Double Precision Research Articles

Dissociation temperature of gas hydrates through isenthalpic-isobaric molecular dynamics simulations.

Molecular simulations are a powerful tool to understand phenomena and obtain properties of gas hydrate systems. The direct coexistence method (DCM) in the NVT or NPT ensembles, the most commonly used method to determine hydrate dissociation temperatures, can be computationally expensive due to the need for several long simulations. Through an extensive set of simulations, we report here the details of the DCM within the NPH (isobaric-isenthalpic) ensemble, which require fewer and shorter trajectories. The dissociation pressure of methane hydrates is obtained for pressures of 4, 8, 15, 30, and 50 MPa. The values are in agreement with other literature simulations and experimental data. The results are further validated with the calculation of the enthalpy of dissociation, with a value of 50kJ/mol of methane, also in agreement with the literature. The complexity of a multiphase and multicomponent system presents challenges lacking in simpler water/ice systems. These are found to be dependent on energy conservation. The optimal set of parameters to achieve it is also reported, including a smaller time step and the use of double precision, along with an analysis of some factors that could affect the convergence of the method. Although these parameters require more computational cost, the NPH ensemble is successful in providing the dissociation temperature of gas hydrates in fewer simulations than other ensembles and with productions lasting only 500ns.

Approximate single precision floating point adder for low power applications

With an increasing demand for power-hungry data-intensive computing, design methodologies with low power consumption are increasingly gaining prominence in the industry. Most of the systems operate on critical and noncritical data both. An attempt to generate a precision result results in excessive power consumption and results in a slower system. For noncritical data, approximate computing circuits significantly reduce the circuit complexity and hence power consumption. In this paper, a novel approximate single precision floating point adder is proposed with an approximate mantissa adder. The mantissa adder is designed with three 8-bit full adder blocks. In this paper, a detailed mathematical background, and proposed design approach in terms of the circuit configuration and truth tables are discussed. Additionally, a concept of switching between exact computing and approximate computing is analysed considering an approximate carry look-ahead adder. The delay and power consumption for the exact operating mode and approximate operation mode considering varied window sizes is observed. Performance of the approximate computation is compared against exact computation and varied approximate computing approaches.

Mixed-precision pre-pivoting strategy for the LU factorization

This paper investigates the efficient application of half-precision floating-point (FP16) arithmetic on GPUs for boosting LU decompositions in double (FP64) precision. Addressing the motivation to enhance computational efficiency, we introduce two novel algorithms: Pre-Pivoted LU (PRP) and Mixed-precision Panel Factorization (MPF). Deployed in both hybrid CPU-GPU setups and native GPU-only configurations, PRP identifies pivot lists through LU decomposition computed in reduced precision and subsequently reorders matrix rows in FP64 precision before executing LU decomposition without pivoting. Two variants of PRP, namely hPRP and xPRP, are introduced, differing in their computation of pivot lists in full half-precision or mixed half-single precision. The MPF algorithm generates FP64 LU factorization while internally utilizing hPRP for panel factorization, showcasing accuracy on par with standard DGETRF but with superior speed. The study further explores auxiliary functions required for the native mode implementation of PRP variants and MPF.

Topological transport of a classical droplet in a lattice of time

AbstractThouless charge pumps are quantum mechanical devices whose operation relies on topology. They provide the means for transporting quantum matter in space lattices with a single quantum precision. Contrasting space crystals that spontaneously break a continuous spatial translation symmetry and form crystals in space, time crystals have emerged as novel states of matter that organise into time lattices and spontaneously break a discrete time translation symmetry. The utility of Thouless pumps that enable topologically protected quantised transport of electrons and neutral atoms in spatial superlattices leads to the question if corresponding devices exist for time crystals. Here we show that topological pumps can be realised for time solids by transporting droplets of a liquid forward and backward in time lattices and we measure the topological index that characterises such pumping processes. By exploiting a synthetic time dimension, classical time crystals can circumvent the quantum tunnelling that underpins Thouless charge pumps. Our results establish topological pumping through time instead of space and pave the way for applications of time crystals.Key Points Periodically driven fluid droplets can be made to bounce persistently in a gravitational field and may undergo spontaneous time translation symmetry breaking forming classical time crystals. These classical time crystals can be pumped from one time site to another, and this pumping process is characterised by a topological index. Our observations establish a classical temporal counterpart to Thouless pumping of quantum particles in space lattices.

B-043 Q-nab™ Igg kit: verification and validation of a novel cell-free quantitative neutralizing antibody assay

Abstract Background As the world exits the COVID-19 pandemic era, it is very important to assess population-level and individual seroprotectivity against multiple circulating and emerging SARS-CoV-2 strains, especially in vulnerable populations. Current serological measurements for neutralizing antibodies (NAbs) are either done using complex and time-intensive pseudovirus- or micro-neutralization assays or as a high throughput competitive ELISA assay. These assays, by design, cannot be multiplexed to measure NAb titers against multiple strains and suffer from a limited dynamic range. The Q-NAb IgG Tests, which rely on blocking the non-neutralizing antibodies (NNAbs) using a novel fusion protein, were developed and validated to quantitatively measure NAb titer against the SARS-CoV-2 strains used to develop the first two mRNA vaccines viz. ancestral and BA.4/5 strains. Methods The Q-NAb assay measures the NAb titer in a sample by first sequestering the NNAbs using the fusion protein (made by covalently coupling SARS-CoV-2 BA.4/5 RBD and h-ACE2 protein) thereby enriching the NAbs in the sample. Variant-specific NAbs are then detected using an indirect ELISA where the wells are coated with the variant RBD and yield a signal that is directly proportional to the concentration of NAb. Clinical samples from subjects that have either had a SARS-CoV-2 infection (N=27), taken the initial vaccine series (N=123) and/or the bivalent booster (N=151) and pre-pandemic samples (N=29) were commercially sourced. Q-NAb NAb titers were measured using the ancestral and BA.4/5 kits and were compared against corresponding MN50 values using both parametric and non-parametric models. Single-site precision, low-end sensitivity, dilutional linearity, and interference screening of Q-NAb IgG Kits were evaluated in accordance with appropriate CLSI guidelines. Traceability and total uncertainty of the Q-NAb IgG Kits (Ancestral and BA.4/5) to WHO International Standard 21/340 and 21/338 respectively was established using a calibration hierarchy adapted from ISO 17511:2020. Results Both the ancestral and BA.4/5 Q-NAb IgG Test Kits correlated with MN50 titers with a Spearman correlation coefficient of 0.90 and 0.89 respectively. The total uncertainty in traceability of calibration to WHO International standards 21/340 and 21/338 was found to be less than 10% for both tests. Single site precision across the assay range for BA.4/5 kit was found to be 6-10% with a limit of quantitation of 312 IU/mL. Assays exhibited dilutional linearity across their respective dynamic range and had no significant interference from common endogenous and exogenous substances. Conclusions We have successfully validated the novel Q-NAb IgG neutralization assay platform as individual assays for measuring neutralizing titers against ancestral and BA.4/5 SARS-CoV-2 strains. The assays have expanded dynamic range, excellent analytical performance and correlation to accepted gold-standard microneutralization assays. Both assays are calibrated and traceable to WHO international standards, thereby enabling the compilation and comparison of data from multiple clinical cohorts run across multiple laboratories. The design of the Q-NAb IgG Test kit makes it amenable to multiplexing (for example, on an MSD platform) to assess neutralization titers against multiple variants at the same time. Such a multiplexed platform can be used to assess vaccine efficacy and seroprotection against multiple variants including emerging variants.

A novel technique to resolve directional ambiguity for Particle Streak Velocimetry

In this study, a novel streak direction-resolving algorithm is introduced to determine the direction of the flow field for a single image frame with multiple streaks. The streak direction was resolved by varying its intensity from 100% to 50% of the total intensity of the streak image. This technique was applied to two main types of flows parallel and Hill’s vortex flows, parallel flows were divided into further two types constant velocity parallel flow and accelerating parallel flow. The purpose of using different types of flows was to test the robustness of the algorithm with easy and complex flows. The performance of the algorithm was checked by the angular deviation between the true and least-square fitted velocity vectors. The number of correct synthetic streak directions was measured with the success rate in percentage. A high success rate means low angular deviation and a high number of velocity vectors with correct direction was obtained. In this research, four different types of image formats were considered DP (double precision), 16-bit (without noise), 16-bit (1.0% noise), 16-bit (5.0% noise), and 8-bit (without noise) and the best results were obtained for DP and 16-bit image formats. The results of the parallel flows indicated a 100% streak direction success rate, Hill’s vortex was a type of complex flow therefore, the algorithm hard to resolve some streak directions due to very low velocities (less than 0.1 px (pixel)/interval) and very high-velocity gradients (greater than 52 px/interval). The observation shows that Hill’s vortex synthetic streak images were resolved with a success rate of 92.76% which means that the majority of the synthetic streaks were resolved. Experimental analysis was also done by using the PDMS microchannel setup. Intensity variation streaks were recorded with long camera exposure time and for maintaining the intensity variation, as the LED switched on maximum illumination was started, and continuous decrement in illumination was set by switching off at 50% of the total illumination intensity. The best results were achieved with a 94% success rate. Therefore, the proposed novel approach can be used with less expensive hardware for image processing with a single image frame and is useful for multiple applications.

GPU parallel implementation of a finite volume lattice Boltzmann method for incompressible flows

This work presents a graphics processing units (GPU) parallel algorithm of a cell-centered finite volume lattice Boltzmann method (FVLBM) on unstructured meshes. In the present GPU parallel algorithm, the parallelization is performed in the physical space. To reduce the frequency of GPU memory accesses, this algorithm develops coalesced access to GPU memory. In addition, to avoid the race for resources leading to data anomalies, such as dirty read or phantom read etc., and the double counting for flux calculation, the efficient face-based data structure often used for flux calculation in cells in the central processing unit (CPU) version of FVLBM is modified into a face-based data structure used for the fluxes on all faces, followed by a cell-based loop for the final residuals in all cells. Therefore, the proposed GPU parallel algorithm does not need to use the resource lock and retains the high efficiency of the face-based data structure in the fluxes computation to enhance its’ parallel efficiency. Additionally, to demonstrate the computational efficiency of the proposed GPU parallel algorithm, various benchmark studies are performed in this work by the proposed parallel scheme on a double precision NVIDIA GeForce RTX 3090Ti GPU card, including (a) the lid-driven flow in a two-dimensional (2D) square cavity, (b) a 2D flow past a cylinder, and (c) the lid-driven flow in a three-dimensional (3D) cubic cavity. The numerical results show that the proposed GPU parallel algorithm can be as accurate as the original CPU serial scheme with 1 to 2 orders of speedup.

CHARM 2.0: Composing Heterogeneous Accelerators for Deep Learning on Versal ACAP Architecture

Dense matrix multiply (MM) serves as one of the most heavily used kernels in deep learning applications. To cope with the high computation demands of these applications, heterogeneous architectures featuring both FPGA and dedicated ASIC accelerators have emerged as promising platforms. For example, the AMD/Xilinx Versal ACAP architecture combines general-purpose CPU cores and programmable logic with AI Engine processors optimized for AI/ML. An array of 400 AI Engine processors executing at 1 GHz can provide up to 6.4 TFLOPS performance for 32-bit floating-point (FP32) data. However, machine learning models often contain both large and small MM operations. While large MM operations can be parallelized efficiently across many cores, small MM operations typically cannot. We observe that executing some small MM layers from the BERT natural language processing model on a large, monolithic MM accelerator in Versal ACAP achieved less than 5% of the theoretical peak performance. Therefore, one key question arises: How can we design accelerators to fully use the abundant computation resources under limited communication bandwidth for end-to-end applications with multiple MM layers of diverse sizes? We identify the biggest system throughput bottleneck resulting from the mismatch between the massive computation resources of one monolithic accelerator and the various MM layers of small sizes in the application. To resolve this problem, we propose the CHARM framework to compose multiple diverse MM accelerator architectures working concurrently on different layers within one application. CHARM includes analytical models that guide design space exploration to determine accelerator partitions and layer scheduling. To facilitate system designs, CHARM automatically generates code, enabling thorough onboard design verification. We deploy the CHARM framework on four different deep learning applications in FP32, INT16, and INT8 data types, including BERT, ViT, NCF, and MLP, on the AMD/Xilinx Versal ACAP VCK190 evaluation board. Our experiments show that we achieve 1.46 TFLOPS, 1.61 TFLOPS, 1.74 TFLOPS, and 2.94 TFLOPS inference throughput for BERT, ViT, NCF, and MLP in FP32 data type, respectively, which obtain 5.29 \(\times\) , 32.51 \(\times\) , 1.00 \(\times\) , and 1.00 \(\times\) throughput gains compared to one monolithic accelerator. CHARM achieves the maximum throughput of 1.91 TOPS, 1.18 TOPS, 4.06 TOPS, and 5.81 TOPS in the INT16 data type for the four applications. The maximum throughput achieved by CHARM in the INT8 data type is 3.65 TOPS, 1.28 TOPS, 10.19 TOPS, and 21.58 TOPS, respectively. We have open-sourced our tools, including detailed step-by-step guides to reproduce all the results presented in this article and to enable other users to learn and leverage CHARM framework and tools in their end-to-end systems: https://github.com/arc-research-lab/CHARM .

Numerical conformal bootstrap with analytic functionals and outer approximation

This paper explores the numerical conformal bootstrap in general spacetime dimensions through the lens of a distinct category of analytic functionals, previously employed in two-dimensional studies. We extend the application of these functionals to a more comprehensive backdrop, demonstrating their adaptability and efficacy in general spacetime dimensions above two. The bootstrap is implemented using the outer approximation methodology, with computations conducted in double precision. The crux of our study lies in comparing the performance of this category of analytic functionals with conventional derivatives at crossing symmetric points. It is worth highlighting that in our study, we identified some novel kinks in the scalar channel during the maximization of the gap in two-dimensional conformal field theory. Our numerical analysis indicates that these analytic functionals offer a superior performance, thereby revealing a potential alternative paradigm in the application of conformal bootstrap.

Single Layer Control of Nanoscale Metal-Insulator Transition at the LaAlO3/SrTiO3 Interface.

Modern quantum device fabrication often requires precisely adding and removing materials in situ at nanoscales, which is challenging for high-quality correlated oxide devices. In this work, we present a novel nanofabrication method that remotely controls the interfacial metal-insulator transition at the LaAlO3/SrTiO3 interface by selectively removing an LaAlO3 overlayer using a diamond tip. Remarkably, we observe a large force window within which single atomic layer precision of control is achievable. Our results confirm the critical thickness and charge transfer mechanism through a layer-by-layer removal process at the interface. Additionally, high-quality nanodevices, including nanochannels and single electron transistors, are successfully fabricated using this method. This nonvolatile and high-precision nanofabrication method provides a promising oxide platform for quantum engineering by harnessing the rich electron correlations at the nanoscale.

On-Edge Deployment of Vision Transformers for Medical Diagnostics Using the Kvasir-Capsule Dataset

This paper aims to explore the possibility of utilizing vision transformers (ViTs) for on-edge medical diagnostics by experimenting with the Kvasir-Capsule image classification dataset, a large-scale image dataset of gastrointestinal diseases. Quantization techniques made available through TensorFlow Lite (TFLite), including post-training float-16 (F16) quantization and quantization-aware training (QAT), are applied to achieve reductions in model size, without compromising performance. The seven ViT models selected for this study are EfficientFormerV2S2, EfficientViT_B0, EfficientViT_M4, MobileViT_V2_050, MobileViT_V2_100, MobileViT_V2_175, and RepViT_M11. Three metrics are considered when analyzing a model: (i) F1-score, (ii) model size, and (iii) performance-to-size ratio, where performance is the F1-score and size is the model size in megabytes (MB). In terms of F1-score, we show that MobileViT_V2_175 with F16 quantization outperforms all other models with an F1-score of 0.9534. On the other hand, MobileViT_V2_050 trained using QAT was scaled down to a model size of 1.70 MB, making it the smallest model amongst the variations this paper examined. MobileViT_V2_050 also achieved the highest performance-to-size ratio of 41.25. Despite preferring smaller models for latency and memory concerns, medical diagnostics cannot afford poor-performing models. We conclude that MobileViT_V2_175 with F16 quantization is our best-performing model, with a small size of 27.47 MB, providing a benchmark for lightweight models on the Kvasir-Capsule dataset.

Acceleration of Tensor-Product Operations with Tensor Cores

In this paper, we explore the acceleration of tensor product operations in finite element methods, leveraging the computational power of the NVIDIA A100 GPU Tensor Cores. We provide an accessible overview of the necessary mathematical background and discuss our implementation strategies. Our study focuses on two common programming approaches for NVIDIA Tensor Cores: the C++ Warp Matrix Functions in nvcuda::wmma and the inline Parallel Thread Execution (PTX) instructions mma.sync.aligned . A significant focus is placed on the adoption of the versatile inline PTX instructions combined with a conflict-free shared memory access pattern, a key to unlocking superior performance. When benchmarked against traditional CUDA Cores, our approach yields a remarkable 2.3-fold increase in double precision performance, achieving 8 TFLOPS/s—45% of the theoretical maximum. Furthermore, in half-precision computations, numerical experiments demonstrate a fourfold enhancement in solving the Poisson equation using the flexible GMRES (FGMRES) method, preconditioned by a multigrid method in 3D. This is achieved while maintaining the same discretization error as observed in double precision computations. These results highlight the considerable benefits of using Tensor Cores for finite element operators with tensor products, achieving an optimal balance between computational speed and precision.

On explicit ninth‐order, two‐step methods addressing y″=f(x,y)

We present a new family of ninth‐order hybrid explicit Numerov‐type methods, effectively utilizing only eight stages, for solving the special second‐order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well‐known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm.

MicroMagnetic.jl: A Julia package for micromagnetic and atomistic simulations with GPU support

Abstract MicroMagnetic.jl is an open-source Julia package for micromagnetic and atomistic simulations. Using the features of the Julia programming language, MicroMagnetic.jl supports CPU and various GPU platforms, including NVIDIA, AMD, Intel, and Apple GPUs. Moreover, MicroMagnetic.jl supports Monte Carlo simulations for atomistic models and implements the nudged-elastic-band method for energy barrier computations. With built-in support for double and single precision modes and a design allowing easy extensibility to add new features, MicroMagnetic.jl provides a versatile toolset for researchers in micromagnetics and atomistic simulations.

Development and deployment of data-driven turbulence model for three-dimensional complex configurations

Abstract In recent years, the synergy between artificial intelligence and turbulence big data has given rise to a new data-driven paradigm in turbulence research. Data-driven turbulence modeling has emerged as one of the forefront directions in fluid mechanics. Most existing studies focus on feature construction, selection, and the development of modeling frameworks, often overlooking the practical deployment and application of trained models. This paper examines the entire process from model construction to real-world deployment, using data-driven turbulence modeling for high Reynolds number flows over complex three-dimensional configurations as a case study. Key stages include data generation, input-output feature construction, model training, model compilation and optimization, deployment, and validation. We successfully implemented the entire workflow in a heterogeneous supercomputing environment and, through mixed programming techniques, integrated the resulting turbulence model into the Platform for Hybrid Engineering Simulation of Flows (PHengLEI) open-source software framework. This allowed for mixed-precision simulations, with the main equations solved in double precision and the turbulence model in half precision. The new computational framework was validated through large-scale parallel numerical simulations on grids with tens of millions of elements for three-dimensional complex configurations. The results highlight the efficiency of our model deployment, with overall computational efficiency improving by 13.35% and the turbulence model’s solution speed increasing by approximately 3.9 times. The accuracy of the computations was also confirmed, with the average relative error in the lift and drag coefficients calculated by the data-driven turbulence model within 3%. Across various computing nodes, the relative error in the computed aerodynamic coefficients remained within 1%, demonstrating the framework’s scalability. Notably, our contributions have been incorporated as a case study in the latest PHengLEI open-source project5 5 https://forge.osredm.com/PHengLEI/PHengLEI-TestCases/tree/master/Y02_ThreeD_M6_Unstruct_Branch_Ascend. .

Tuning high-level synthesis SpMV kernels in Alveo FPGAs

Sparse Matrix-Vector Multiplication (SpMV) is an essential operation in scientific and engineering fields, with applications in areas like finite element analysis, image processing, and machine learning. To address the need for faster and more energy-efficient computing, this paper investigates the acceleration of SpMV through Field-Programmable Gate Arrays (FPGAs), leveraging High-Level Synthesis (HLS) for design simplicity. Our study focuses on the AMD-Xilinx Alveo U280 FPGA, assessing the performance of the SpMV kernel from Vitis Libraries, which is the state of the art on SpMV acceleration on FPGAs. We explore kernel modifications, transition to single precision, and varying partition sizes, demonstrating the impact of these changes on execution time. Furthermore, we investigate matrix preprocessing techniques, including Reverse Cuthill-McKee (RCM) reordering and a hybrid sparse storage format, to enhance efficiency. Our findings reveal that the performance of FPGA-accelerated SpMV is influenced by matrix characteristics, by smaller partition sizes, and by specific preprocessing techniques delivering notable performance improvements. By selecting the best results from these experiments, we achieved execution time enhancements of up to 3.2×. This study advances the understanding of FPGA-accelerated SpMV, providing insights into key factors that impact performance and potential avenues for further improvement.

DESIGN AND PERFORMANCE ANALYSIS OF TERNARY LOGIC BASED ALU USING DOUBLE PRECISION FLOATING POINT

DESIGN AND PERFORMANCE ANALYSIS OF TERNARY LOGIC BASED ALU USING DOUBLE PRECISION FLOATING POINT

Mixed-precision computing in the GRIST dynamical core for weather and climate modelling

Abstract. Atmosphere modelling applications are becoming increasingly memory-bound due to the inconsistent development rates between processor speeds and memory bandwidth. In this study, we mitigate memory bottlenecks and reduce the computational load of the Global–Regional Integrated Forecast System (GRIST) dynamical core by adopting a mixed-precision computing strategy. Guided by an application of the iterative development principle, we identify the coded equation terms that are precision insensitive and modify them from double to single precision. The results show that most precision-sensitive terms are predominantly linked to pressure gradient and gravity terms, while most precision-insensitive terms are advective terms. Without using more computing resources, computational time can be saved, and the physical performance of the model is largely kept. In the standard computational test, the reference runtime of the model's dry hydrostatic core, dry nonhydrostatic core, and the tracer transport module is reduced by 24 %, 27 %, and 44 %, respectively. A series of idealized tests, real-world weather and climate modelling tests, was performed to assess the optimized model performance qualitatively and quantitatively. In particular, in the long-term coarse-resolution climate simulation, the precision-induced sensitivity can manifest at the large scale, while in the kilometre-scale weather forecast simulation, the model's sensitivity to the precision level is mainly limited to small-scale features, and the wall-clock time is reduced by 25.5 % from the double- to mixed-precision full-model simulations.

Clustering one million molecular structures on GPU within seconds.

Structure clustering is a general but time-consuming work in the study of life science. Up to now, most published tools do not support the clustering analysis on graphics processing unit (GPU) with root mean square deviation metric. In this work, we specially write codes to do the work. It supports multiple threads on multiple GPUs. To show the performance, we apply the program to a 33-residue fragment in protein Pin1 WW domain mutant. The dataset contains 1,400,000 snapshots, which are extracted from an enhanced sampling simulation and distribute widely in the conformational space. Various testing results present that our program is quite efficient. Particularly, with two NVIDIA RTX4090 GPUs and single precision data type, the clustering calculation on 1 million snapshots is completed in a few seconds (including the uploading time of data from memory to GPU and neglecting the reading time from hard disk). This is hundreds of times faster than central processing unit. Our program could be a powerful tool for fast extraction of representative states of a molecule among its thousands to millions of candidate structures.

Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs.

In recent years, a new kind of accelerated hardware has gained popularity in the artificial intelligence (AI) community which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI-hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, S-1. This factorization of S-1, written as ZZT = S-1, is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where Z is given recursively using matrix-matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a nonparametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to Z, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization.