MOTIVATION: Advances in DNA sequencing have outpaced advances in computation, making sequence alignment a major bottleneck in genome data analyses. Classical dynamic programming (DP) algorithms are particularly memory-intensive, especially when computing gap-affine and dual gap-affine alignments. Existing strategies to reduce memory consumption often sacrifice speed or alignment accuracy. RESULTS: We present Singletrack, an efficient algorithm for backtrace gap-affine and dual gap-affine alignments that requires storing a single DP matrix while preserving optimal alignment results. Compared to classical DP algorithms, Singletrack removes the need to store additional matrices (i.e. 2 for gap-affine and 4 for dual gap-affine), significantly reducing memory consumption and, in turn, reducing pressure on the memory hierarchy and improving overall performance. Most importantly, Singletrack is a general backtrace method compatible with state-of-the-art DP-based algorithms and heuristics, such as the Suzuki-Kasahara (SK) and the Wavefront Alignment (WFA) algorithms. We demonstrate that Singletrack reduces memory consumption for both SK and WFA algorithms, lowering SK usage by 2× and 4× and WFA usage by 3× and 5× for gap-affine and dual gap-affine alignments, respectively. Moreover, replacing KSW2's memory-reduction technique with Singletrack accelerates its SK implementation by up to 1.4× at the cost of doubling memory consumption, while Singletrack increases the performance of the WFA implementation in WFA2-lib by 1.2-2.1×. Compared to the efficient linear-memory BiWFA algorithm, the Singletrack-accelerated version of WFA trades a practical increase in memory usage for up to 5.2× higher performance. AVAILABILITY AND IMPLEMENTATION: The Singletrack implementations presented in this work are available on Zenodo (DOI: 10.5281/zenodo.18770585) and GitHub (https://github.com/LorienLV/singletrack).

ABSTRACT We consider the fundamental problem of estimating the difference between the exact value and approximations that depend on a single real parameter . It is well‐known that if the error satisfies an asymptotic expansion, then we can use Richardson extrapolation to approximate . In this paper, our primary concern is the accuracy of Richardson's error estimate , that is, the size of the relative error . In practice, the computed value is different from the exact value . We show how to determine when the computational error is irrelevant and how to estimate the accuracy of Richardson's error estimate in terms of Richardson's fraction . We establish monotone convergence theorems and derive upper and lower bounds for in terms of and . We classify asymptotic error expansions according to their practical value rather than the order of the primary error term. We present a sequence of numerical experiments that illustrate the theory. Weierstrass's function is used to define a sequence of smooth problems for which it is impractical to apply Richardson's techniques.

Summary We analyze the convergence of quasi‐Newton methods in exact and finite precision arithmetic using three different techniques. We derive an upper bound for the stagnation level and we show that any sufficiently exact quasi‐Newton method will converge quadratically until stagnation. In the absence of sufficient accuracy, we are likely to retain rapid linear convergence. We confirm our analysis by computing square roots and solving bond constraint equations in the context of molecular dynamics. In particular, we apply both a symmetric variant and Forsgren's variant of the simplified Newton method. This work has implications for the implementation of quasi‐Newton methods regardless of the scale of the calculation or the machine.

Arm usage has substantially grown in the High-Performance Computing (HPC) community. Japanese supercomputer Fugaku, powered by Arm-based A64FX processors, held the top position on the Top500 list between June 2020 and June 2022, currently sitting in the fourth position. The recently released 7th generation of Amazon EC2 instances for compute-intensive workloads (C7 g) is also powered by Arm Graviton3 processors. Projects like European Mont-Blanc and U.S. DOE/NNSA Astra are further examples of Arm irruption in HPC. In parallel, over the last decade, the rapid improvement of genomic sequencing technologies and the exponential growth of sequencing data has placed a significant bottleneck on the computational side. While most genomics applications have been thoroughly tested and optimized for x86 systems, just a few are prepared to perform efficiently on Arm machines. Moreover, these applications do not exploit the newly introduced Scalable Vector Extensions (SVE). This paper presents GenArchBench, the first genome analysis benchmark suite targeting Arm architectures. We have selected computationally demanding kernels from the most widely used tools in genome data analysis and ported them to Arm-based A64FX and Graviton3 processors. Overall, the GenArch benchmark suite comprises 13 multi-core kernels from critical stages of widely-used genome analysis pipelines, including base-calling, read mapping, variant calling, and genome assembly. Our benchmark suite includes different input data sets per kernel (small and large), each with a corresponding regression test to verify the correctness of each execution automatically. Moreover, the porting features the usage of the novel Arm SVE instructions, algorithmic and code optimizations, and the exploitation of Arm-optimized libraries. We present the optimizations implemented in each kernel and a detailed performance evaluation and comparison of their performance on four different HPC machines (i.e., A64FX, Graviton3, Intel Xeon Skylake Platinum, and AMD EPYC Rome). Overall, the experimental evaluation shows that Graviton3 outperforms other machines on average. Moreover, we observed that the performance of the A64FX is significantly constrained by its small memory hierarchy and latencies. Additionally, as proof of concept, we study the performance of a production-ready tool that exploits two of the ported and optimized genomic kernels.