Wheldon, Kirk, and Finlay Work-Precision Diagrams
Wheldon, Kirk, and Finlay
We study algorithms for solving constant delay differential equations with a test problem from W.H. Enright and H. Hayashi, "The evaluation of numerical software for delay differential equations", 1997. It is a model of chronic granulocytic leukemia that was published by T. Wheldon, J. Kirk and H. Finlay in "Cyclical granulopoiesis in chronic granulocytic leukemia: A simulation study", 1974, and is given by
\[\begin{align} y_1'(t) &= \frac{1.1}{1 + \sqrt{10}y_1(t-20)^{5/4}} - \frac{10y_1(t)}{1 + 40y_2(t)} \\ y_2'(t) &= \frac{100y_1(t)}{1 + 40y_2(t)} - 2.43y_2(t) \end{align}\]
using DelayDiffEq, DiffEqDevTools, Plots
using OrdinaryDiffEqLowOrderRK, OrdinaryDiffEqTsit5, OrdinaryDiffEqVerner
using OrdinaryDiffEqNonlinearSolve: NLFunctional
using DDEProblemLibrary: prob_dde_DDETST_A2 as prob
gr()
sol = solve(prob, MethodOfSteps(Vern9(); fpsolve = NLFunctional(; max_iter = 1000));
reltol = 1e-14, abstol = 1e-14)
test_sol = TestSolution(sol)
plot(sol)
Each diagram is followed by a summary computed from its runs. It lists the tolerances at which a method produced no finite error and time (the solve failed, timed out or diverged), the smallest error each method reached, and each method's unbeaten runs with their errors and times. A run is beaten when another run on the same diagram is at least as accurate, comparing errors as printed to 3 significant digits, and more than 1.2× faster; the factor keeps timing noise from deciding a comparison, so two methods within it of each other both keep their runs. In the smallest-error list, ≈ marks an error within 1.2× of the one listed before it and < one further away.
function wp_verdict(wp; estimate = wp.error_estimate, margin = 1.2)
fmt(x) = string(round(x; sigdigits = 3))
println("Summary computed from the $estimate errors and times above:")
if !all(w -> hasproperty(w.errors, estimate), wp.wps)
println(" No $estimate errors were recorded, so nothing is compared.")
return nothing
end
runs = map(wp.wps) do w
errors = getproperty(w.errors, estimate)
bad = [i for i in eachindex(w.times) if !(isfinite(errors[i]) && isfinite(w.times[i]))]
good = setdiff(eachindex(w.times), bad)
steps = w.dts === nothing ? ("abstol", w.abstols) : ("dt", w.dts)
(; name = w.name, errors = errors[good], times = w.times[good], bad = steps[2][bad], label = steps[1])
end
allunique(r.name for r in runs) ||
println(" Note: several setups share a legend name, so their lines below cannot be told apart.")
failed = [r for r in runs if !isempty(r.bad)]
println(
" Runs without a finite error and time (failed, timed out or diverged): ",
isempty(failed) ? "none" :
join(("$(r.name) at $(r.label) $(join(fmt.(r.bad), ", "))" for r in failed), "; ")
)
ok = [r for r in runs if !isempty(r.errors)]
if length(ok) < 2
who = isempty(ok) ? "No method" : "Only $(only(ok).name)"
println(" $who produced a usable run, so nothing is compared.")
return nothing
end
best = sort!([(r.name, minimum(r.errors)) for r in ok]; by = last)
parts = String[]
for (i, (name, e)) in enumerate(best)
i > 1 && push!(parts, e <= margin * best[i - 1][2] ? "≈" : "<")
push!(parts, "$name ($(fmt(e)))")
end
println(" Smallest error reached, most accurate first: ", join(parts, " "))
points = [(e, t) for r in ok for (e, t) in zip(r.errors, r.times)]
shown(x) = round(x; sigdigits = 3)
beaten(e, t) = any(p -> shown(p[1]) <= shown(e) && margin * p[2] < t, points)
front = map(ok) do r
kept = sort!([(e, t) for (e, t) in zip(r.errors, r.times) if !beaten(e, t)]; by = first)
(; r.name, kept, n = length(r.errors))
end
sort!(front; by = f -> isempty(f.kept) ? Inf : first(f.kept[1]))
println(
" Unbeaten runs by method, as error (time); a run is beaten when another run",
" is at least as accurate (as printed) and more than $(margin)x faster:"
)
for f in front
runs_of = "of $(f.n) usable $(f.n == 1 ? "run" : "runs")"
println(
" $(f.name): ",
isempty(f.kept) ? "none $runs_of (every run is beaten)" :
"$(length(f.kept)) $runs_of: " * join(("$(fmt(e)) ($(fmt(t)) s)" for (e, t) in f.kept), ", ")
)
end
return nothing
endwp_verdict (generic function with 1 method)Low order RK methods
High tolerances
First we compare final errors of solutions with low order RK methods at high tolerances.
abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp)
wp_verdict(wp)Summary computed from the final errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: OwrenZen4 (5.21e-7) ≈ OwrenZen5 (5.74e-7) < RK4 (1.36e-6) < OwrenZen3 (1.94e-6) < DP5 (2.51e-6) < Tsit5 (3.01e-6) < BS3 (1.15e-5)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
OwrenZen4: 1 of 4 usable runs: 5.21e-7 (0.000899 s)
OwrenZen5: 1 of 4 usable runs: 5.74e-7 (0.000963 s)
RK4: 1 of 4 usable runs: 1.36e-6 (0.000671 s)
OwrenZen3: 3 of 4 usable runs: 1.94e-6 (0.000743 s), 5.61e-5 (0.00034 s), 0.000511 (0.000317 s)
DP5: 3 of 4 usable runs: 2.51e-6 (0.000419 s), 6.92e-5 (0.000336 s), 0.000508 (0.000317 s)
Tsit5: 1 of 4 usable runs: 3.01e-6 (0.000497 s)
BS3: 4 of 4 usable runs: 1.15e-5 (0.000423 s), 1.24e-5 (0.000329 s), 0.00055 (0.0003 s), 0.0119 (0.000285 s)Next we test interpolation errors:
abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp)
wp_verdict(wp)Summary computed from the L2 errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: OwrenZen5 (6.47e-6) < OwrenZen4 (1.04e-5) ≈ RK4 (1.08e-5) < OwrenZen3 (2.47e-5) < Tsit5 (3.17e-5) < DP5 (4.41e-5) < BS3 (0.000106)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
OwrenZen5: 1 of 4 usable runs: 6.47e-6 (0.000899 s)
OwrenZen4: 1 of 4 usable runs: 1.04e-5 (0.000871 s)
RK4: 3 of 4 usable runs: 1.08e-5 (0.000644 s), 6.91e-5 (0.000513 s), 0.000517 (0.000389 s)
OwrenZen3: 3 of 4 usable runs: 2.47e-5 (0.000738 s), 0.000202 (0.00041 s), 0.0017 (0.000338 s)
Tsit5: 2 of 4 usable runs: 3.17e-5 (0.000476 s), 0.000342 (0.000416 s)
DP5: 3 of 4 usable runs: 4.41e-5 (0.000523 s), 0.000381 (0.000338 s), 0.0051 (0.000338 s)
BS3: 4 of 4 usable runs: 0.000106 (0.00043 s), 0.000805 (0.000319 s), 0.00788 (0.000267 s), 0.095 (0.000272 s)Low tolerances
We repeat our tests at low tolerances.
abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)
setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp)
wp_verdict(wp)Summary computed from the final errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: OwrenZen5 (4.62e-11) < OwrenZen4 (7.5e-11) < OwrenZen3 (2.06e-10) < Tsit5 (3.33e-10) < DP5 (4.73e-10) < RK4 (1.01e-9) < BS3 (2.02e-9)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
OwrenZen5: 1 of 4 usable runs: 4.62e-11 (0.00454 s)
Tsit5: 4 of 4 usable runs: 3.33e-10 (0.00224 s), 2.92e-9 (0.00146 s), 2.87e-8 (0.00102 s), 2.1e-7 (0.000745 s)
DP5: 4 of 4 usable runs: 4.73e-10 (0.00198 s), 4.93e-9 (0.00131 s), 4.5e-8 (0.000906 s), 4.59e-7 (0.000604 s)
BS3: 1 of 4 usable runs: 1.95e-6 (0.000602 s)
RK4: none of 4 usable runs (every run is beaten)
OwrenZen3: none of 4 usable runs (every run is beaten)
OwrenZen4: none of 4 usable runs (every run is beaten)abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)
setups = [Dict(:alg=>MethodOfSteps(BS3())),
Dict(:alg=>MethodOfSteps(Tsit5())),
Dict(:alg=>MethodOfSteps(RK4())),
Dict(:alg=>MethodOfSteps(DP5())),
Dict(:alg=>MethodOfSteps(OwrenZen3())),
Dict(:alg=>MethodOfSteps(OwrenZen4())),
Dict(:alg=>MethodOfSteps(OwrenZen5()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp)
wp_verdict(wp)Summary computed from the L2 errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: OwrenZen4 (5.22e-10) ≈ OwrenZen5 (5.25e-10) < OwrenZen3 (1.88e-9) < Tsit5 (2.32e-9) ≈ DP5 (2.34e-9) < RK4 (5.88e-9) < BS3 (7.32e-8)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
OwrenZen4: 1 of 4 usable runs: 5.22e-10 (0.00745 s)
OwrenZen5: 1 of 4 usable runs: 5.25e-10 (0.00483 s)
Tsit5: 3 of 4 usable runs: 2.32e-9 (0.00241 s), 2.47e-8 (0.00189 s), 2.23e-7 (0.001 s)
DP5: 4 of 4 usable runs: 2.34e-9 (0.00198 s), 2.49e-8 (0.00137 s), 2.61e-7 (0.00112 s), 3.42e-6 (0.000666 s)
RK4: 1 of 4 usable runs: 1.62e-6 (0.000951 s)
BS3: 1 of 4 usable runs: 1.67e-5 (0.000708 s)
OwrenZen3: none of 4 usable runs (every run is beaten)Lazy interpolants
High tolerances
We compare the Verner methods, which use lazy interpolants, at high tolerances. As reference we include OwrenZen4.
abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp)
wp_verdict(wp)Summary computed from the final errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: Vern9 (2.33e-8) < Vern6 (1.68e-7) ≈ Vern7 (1.95e-7) < OwrenZen4 (5.21e-7) < Vern8 (4.0e-6)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
Vern9: 1 of 4 usable runs: 2.33e-8 (0.00191 s)
Vern6: 2 of 4 usable runs: 1.68e-7 (0.00113 s), 2.98e-6 (0.000855 s)
Vern7: 2 of 4 usable runs: 1.95e-7 (0.00105 s), 6.24e-7 (0.000757 s)
OwrenZen4: 4 of 4 usable runs: 5.21e-7 (0.000882 s), 9.07e-6 (0.000544 s), 0.000227 (0.000526 s), 0.000739 (0.000487 s)
Vern8: none of 4 usable runs (every run is beaten)abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (1:4)
setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp)
wp_verdict(wp)Summary computed from the L2 errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: Vern9 (2.23e-6) < Vern7 (2.78e-6) < OwrenZen4 (1.04e-5) < Vern8 (5.32e-5) < Vern6 (0.000101)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
Vern9: 1 of 4 usable runs: 2.23e-6 (0.00176 s)
Vern7: 2 of 4 usable runs: 2.78e-6 (0.000916 s), 9.14e-5 (0.000717 s)
OwrenZen4: 4 of 4 usable runs: 1.04e-5 (0.000777 s), 0.000104 (0.000503 s), 0.000852 (0.000403 s), 0.00971 (0.000408 s)
Vern6: none of 4 usable runs (every run is beaten)
Vern8: none of 4 usable runs (every run is beaten)Low tolerances
We repeat these tests and compare the Verner methods also at low tolerances.
abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)
setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :final)
plot(wp)
wp_verdict(wp)Summary computed from the final errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: Vern7 (6.36e-12) < Vern9 (8.0e-12) < Vern6 (3.33e-11) < OwrenZen4 (7.5e-11) < Vern8 (7.3e-10)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
Vern7: 4 of 4 usable runs: 6.36e-12 (0.00282 s), 6.45e-11 (0.0021 s), 6.73e-10 (0.00174 s), 6.82e-9 (0.00125 s)
Vern9: 1 of 4 usable runs: 1.04e-9 (0.00207 s)
OwrenZen4: 1 of 4 usable runs: 5.9e-8 (0.00123 s)
Vern6: none of 4 usable runs (every run is beaten)
Vern8: none of 4 usable runs (every run is beaten)abstols = 1.0 ./ 10.0 .^ (8:11)
reltols = 1.0 ./ 10.0 .^ (5:8)
setups = [Dict(:alg=>MethodOfSteps(Vern6())),
Dict(:alg=>MethodOfSteps(Vern7())),
Dict(:alg=>MethodOfSteps(Vern8())),
Dict(:alg=>MethodOfSteps(Vern9())),
Dict(:alg=>MethodOfSteps(OwrenZen4()))]
wp = WorkPrecisionSet(prob, abstols, reltols, setups;
appxsol = test_sol, maxiters = Int(1e5), error_estimate = :L2)
plot(wp)
wp_verdict(wp)Summary computed from the L2 errors and times above:
Runs without a finite error and time (failed, timed out or diverged): none
Smallest error reached, most accurate first: Vern7 (1.55e-10) < Vern9 (2.51e-10) < OwrenZen4 (5.22e-10) ≈ Vern6 (6.04e-10) < Vern8 (6.88e-9)
Unbeaten runs by method, as error (time); a run is beaten when another run is at least as accurate (as printed) and more than 1.2x faster:
Vern7: 4 of 4 usable runs: 1.55e-10 (0.00291 s), 1.92e-9 (0.00219 s), 2.49e-8 (0.00179 s), 2.52e-7 (0.00128 s)
Vern9: 3 of 4 usable runs: 2.64e-10 (0.00331 s), 3.62e-9 (0.00255 s), 4.59e-8 (0.0021 s)
OwrenZen4: 1 of 4 usable runs: 8.34e-7 (0.00134 s)
Vern6: none of 4 usable runs (every run is beaten)
Vern8: none of 4 usable runs (every run is beaten)Appendix
These benchmarks are a part of the SciMLBenchmarks.jl repository, found at: https://github.com/SciML/SciMLBenchmarks.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.
To locally run this benchmark, do the following commands:
using SciMLBenchmarks
SciMLBenchmarks.weave_file("benchmarks/NonStiffDDE","Wheldon_Kirk_Finlay_wpd.jmd")Computer Information:
Julia Version 1.11.9
Commit 53a02c0720c (2026-02-06 00:27 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: 128 × AMD EPYC 7502 32-Core Processor
WORD_SIZE: 64
LLVM: libLLVM-16.0.6 (ORCJIT, znver2)
Threads: 128 default, 0 interactive, 64 GC (on 128 virtual cores)
Environment:
JULIA_NUM_THREADS = auto
Package Information:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NonStiffDDE/Project.toml`
⌃ [f42792ee] DDEProblemLibrary v0.1.9
⌃ [bcd4f6db] DelayDiffEq v6.4.0
[f3b72e0c] DiffEqDevTools v3.6.3
[2ee39098] LabelledArrays v1.20.5
⌃ [1344f307] OrdinaryDiffEqLowOrderRK v2.2.5
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.8
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.4
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.4.1
[91a5bcdd] Plots v1.41.7
[31c91b34] SciMLBenchmarks v0.2.1
⌃ [90137ffa] StaticArrays v1.9.20
Info Packages marked with ⌃ have new versions available and may be upgradable.And the full manifest:
Status `~/github-runners/amdci3-1/_work/SciMLBenchmarks.jl/SciMLBenchmarks.jl/benchmarks/NonStiffDDE/Manifest.toml`
[47edcb42] ADTypes v1.24.0
[14f7f29c] AMD v0.5.4
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⌃ [79e6a3ab] Adapt v4.7.0
[66dad0bd] AliasTables v1.1.3
[4fba245c] ArrayInterface v7.30.2
[b2a6c25c] BinaryHeaps v1.1.0
⌃ [70df07ce] BracketingNonlinearSolve v1.12.7
[d360d2e6] ChainRulesCore v1.26.1
[35d6a980] ColorSchemes v3.31.0
⌃ [3da002f7] ColorTypes v0.12.1
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[34da2185] Compat v4.18.1
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[187b0558] ConstructionBase v1.6.0
[d38c429a] Contour v0.6.3
[a8cc5b0e] Crayons v4.2.0
⌃ [f42792ee] DDEProblemLibrary v0.1.9
[9a962f9c] DataAPI v1.16.0
[864edb3b] DataStructures v0.19.6
[e2d170a0] DataValueInterfaces v1.0.0
⌃ [bcd4f6db] DelayDiffEq v6.4.0
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[f3b72e0c] DiffEqDevTools v3.6.3
⌃ [77a26b50] DiffEqNoiseProcess v5.36.3
[163ba53b] DiffResults v1.1.0
[b552c78f] DiffRules v1.16.0
[a0c0ee7d] DifferentiationInterface v0.7.21
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[ffbed154] DocStringExtensions v0.9.5
[4e289a0a] EnumX v1.0.7
⌃ [f151be2c] EnzymeCore v0.8.21
[e2ba6199] ExprTools v0.1.11
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[7034ab61] FastBroadcast v1.4.0
[9aa1b823] FastClosures v0.3.2
[a4df4552] FastPower v1.5.0
⌃ [1a297f60] FillArrays v1.17.0
⌃ [64ca27bc] FindFirstFunctions v3.2.1
[6a86dc24] FiniteDiff v2.33.0
⌅ [53c48c17] FixedPointNumbers v0.8.6
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[e1d29d7a] Missings v1.2.0
[46d2c3a1] MuladdMacro v0.2.7
⌃ [ffc61752] Mustache v1.0.21
[77ba4419] NaNMath v1.1.4
⌃ [8913a72c] NonlinearSolve v4.30.0
⌃ [be0214bd] NonlinearSolveBase v2.49.5
⌃ [5959db7a] NonlinearSolveFirstOrder v2.6.1
[9a2c21bd] NonlinearSolveQuasiNewton v1.15.3
[26075421] NonlinearSolveSpectralMethods v1.8.3
⌃ [bac558e1] OrderedCollections v2.0.1
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.9
⌃ [bbf590c4] OrdinaryDiffEqCore v4.17.2
[50262376] OrdinaryDiffEqDefault v2.6.2
⌃ [4302a76b] OrdinaryDiffEqDifferentiation v3.12.0
[d3585ca7] OrdinaryDiffEqFunctionMap v2.3.0
⌃ [1344f307] OrdinaryDiffEqLowOrderRK v2.2.5
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.8
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.3
[b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.2
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.4
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.4
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.4.1
[90014a1f] PDMats v0.11.41
⌅ [69de0a69] Parsers v2.8.8
[ccf2f8ad] PlotThemes v3.3.0
⌃ [995b91a9] PlotUtils v1.4.4
[91a5bcdd] Plots v1.41.7
[e409e4f3] PoissonRandom v0.4.13
[d236fae5] PreallocationTools v1.7.1
⌅ [aea7be01] PrecompileTools v1.2.1
[21216c6a] Preferences v1.6.0
⌃ [08abe8d2] PrettyTables v3.4.8
[43287f4e] PtrArrays v1.4.0
⌃ [0c0d3e7f] PureKLU v1.5.0
[1fd47b50] QuadGK v2.11.3
⌅ [3cdcf5f2] RecipesBase v1.3.4
[01d81517] RecipesPipeline v0.6.12
⌃ [731186ca] RecursiveArrayTools v4.5.1
[189a3867] Reexport v1.2.2
[05181044] RelocatableFolders v1.0.1
[ae029012] Requires v1.3.1
[ae5879a3] ResettableStacks v1.4.0
[9fe22ead] RespecializeParams v1.3.0
[79098fc4] Rmath v0.9.0
[47965b36] RootedTrees v2.27.0
⌃ [f2b01f46] Roots v3.0.8
⌃ [7e49a35a] RuntimeGeneratedFunctions v0.5.26
⌃ [0bca4576] SciMLBase v3.54.0
[31c91b34] SciMLBenchmarks v0.2.1
[19f34311] SciMLJacobianOperators v0.1.19
[a6db7da4] SciMLLogging v2.1.0
⌃ [c0aeaf25] SciMLOperators v1.30.0
[431bcebd] SciMLPublic v1.3.0
[53ae85a6] SciMLStructures v1.10.5
[6c6a2e73] Scratch v1.3.0
[efcf1570] Setfield v1.1.2
[992d4aef] Showoff v1.1.1
⌃ [727e6d20] SimpleNonlinearSolve v2.14.5
[a2af1166] SortingAlgorithms v1.2.3
[a57abbd0] SparseColumnPivotedQR v2.1.8
[0a514795] SparseMatrixColorings v0.4.28
[276daf66] SpecialFunctions v2.9.0
[860ef19b] StableRNGs v1.0.4
⌃ [90137ffa] StaticArrays v1.9.20
[1e83bf80] StaticArraysCore v1.4.4
[10745b16] Statistics v1.11.5
[82ae8749] StatsAPI v1.8.0
[2913bbd2] StatsBase v0.34.13
[4c63d2b9] StatsFuns v2.2.1
[69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
[09ab397b] StructArrays v0.7.3
[2efcf032] SymbolicIndexingInterface v0.3.55
[3783bdb8] TableTraits v1.0.1
[bd369af6] Tables v1.14.0
[62fd8b95] TensorCore v0.1.1
⌃ [a759f4b9] TimerOutputs v1.2.1
[781d530d] TruncatedStacktraces v1.4.0
[1cfade01] UnicodeFun v0.4.1
[41fe7b60] Unzip v0.2.0
[44d3d7a6] Weave v0.10.12
⌃ [ddb6d928] YAML v0.4.16
[6e34b625] Bzip2_jll v1.0.9+0
⌃ [83423d85] Cairo_jll v1.18.7+0
[ee1fde0b] Dbus_jll v1.16.2+0
[2702e6a9] EpollShim_jll v0.0.20230411+1
[2e619515] Expat_jll v2.8.4+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
[a3f928ae] Fontconfig_jll v2.17.1+0
[d7e528f0] FreeType2_jll v2.14.3+1
[559328eb] FriBidi_jll v1.0.17+0
[0656b61e] GLFW_jll v3.5.1+0
[d2c73de3] GR_jll v0.73.27+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
[61579ee1] Ghostscript_jll v9.55.1+0
[7746bdde] Glib_jll v2.88.3+0
[3b182d85] Graphite2_jll v1.3.16+0
[2e76f6c2] HarfBuzz_jll v100.14004.0+0
[1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
[aacddb02] JpegTurbo_jll v3.2.0+1
[c1c5ebd0] LAME_jll v3.100.3+0
[88015f11] LERC_jll v4.2.0+0
[1d63c593] LLVMOpenMP_jll v23.1.1+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 v2.42.0+0
[89763e89] Libtiff_jll v4.7.3+0
[38a345b3] Libuuid_jll v2.42.0+0
[856f044c] MKL_jll v2025.2.0+0
[e7412a2a] Ogg_jll v1.3.6+0
⌃ [458c3c95] OpenSSL_jll v3.5.8+0
[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.6.1+0
[36c8627f] Pango_jll v1.58.2+0
[30392449] Pixman_jll v0.46.4+0
[c0090381] Qt6Base_jll v6.10.2+2
[629bc702] Qt6Declarative_jll v6.10.2+2
[ce943373] Qt6ShaderTools_jll v6.10.2+1
[6de9746b] Qt6Svg_jll v6.10.2+0
[e99dba38] Qt6Wayland_jll v6.10.2+1
[f50d1b31] Rmath_jll v0.5.2+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.4+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 v1.8.4+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
[a65dc6b1] Xorg_libpciaccess_jll v0.19.0+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 v2.47.0+2
[c5fb5394] Xorg_xtrans_jll v1.6.0+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 v3.14.1+0
[0ac62f75] libass_jll v0.17.5+0
[1183f4f0] libdecor_jll v0.2.2+0
[8e53e030] libdrm_jll v2.4.134+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 v1.6.58+0
[9a156e7d] libva_jll v2.23.0+0
[f27f6e37] libvorbis_jll v1.3.8+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.3.0+0
⌅ [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 v1.6.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.11.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.11.0
[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 v1.11.0
[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 v1.1.1+0
[deac9b47] LibCURL_jll v8.6.0+0
[e37daf67] LibGit2_jll v1.7.2+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.6+0
[14a3606d] MozillaCACerts_jll v2023.12.12
[4536629a] OpenBLAS_jll v0.3.27+1
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.7.0+0
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.59.0+0
[3f19e933] p7zip_jll v17.4.0+2
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`