Mackey and Glass Work-Precision Diagrams

Mackey and Glass

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 blood production that was published by M. C. Mackey and L. Glass in "Oscillation and chaos in physiological control systems", 1977, and is given by

\[\begin{equation} y'(t) = \frac{0.2y(t-14)}{1 + y(t-14)^{10}} - 0.1y(t) \end{equation}\]

using DelayDiffEq, DiffEqDevTools, Plots
using OrdinaryDiffEqLowOrderRK, OrdinaryDiffEqTsit5, OrdinaryDiffEqVerner
using OrdinaryDiffEqNonlinearSolve: NLFunctional
using DDEProblemLibrary: prob_dde_DDETST_A1 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
end

wp_verdict (generic function with 1 method)

Low order RK methods

High tolerances

First we test final errors of continuous RK methods of low order 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 (5.65e-6) < OwrenZen5 (3.53e-5) ≈ RK4 (3.96e-5) < DP5 (0.000354) ≈ BS3 (0.000368) < OwrenZen3 (0.000818) < Tsit5 (0.00211)
  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: 2 of 4 usable runs: 5.65e-6 (0.000375 s), 0.000105 (0.000237 s)
    OwrenZen5: 2 of 4 usable runs: 3.53e-5 (0.000441 s), 0.00177 (0.000148 s)
    RK4: 2 of 4 usable runs: 3.96e-5 (0.000435 s), 0.00032 (0.000272 s)
    DP5: 4 of 4 usable runs: 0.000354 (0.000126 s), 0.0179 (8.27e-5 s), 0.0605 (5.66e-5 s), 0.105 (6.4e-5 s)
    Tsit5: 3 of 4 usable runs: 0.00211 (0.000137 s), 0.00385 (7.35e-5 s), 0.0598 (7.35e-5 s)
    BS3: 1 of 4 usable runs: 0.116 (4.4e-5 s)
    OwrenZen3: none of 4 usable runs (every run is beaten)

Next we test average 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: OwrenZen4 (2.38e-6) < RK4 (1.94e-5) < OwrenZen5 (2.47e-5) < BS3 (0.000236) < OwrenZen3 (0.00043) ≈ DP5 (0.000489) < Tsit5 (0.00123)
  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: 3 of 4 usable runs: 2.38e-6 (0.000392 s), 9.36e-5 (0.000279 s), 0.00477 (0.000124 s)
    RK4: 2 of 4 usable runs: 1.94e-5 (0.000453 s), 0.000133 (0.000246 s)
    OwrenZen5: 2 of 4 usable runs: 2.47e-5 (0.00043 s), 0.000164 (0.000271 s)
    DP5: 3 of 4 usable runs: 0.000489 (0.000118 s), 0.0135 (7.59e-5 s), 0.04 (6.17e-5 s)
    Tsit5: 3 of 4 usable runs: 0.00562 (9.98e-5 s), 0.0763 (7.06e-5 s), 0.268 (7.07e-5 s)
    BS3: 1 of 4 usable runs: 0.279 (4.64e-5 s)
    OwrenZen3: none of 4 usable runs (every run is beaten)

Low tolerances

We repeat our tests with 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 (1.12e-9) < OwrenZen4 (1.44e-9) < RK4 (4.77e-9) < DP5 (7.9e-9) < Tsit5 (1.1e-8) < BS3 (1.99e-8) < OwrenZen3 (1.17e-7)
  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: 1.12e-9 (0.0023 s)
    DP5: 4 of 4 usable runs: 7.9e-9 (0.000627 s), 7.78e-8 (0.000401 s), 3.83e-7 (0.000266 s), 1.07e-5 (0.000183 s)
    Tsit5: 1 of 4 usable runs: 7.5e-8 (0.000382 s)
    BS3: none of 4 usable runs (every run is beaten)
    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)

And once again we also test the interpolation errors:

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 (5.69e-10) ≈ OwrenZen5 (6.6e-10) < RK4 (2.12e-9) < Tsit5 (7.18e-9) < DP5 (1.73e-8) < BS3 (2.43e-8) < OwrenZen3 (6.31e-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: 5.69e-10 (0.004 s)
    OwrenZen5: 2 of 4 usable runs: 6.6e-10 (0.00237 s), 5.54e-9 (0.00139 s)
    Tsit5: 4 of 4 usable runs: 7.18e-9 (0.000937 s), 8.84e-8 (0.000472 s), 8.33e-7 (0.000371 s), 2.5e-5 (0.000243 s)
    DP5: 4 of 4 usable runs: 1.73e-8 (0.000633 s), 1.3e-7 (0.000409 s), 1.02e-6 (0.000267 s), 2.88e-5 (0.00018 s)
    BS3: none of 4 usable runs (every run is beaten)
    RK4: none of 4 usable runs (every run is beaten)
    OwrenZen3: none of 4 usable runs (every run is beaten)

Lazy interpolants

High tolerances

We repeat our tests with the Verner methods which, in contrast to the methods above, use lazy interpolants. 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: OwrenZen4 (5.65e-6) < Vern9 (1.72e-5) < Vern7 (0.000122) < Vern6 (0.000238) < Vern8 (0.00188)
  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: 4 of 4 usable runs: 5.65e-6 (0.000375 s), 0.000105 (0.00025 s), 0.0154 (0.000129 s), 0.252 (8.96e-5 s)
    Vern9: 1 of 4 usable runs: 1.72e-5 (0.000288 s)
    Vern7: 4 of 4 usable runs: 0.000122 (0.00025 s), 0.000444 (0.000201 s), 0.000518 (0.000154 s), 0.0251 (0.000136 s)
    Vern6: none of 4 usable runs (every run is beaten)
    Vern8: none of 4 usable runs (every run is beaten)

And we obtain the following interpolation errors:

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: OwrenZen4 (2.38e-6) < Vern9 (7.2e-5) ≈ Vern7 (7.24e-5) ≈ Vern6 (7.5e-5) < Vern8 (0.00141)
  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: 4 of 4 usable runs: 2.38e-6 (0.000409 s), 9.36e-5 (0.000262 s), 0.00477 (0.000126 s), 0.0963 (8.7e-5 s)
    Vern9: 1 of 4 usable runs: 7.2e-5 (0.000397 s)
    Vern7: 4 of 4 usable runs: 7.24e-5 (0.000232 s), 0.000164 (0.000184 s), 0.0016 (0.000149 s), 0.0125 (0.000143 s)
    Vern6: none of 4 usable runs (every run is beaten)
    Vern8: none of 4 usable runs (every run is beaten)

Low tolerances

Again, 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(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: Vern6 (6.21e-10) < Vern7 (1.44e-9) ≈ OwrenZen4 (1.44e-9) ≈ Vern9 (1.56e-9) < Vern8 (8.37e-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:
    Vern6: 1 of 4 usable runs: 6.21e-10 (0.00166 s)
    Vern7: 4 of 4 usable runs: 1.44e-9 (0.000738 s), 6.28e-8 (0.000624 s), 1.16e-7 (0.000453 s), 3.06e-7 (0.000323 s)
    Vern9: 2 of 4 usable runs: 1.56e-9 (0.000862 s), 5.07e-8 (0.00068 s)
    Vern8: 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(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: Vern6 (2.6e-10) < OwrenZen4 (5.69e-10) < Vern7 (7.8e-10) < Vern9 (1.64e-9) < Vern8 (5.17e-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:
    Vern6: 1 of 4 usable runs: 2.6e-10 (0.00167 s)
    Vern7: 4 of 4 usable runs: 7.8e-10 (0.000931 s), 3.02e-8 (0.000635 s), 1.59e-7 (0.000433 s), 3.81e-6 (0.000323 s)
    Vern9: 2 of 4 usable runs: 1.64e-9 (0.000862 s), 3.44e-8 (0.000686 s)
    Vern8: none of 4 usable runs (every run is beaten)
    OwrenZen4: 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","Mackey_Glass_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`
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⌃ [f42792ee] DDEProblemLibrary v0.1.9
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⌃ [c87230d0] FFMPEG v0.4.5
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⌃ [1a297f60] FillArrays v1.17.0
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  [2ee39098] LabelledArrays v1.20.5
  [23fbe1c1] Latexify v0.16.12
⌃ [87fe0de2] LineSearch v0.1.18
⌃ [7ed4a6bd] LinearSolve v5.17.3
⌃ [2ab3a3ac] LogExpFunctions v1.0.1
  [e6f89c97] LoggingExtras v1.2.0
  [1914dd2f] MacroTools v0.5.16
  [bb5d69b7] MaybeInplace v0.1.8
  [442fdcdd] Measures v0.3.3
  [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`