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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Non-stiff ODE: Lotka-Volterra · ProbNumDiffEq.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">ProbNumDiffEq.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../../tutorials/getting_started/">Getting Started</a></li><li><a class="tocitem" href="../../tutorials/dynamical_odes/">Second Order ODEs and Energy Preservation</a></li><li><a class="tocitem" href="../../tutorials/dae/">Differential Algebraic Equations</a></li><li><a class="tocitem" href="../../tutorials/exponential_integrators/">Probabilistic Exponential Integrators</a></li><li><a class="tocitem" href="../../tutorials/fenrir/">Parameter Inference</a></li></ul></li><li><span class="tocitem">Solvers and Options</span><ul><li><a class="tocitem" href="../../solvers/">Solvers</a></li><li><a class="tocitem" href="../../priors/">Priors</a></li><li><a class="tocitem" href="../../initialization/">Initialization</a></li><li><a class="tocitem" href="../../diffusions/">Diffusion models and calibration</a></li></ul></li><li><span class="tocitem">Benchmarks</span><ul><li><a class="tocitem" href="../multi-language-wrappers/">Multi-Language Wrapper Benchmark</a></li><li class="is-active"><a class="tocitem" href>Non-stiff ODE: Lotka-Volterra</a><ul class="internal"><li><a class="tocitem" href="#EK0-accross-orders"><span>EK0 accross orders</span></a></li><li><a class="tocitem" href="#EK1-accross-orders"><span>EK1 accross orders</span></a></li><li><a class="tocitem" href="#EK0-vs.-EK1"><span>EK0 vs. EK1</span></a></li><li><a class="tocitem" href="#DynamicDiffusion-vs-FixedDiffusion"><span>DynamicDiffusion vs FixedDiffusion</span></a></li><li><a class="tocitem" href="#TaylorModeInit-vs-ClassicSolverInit"><span>TaylorModeInit vs ClassicSolverInit</span></a></li><li><a class="tocitem" href="#Conclusion"><span>Conclusion</span></a></li><li><a class="tocitem" href="#Appendix"><span>Appendix</span></a></li></ul></li><li><a class="tocitem" href="../vanderpol/">Stiff ODE: Van der Pol</a></li><li><a class="tocitem" href="../rober/">DAE: ROBER</a></li></ul></li><li><span class="tocitem">Internals</span><ul><li><a class="tocitem" href="../../filtering/">Filtering and Smoothing</a></li><li><a class="tocitem" href="../../implementation/">Implementation via OrdinaryDiffEq.jl</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Benchmarks</a></li><li class="is-active"><a href>Non-stiff ODE: Lotka-Volterra</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Non-stiff ODE: Lotka-Volterra</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/nathanaelbosch/ProbNumDiffEq.jl/blob/main/docs/src/benchmarks/lotkavolterra.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Lotka-Volterra-benchmark"><a class="docs-heading-anchor" href="#Lotka-Volterra-benchmark">Lotka-Volterra benchmark</a><a id="Lotka-Volterra-benchmark-1"></a><a class="docs-heading-anchor-permalink" href="#Lotka-Volterra-benchmark" title="Permalink"></a></h1><p>Adapted from <a href="https://docs.sciml.ai/SciMLBenchmarksOutput/stable/NonStiffODE/LotkaVolterra_wpd/">SciMLBenchmarks.jl</a>.</p><pre><code class="language-julia hljs">using LinearAlgebra, Statistics, InteractiveUtils
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Non-stiff ODE: Lotka-Volterra · ProbNumDiffEq.jl</title><script data-outdated-warner src="../../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.045/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.15.4/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL="../.."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../../assets/documenter.js"></script><script src="../../siteinfo.js"></script><script src="../../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../../assets/themeswap.js"></script><link href="../../assets/citations.css" rel="stylesheet" type="text/css"/></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../../">ProbNumDiffEq.jl</a></span></div><form class="docs-search" action="../../search/"><input class="docs-search-query" id="documenter-search-query" name="q" type="text" placeholder="Search docs"/></form><ul class="docs-menu"><li><a class="tocitem" href="../../">Home</a></li><li><span class="tocitem">Tutorials</span><ul><li><a class="tocitem" href="../../tutorials/getting_started/">Getting Started</a></li><li><a class="tocitem" href="../../tutorials/dynamical_odes/">Second Order ODEs and Energy Preservation</a></li><li><a class="tocitem" href="../../tutorials/dae/">Differential Algebraic Equations</a></li><li><a class="tocitem" href="../../tutorials/exponential_integrators/">Probabilistic Exponential Integrators</a></li><li><a class="tocitem" href="../../tutorials/fenrir/">Parameter Inference</a></li></ul></li><li><span class="tocitem">Solvers and Options</span><ul><li><a class="tocitem" href="../../solvers/">Solvers</a></li><li><a class="tocitem" href="../../priors/">Priors</a></li><li><a class="tocitem" href="../../initialization/">Initialization</a></li><li><a class="tocitem" href="../../diffusions/">Diffusion models and calibration</a></li></ul></li><li><span class="tocitem">Benchmarks</span><ul><li><a class="tocitem" href="../multi-language-wrappers/">Multi-Language Wrapper Benchmark</a></li><li class="is-active"><a class="tocitem" href>Non-stiff ODE: Lotka-Volterra</a><ul class="internal"><li><a class="tocitem" href="#EK0-accross-orders"><span>EK0 accross orders</span></a></li><li><a class="tocitem" href="#EK1-accross-orders"><span>EK1 accross orders</span></a></li><li><a class="tocitem" href="#EK0-vs.-EK1"><span>EK0 vs. EK1</span></a></li><li><a class="tocitem" href="#DynamicDiffusion-vs-FixedDiffusion"><span>DynamicDiffusion vs FixedDiffusion</span></a></li><li><a class="tocitem" href="#TaylorModeInit-vs-ClassicSolverInit"><span>TaylorModeInit vs ClassicSolverInit</span></a></li><li><a class="tocitem" href="#Conclusion"><span>Conclusion</span></a></li><li><a class="tocitem" href="#Appendix"><span>Appendix</span></a></li></ul></li><li><a class="tocitem" href="../vanderpol/">Stiff ODE: Van der Pol</a></li><li><a class="tocitem" href="../rober/">DAE: ROBER</a></li></ul></li><li><span class="tocitem">Internals</span><ul><li><a class="tocitem" href="../../filtering/">Filtering and Smoothing</a></li><li><a class="tocitem" href="../../implementation/">Implementation via OrdinaryDiffEq.jl</a></li></ul></li><li><a class="tocitem" href="../../references/">References</a></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Benchmarks</a></li><li class="is-active"><a href>Non-stiff ODE: Lotka-Volterra</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Non-stiff ODE: Lotka-Volterra</a></li></ul></nav><div class="docs-right"><a class="docs-edit-link" href="https://github.com/nathanaelbosch/ProbNumDiffEq.jl/blob/main/docs/src/benchmarks/lotkavolterra.md" title="Edit on GitHub"><span class="docs-icon fab"></span><span class="docs-label is-hidden-touch">Edit on GitHub</span></a><a class="docs-settings-button fas fa-cog" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-sidebar-button fa fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a></div></header><article class="content" id="documenter-page"><h1 id="Lotka-Volterra-benchmark"><a class="docs-heading-anchor" href="#Lotka-Volterra-benchmark">Lotka-Volterra benchmark</a><a id="Lotka-Volterra-benchmark-1"></a><a class="docs-heading-anchor-permalink" href="#Lotka-Volterra-benchmark" title="Permalink"></a></h1><p>Adapted from <a href="https://docs.sciml.ai/SciMLBenchmarksOutput/stable/NonStiffODE/LotkaVolterra_wpd/">SciMLBenchmarks.jl</a>.</p><pre><code class="language-julia hljs">using LinearAlgebra, Statistics, InteractiveUtils
using DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Plots
using ProbNumDiffEq

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Threads: 12 on 12 virtual cores
Environment:
JULIA_NUM_THREADS = auto
JULIA_STACKTRACE_MINIMAL = true</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi-language-wrappers/">« Multi-Language Wrapper Benchmark</a><a class="docs-footer-nextpage" href="../vanderpol/">Stiff ODE: Van der Pol »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 4 July 2023 12:40">Tuesday 4 July 2023</span>. Using Julia version 1.9.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
JULIA_STACKTRACE_MINIMAL = true</code></pre></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../multi-language-wrappers/">« Multi-Language Wrapper Benchmark</a><a class="docs-footer-nextpage" href="../vanderpol/">Stiff ODE: Van der Pol »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 1 August 2023 09:46">Tuesday 1 August 2023</span>. Using Julia version 1.9.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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