[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83185-en":3,"doc-seo-83185-105":29,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83185,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Mixed Precision Explicit Numerical Methods for Ordinary Differential Equations","Objective: accelerate the numerical solution of large, nonlinear, high-dimensional ordinary differential equation (ODE) systems used to model biological processes. The study targets explicit numerical methods, whose limited stability can increase computational cost, by introducing mixed precision algorithms that execute selected parts of the method in lower arithmetic precision. Several mixed precision explicit schemes are developed and tested on two large-scale biological benchmark models. Analysis and MPI-based experiments show up to 2× speed using float/double combinations while maintaining accuracy; smaller time steps improve robustness, whereas full single precision can fail to converge.","Mixed precision explicit numerical methods for ordinary differential equations⋆  \nM. Al Sayed Alia , S. Bernardb , A. Marzoratic and J. Rouzaud-Cornabasd  \naIRMAR, Univ. Rennes, Rennes, 35000, France  \nb Université Claude Bernard Lyon 1, CNRS, École Centrale de Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, Inria, Villeurbanne, 69622, France  \ncInria, Lyon, France  \nd CITI, INSA Lyon, CNRS, Inria, LIRIS UMR5205, Université Claude Bernard Lyon 1, ECL, Université Lumière Lyon 2, Lyon, France  \n8 Jul 2026  \nARTICLE INFO  \nKeywords: Arithmetic precision Mixed precision  \nODE  \nExplicit numerical methods Sequential and parallel computing  \nAB STRACT  \nOur objective is to solve large systems of ordinary differential equations (ODEs) commonly used to model biological processes. These equations are typically nonlinear, complex, and high-dimensional. In computational biology, such ODEs are generally solved using numerical methods. In this work, we focus on explicit numerical methods because of their flexibility. However, their limited stability regions may result in high computational costs. To mitigate this issue, we investigate mixed precision algorithms designed to reduce computational effort by performing selected parts of the numerical method in lower arithmetic precision. We develop several mixed precision explicit methods and assess their performance on two largescale biological benchmark ODE models. Our theoretical analysis highlights the effectiveness of partially reducing arithmetic precision within explicit methods. Numerical experiments demonstrate that our mixed methods—implemented in both sequential and parallel versions using MPI—combining single (float) and double precision arithmetic can achieve up to twice the speed of a fully double precision implementation while preserving the same level of accuracy. Furthermore, the results indicate that decreasing the timestep improves the performance and  \nrobustness of our mixed methods, while the single precision method fails to converge.  \n1. Introduction  \nOn modern architectures, the performance of single precision (float, usually occupying 32 bits in memory) operations is often at least twice as fast as the performance of double precision (64 bits) operations ([1, 18]) . Because  \n[math .NA]  \narXiv :2607 .07080v1  \nsingle precision is limited in accuracy, double precision has become the de facto standard in scientific computing. Double precision algorithms are less susceptible to numerical instabilities at the expense of higher computational cost. Lowering the arithmetic precision could speed up computations and communications without compromising accuracy. Lower precision (e.g., single precision) algorithms may be employed; however, this typically comes at the cost of reduced accuracy or may even lead to numerical divergence due to stability issues. Mixed precision algorithms, which combine lower and higher arithmetic precisions, could therefore be used to increase performances while maintaining high accuracy. Mixed precision algorithms have become popular in numerical linear algebra [1, 2, 29], machine learning [9, 24], climate and weather model simulation [3, 10, 11, 21, 25, 27] and for numerical integration [6, 14, 15] .  \nIn this paper, we are interested in using mixed precision in explicit numerical methods for solving large systems of ordinary differential equations (ODEs) . Our approach work for all size of ODE, but we are mainly interested in the ODEs obtained from biological models that are of large size (see [12, 13, 22]). To our knowledge, few authors [23] have studied mixed precision in an explicit numerical method. All other authors [5, 6, 14, 20] have used mixed precision in implicit numerical methods. In [23], the authors analyzed the accuracy and the stability of a designed mixed precision explicit Runge-Kutta-Chebyshev (RKC). They showed that they can preserve the order 􀁰 of RKC by 􀁰 higher precision evaluations of the right-hand side (RHS) of the ","cbCaikwHffLOZuav","https://ap.wps.com/l/cbCaikwHffLOZuav","pdf",419084,1,16,"English","en",105,"# Introduction\n## Mixed precision motivation on modern architectures\n## Background and related work\n## Problem focus: explicit ODE solvers\n## Opportunities for reduced precision in explicit methods","[{\"question\":\"How do time step size and single precision affect convergence and robustness?\",\"answer\":\"Decreasing the timestep improves both performance and robustness of the mixed methods. In contrast, the single precision method fails to converge in the reported results.\"}]",1784185834,40,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"mixed-precision-explicit-numerical-methods-for-ordinary-differential-equations","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/mixed-precision-explicit-numerical-methods-for-ordinary-differential-equations/83185/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How do time step size and single precision affect convergence and robustness?","Question",{"text":75,"@type":76},"Decreasing the timestep improves both performance and robustness of the mixed methods. In contrast, the single precision method fails to converge in the reported results.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,111,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":28,"slug":110},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":112,"slug":113},30,"research-report",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":45,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]