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Paper details
Number 4 - December 2015
Volume 25 - 2015
The non-symmetric s-step Lanczos algorithm: Derivation of efficient recurrences and synchronization-reducing variants of BiCG and QMR
Stefan Feuerriegel, H. Martin Bücker
Abstract
The Lanczos algorithm is among the most frequently used iterative techniques for computing a few dominant eigenvalues
of a large sparse non-symmetric matrix. At the same time, it serves as a building block within biconjugate gradient (BiCG)
and quasi-minimal residual (QMR) methods for solving large sparse non-symmetric systems of linear equations. It is well
known that, when implemented on distributed-memory computers with a huge number of processes, the synchronization
time spent on computing dot products increasingly limits the parallel scalability. Therefore, we propose synchronization-reducing variants of the Lanczos, as well as BiCG and QMR methods, in an attempt to mitigate these negative performance
effects. These so-called s-step algorithms are based on grouping dot products for joint execution and replacing time-consuming matrix operations by efficient vector recurrences. The purpose of this paper is to provide a rigorous derivation of the recurrences for the s-step Lanczos algorithm, introduce s-step BiCG and QMR variants, and compare the parallel performance of these new s-step versions with previous algorithms.
Keywords
synchronization-reducing, s-step Lanczos, s-step BiCG, s-step QMR, efficient recurrences