyzed gaps may be dramatically amplified while propagating through the algorithm. This result explains, for example, the well-known behavior of three-term-based versions of the biconjugate gradient method, where large gaps between recursively and explicitly computed residuals are not uncommon. For the conjugate gradient method, however, such a devastating behavior---although possible---is not observed frequently in practical computations, and the difference between two-term and three-term implementations is usually moderate or small. This can also be explained by our results.