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.
The objective of this study was to determine the potential value of electrical impedance myography (EIM) for assessing lumbosacral paraspinal muscle (LPM) condition in lower back pain (LBP) patients. Standard methods for assessing the condition of LPMs, such as magnetic resonance imaging, are inconvenient and expensive. One tool that could be useful for this purpose is electrical impedance myography (EIM) a technique that can be performed rapidly at the bedside. After undergoing a screening history and examination, subjects were studied with the mView EIM device (Myolex, Inc, Boston). Bilateral LPMs were measured three times each and the two closest sets of measurements averaged on each side. Data analysis included non-parametric two-group comparisons between healthy subjects and back pain patients, receiver-operating curve analyses, and correlation analyses to age and body mass index. A total of 86 healthy individuals (median age (interquartile range) (IQR), 45.5 years (30.3–56.0 years), 42 men, 44 women) and 47 LBP (median age 51.0 year (39.5–57.5) years, 21 men, 26 women) were enrolled. Median EIM 100kHz phase was lower in the LBP patients (9.3°(IQR 8.4°–10.6°) versus 11.4°(IQR 9.4°–13.0°), p = 0.0007). Significantly increased normalized side-to-side differences were present for all three EIM variables (e.g., median 100 kHz phase 0.15 (IQR 0.07–0.31 in LBP patients versus 0.09 (IQR 0.04–0.17) in healthy individuals). A significant correlation between 100 kHz EIM phase and reactance was found with age (Rspearman=−0.46, P=0.0002 and Rspearman=−0.440, P=0.0003) but not for resistance. This study provides early evidence supporting that EIM has the potential to serve as a useful tool for evaluating the condition of LPMs.