The Non–Symmetric s –Step Lanczos Algorithm: Derivation of Efficient Recurrences and Synchronization–Reducing Variants of BiCG and QMR
Stefan Feuerriegel, H. Martin Bücker · International Journal of Applied Mathematics and Computer Science · 2015
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.