Pleiades is a medium-dimensional, non-stiff, second-order ODE. We see that:
Code:
using LinearAlgebra, Statistics, Distributions
using DiffEqDevTools, ParameterizedFunctions, SciMLBase, OrdinaryDiffEq, Sundials, Plots, ODEInterfaceDiffEq
using ModelingToolkit
using ProbNumDiffEq
Plots.theme(
:dao;
markerstrokewidth=0.5,
legend=:outertopright,
margin=5Plots.mm,
xticks=10.0 .^ (-16:1:16),
)
function plot_chisq_interval!(df, q=0.01)
dist = Chisq(df)
low, high, mid = quantile(dist, [q, 1-q])..., mean(dist)
hline!([low, high], linestyle=:dash, color=:black, label="",
fill_between=[high nothing], fillcolor=:green, fillalpha=0.15)
hline!([mid], linestyle=:solid, color=:black, label="")
end
Code:
# first-order ODE
@fastmath function pleiades(du, u, p, t)
v = view(u, 1:7) # x
w = view(u, 8:14) # y
x = view(u, 15:21) # x′
y = view(u, 22:28) # y′
du[15:21] .= v
du[22:28] .= w
@inbounds @simd ivdep for i in 1:14
du[i] = zero(eltype(u))
end
@inbounds @simd ivdep for i in 1:7
@inbounds @simd ivdep for j in 1:7
if i != j
r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)
du[i] += j * (x[j] - x[i]) / r
du[7+i] += j * (y[j] - y[i]) / r
end
end
end
end
x0 = [3.0, 3.0, -1.0, -3.0, 2.0, -2.0, 2.0]
y0 = [3.0, -3.0, 2.0, 0, 0, -4.0, 4.0]
dx0 = [0, 0, 0, 0, 0, 1.75, -1.5]
dy0 = [0, 0, 0, -1.25, 1, 0, 0]
u0 = [dx0; dy0; x0; y0]
tspan = (0.0, 3.0)
prob1 = ODEProblem(pleiades, u0, tspan)
# second-order ODE
function pleiades2(ddu, du, u, p, t)
x = view(u, 1:7)
y = view(u, 8:14)
for i in 1:14
ddu[i] = zero(eltype(u))
end
for i in 1:7, j in 1:7
if i != j
r = ((x[i] - x[j])^2 + (y[i] - y[j])^2)^(3 / 2)
ddu[i] += j * (x[j] - x[i]) / r
ddu[7+i] += j * (y[j] - y[i]) / r
end
end
end
u0 = [x0; y0]
du0 = [dx0; dy0]
prob2 = SecondOrderODEProblem(pleiades2, du0, u0, tspan)
probs = [prob1, prob2]
ref_sol1 = solve(prob1, Vern9(), abstol=1/10^14, reltol=1/10^14)
ref_sol2 = solve(prob2, Vern9(), abstol=1/10^14, reltol=1/10^14)
ref_sols = [ref_sol1, ref_sol2]
plot(ref_sol1, idxs=[(14+i,21+i) for i in 1:7], title="Pleiades Solution", legend=false,
xticks=:auto, yticks=:auto)
scatter!(ref_sol1.u[end][15:21], ref_sol1.u[end][22:end], color=1:7)
Code:
DENSE = false;
SAVE_EVERYSTEP = false;
_setups = [
"EK0(3) (1st order ODE)" => Dict(:alg => EK0(order=3, smooth=DENSE), :prob_choice => 1)
"EK0(5) (1st order ODE)" => Dict(:alg => EK0(order=5, smooth=DENSE), :prob_choice => 1)
"EK0(4) (2nd order ODE)" => Dict(:alg => EK0(order=4, smooth=DENSE), :prob_choice => 2)
"EK0(6) (2nd order ODE)" => Dict(:alg => EK0(order=6, smooth=DENSE), :prob_choice => 2)
"EK1(3) (1st order ODE)" => Dict(:alg => EK1(order=3, smooth=DENSE), :prob_choice => 1)
"EK1(5) (1st order ODE)" => Dict(:alg => EK1(order=5, smooth=DENSE), :prob_choice => 1)
"EK1(4) (2nd order ODE)" => Dict(:alg => EK1(order=4, smooth=DENSE), :prob_choice => 2)
"EK1(6) (2nd order ODE)" => Dict(:alg => EK1(order=6, smooth=DENSE), :prob_choice => 2)
]
labels = first.(_setups)
setups = last.(_setups)
abstols = 1.0 ./ 10.0 .^ (6:11)
reltols = 1.0 ./ 10.0 .^ (3:8)
wp = WorkPrecisionSet(
probs, abstols, reltols, setups;
names = labels,
appxsol = ref_sols,
dense = DENSE,
save_everystep = SAVE_EVERYSTEP,
numruns = 5,
maxiters = Int(1e7),
)
color = [1 1 1 1 2 2 2 2]
linestyle = [:solid :solid :dash :dash :solid :solid :dash :dash]
plot(wp; color, linestyle)
_ref_setups = [
"Classic: Tsit5" => Dict(:alg => Tsit5(), :prob_choice => 1)
"Classic: RadauIIA5" => Dict(:alg => RadauIIA5(), :prob_choice => 1)
"Classic: DPRKN6" => Dict(:alg => DPRKN6(), :prob_choice => 2)
]
ref_labels = first.(_ref_setups)
ref_setups = last.(_ref_setups)
ref_wp = WorkPrecisionSet(
probs, abstols ./ 1000, reltols ./ 1000, ref_setups;
names = ref_labels,
appxsol = ref_sols,
dense = false,
save_everystep = false,
maxiters = Int(1e7),
)
plot!(ref_wp, x=:final, color=:gray, alpha=0.7, linestyle=[:solid :solid :dash])
Code:
plot(wp; x=:final, y=:chi2_final, color, linestyle)
plot_chisq_interval!(length(u0)*2)
Computer information:
using InteractiveUtils
InteractiveUtils.versioninfo()
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
Official https://julialang.org release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD Ryzen Threadripper PRO 7985WX 64-Cores
WORD_SIZE: 64
LLVM: libLLVM-18.1.7 (ORCJIT, znver4)
GC: Built with stock GC
Threads: 1 default, 1 interactive, 1 GC (on 128 virtual cores)
Environment:
LD_LIBRARY_PATH = /.singularity.d/libs
Package information:
using Pkg
Pkg.status()
Status `/home/nrbosch/.julia/dev/ProbNumDiffEq/benchmarks/Project.toml`
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[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.13.4+0
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.4.1+0
[d2c73de3] GR_jll v0.73.22+0
[b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.86.3+0
[3b182d85] Graphite2_jll v1.3.15+0
[2e76f6c2] HarfBuzz_jll v8.5.1+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.1.4+0
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.0.1+0
[1d63c593] LLVMOpenMP_jll v18.1.8+0
[aae0fff6] LSODA_jll v0.1.2+0
[dd4b983a] LZO_jll v2.10.3+0
⌅ [e9f186c6] Libffi_jll v3.4.7+0
[7e76a0d4] Libglvnd_jll v1.7.1+1
[94ce4f54] Libiconv_jll v1.18.0+0
[4b2f31a3] Libmount_jll v2.41.3+0
[89763e89] Libtiff_jll v4.7.2+0
[38a345b3] Libuuid_jll v2.41.3+0
[856f044c] MKL_jll v2025.2.0+0
[c8ffd9c3] MbedTLS_jll v2.28.1010+0
[c771fb93] ODEInterface_jll v0.0.2+0
[e7412a2a] Ogg_jll v1.3.6+0
[656ef2d0] OpenBLAS32_jll v0.3.30+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.57.0+0
⌅ [30392449] Pixman_jll v0.44.2+0
[c0090381] Qt6Base_jll v6.8.2+2
[629bc702] Qt6Declarative_jll v6.8.2+1
[ce943373] Qt6ShaderTools_jll v6.8.2+1
[e99dba38] Qt6Wayland_jll v6.8.2+2
[f50d1b31] Rmath_jll v0.5.1+0
[ca45d3f4] SuiteSparse32_jll v7.12.1+0
[fb77eaff] Sundials_jll v7.5.0+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.2+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.13+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.8+0
[d091e8ba] Xorg_libXfixes_jll v6.0.2+0
[a51aa0fd] Xorg_libXi_jll v1.8.3+0
[d1454406] Xorg_libXinerama_jll v1.1.7+0
[ec84b674] Xorg_libXrandr_jll v1.5.6+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.2.0+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.6+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[8f1865be] ZeroMQ_jll v4.3.6+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
[214eeab7] fzf_jll v0.61.1+0
[a4ae2306] libaom_jll v3.13.1+0
[0ac62f75] libass_jll v0.17.4+0
[1183f4f0] libdecor_jll v0.2.2+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.55+0
[a9144af2] libsodium_jll v1.0.21+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+1
⌅ [1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.13.0+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.7.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.12.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.3.0
[44cfe95a] Pkg v1.12.1
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.12.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.3.0+1
[deac9b47] LibCURL_jll v8.15.0+0
[e37daf67] LibGit2_jll v1.9.0+0
[29816b5a] LibSSH2_jll v1.11.3+1
[14a3606d] MozillaCACerts_jll v2025.11.4
[4536629a] OpenBLAS_jll v0.3.29+0
[05823500] OpenLibm_jll v0.8.7+0
[458c3c95] OpenSSL_jll v3.5.4+0
[efcefdf7] PCRE2_jll v10.44.0+1
[bea87d4a] SuiteSparse_jll v7.8.3+2
[83775a58] Zlib_jll v1.3.1+2
[8e850b90] libblastrampoline_jll v5.15.0+0
[8e850ede] nghttp2_jll v1.64.0+1
[3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`