GitHub

Medical Akzo Nobel Work-Precision Diagrams

using OrdinaryDiffEq, DiffEqDevTools, Sundials, Plots, ODEInterfaceDiffEq, LSODA
using OrdinaryDiffEqBDF, OrdinaryDiffEqExtrapolation, OrdinaryDiffEqFIRK, OrdinaryDiffEqRosenbrock, OrdinaryDiffEqSDIRK, OrdinaryDiffEqStabilizedRK
using SciMLLogging
using LinearAlgebra, RecursiveFactorization, SparseArrays, Polyester, OrdinaryDiffEqCore
gr()

# Medical Akzo Nobel problem from the IVP Test Set
# Semi-discretization of 2 PDEs describing penetration of radio-labeled
# antibodies into tissue infected by a tumor (oncology application)
# ODE of dimension 400, banded Jacobian (bandwidth 2)
# Exact translation from Fortran feval subroutine at:
# http://archimede.dm.uniba.it/~testset/src/problems/medakzo.f

const k_ma = 100.0
const c_ma = 4.0

function medakzo!(f, y, p, t)
    N = length(y) ÷ 2
    dzeta = 1.0 / N
    dzeta2 = dzeta * dzeta

    if t <= 5.0
        phi = 2.0
    else
        phi = 0.0
    end

    # First grid point (j=1)
    zeta = dzeta
    tmp = (zeta - 1.0)^2 / c_ma
    alpha = 2.0 * (zeta - 1.0) * tmp / c_ma
    beta = tmp * tmp

    f[1] = (phi - 2.0*y[1] + y[3]) * beta / dzeta2 +
           alpha * (y[3] - phi) / (2.0*dzeta) - k_ma*y[1]*y[2]
    f[2] = -k_ma * y[1] * y[2]

    # Interior grid points (j=2,...,N-1)
    for j in 2:N-1
        i = 2*j - 1
        zeta = j * dzeta
        tmp = (zeta - 1.0)^2 / c_ma
        alpha = 2.0 * (zeta - 1.0) * tmp / c_ma
        beta = tmp * tmp
        f[i] = (y[i-2] - 2.0*y[i] + y[i+2]) * beta / dzeta2 +
               alpha * (y[i+2] - y[i-2]) / (2.0*dzeta) - k_ma*y[i]*y[i+1]
        f[i+1] = -k_ma * y[i] * y[i+1]
    end

    # Last grid point (j=N)
    f[2*N-1] = -k_ma * y[2*N-1] * y[2*N]
    f[2*N]   = -k_ma * y[2*N-1] * y[2*N]
end

function medakzo_jac!(dfdy, y, p, t)
    N = length(y) ÷ 2
    dzeta = 1.0 / N
    dzeta2 = dzeta * dzeta

    dfdy .= 0.0

    # First grid point
    zeta = dzeta
    tmp = (zeta - 1.0)^2 / c_ma
    alpha = 2.0 * (zeta - 1.0) * tmp / c_ma
    beta = tmp * tmp

    dfdy[1,1] = -beta * 2.0 / dzeta2 - k_ma*y[2]
    dfdy[1,3] = beta / dzeta2 + alpha / (2.0*dzeta)
    dfdy[2,2] = -k_ma * y[1]
    dfdy[2,1] = -k_ma * y[2]
    dfdy[1,2] = -k_ma * y[1]

    # Interior grid points
    for j in 2:N-1
        i = 2*j - 1
        zeta = j * dzeta
        tmp = (zeta - 1.0)^2 / c_ma
        alpha = 2.0 * (zeta - 1.0) * tmp / c_ma
        beta = tmp * tmp
        bz = beta / dzeta2
        dfdy[i, i-2] = bz - alpha / (2.0*dzeta)
        dfdy[i, i]   = -2.0*bz - k_ma*y[i+1]
        dfdy[i, i+2] = bz + alpha / (2.0*dzeta)
        dfdy[i, i+1] = -k_ma * y[i]
        i2 = 2*j
        dfdy[i2, i2-1] = -k_ma * y[i2]
        dfdy[i2, i2]   = -k_ma * y[i2-1]
    end

    # Last grid point
    dfdy[2*N-1, 2*N-1] = -k_ma * y[2*N]
    dfdy[2*N-1, 2*N]   = -k_ma * y[2*N-1]
    dfdy[2*N, 2*N-1]   = -k_ma * y[2*N]
    dfdy[2*N, 2*N]     = -k_ma * y[2*N-1]
end

N = 200
neqn = 2*N
u0 = zeros(neqn)
for j in 1:N
    u0[2*j-1] = 0.0
    u0[2*j]   = 1.0
end
tspan = (0.0, 20.0)

# The Jacobian is banded with bandwidth 2 - use sparse structure
jac_prototype = spzeros(neqn, neqn)
for i in 1:neqn
    for j in max(1, i-2):min(neqn, i+2)
        jac_prototype[i,j] = 1.0
    end
end

ff = ODEFunction{true, SciMLBase.FullSpecialize}(medakzo!; jac=medakzo_jac!,
    jac_prototype=jac_prototype)
prob = ODEProblem(ff, u0, tspan)

# The parallel extrapolation methods cannot build a W operator from an ODEFunction that
# carries an analytic *sparse* Jacobian - they raise
#     MethodError(SciMLOperators.WOperator,
#                 (ODEFunction{..., typeof(medakzo_jac!), ..., SparseMatrixCSC{Float64,Int64}, ...},))
# That is an uncaught error, not a failed solve, so it aborts the whole weave and
# MedAkzoNobel.md was never written at all (measured: the weave dies in the chunk below
# after 353 s). Give just those setups the same ODE without the analytic sparse Jacobian
# and select it with WorkPrecisionSet's `prob_choice`.
prob_dense = ODEProblem(ODEFunction{true, SciMLBase.FullSpecialize}(medakzo!), u0, tspan)
probs = [prob, prob_dense]

# Generate reference solution
sol = solve(prob, CVODE_BDF(), abstol=1/10^14, reltol=1/10^14)
test_sol = TestSolution(sol)
appxsols = [test_sol, test_sol]

abstols = 1.0 ./ 10.0 .^ (4:11)
reltols = 1.0 ./ 10.0 .^ (1:8);

The Medical Akzo Nobel problem models the penetration of radio-labeled antibodies into tissue infected by a tumor. It is a semi-discretization of 2 coupled PDEs on N=200 grid points, yielding 400 ODEs with a banded Jacobian (bandwidth 2). The solution has a discontinuity at t=5 when the source term switches off.

Reference plots show the u(x,t) and v(x,t) components at selected grid points:

  • y(79)=u(1,t): peaks around t=5, then decays
  • y(133)=u(2,t): smaller peak, later and broader
  • y(171)=u(3,t): even smaller peak
  • y(199)=u(4,t): smallest peak, latest
# Plot selected components matching reference: y(79), y(133), y(171), y(199)
p1 = plot(sol, idxs=[79], title="y(79) (=u(1,t))", legend=false)
p2 = plot(sol, idxs=[133], title="y(133) (=u(2,t))", legend=false)
p3 = plot(sol, idxs=[171], title="y(171) (=u(3,t))", legend=false)
p4 = plot(sol, idxs=[199], title="y(199) (=u(4,t))", legend=false)
plot(p1, p2, p3, p4, layout=(2,2), size=(800,600))
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# Plot v components: y(80), y(134), y(172), y(200)
p5 = plot(sol, idxs=[80], title="y(80) (=v(1,t))", legend=false)
p6 = plot(sol, idxs=[134], title="y(134) (=v(2,t))", legend=false)
p7 = plot(sol, idxs=[172], title="y(172) (=v(3,t))", legend=false)
p8 = plot(sol, idxs=[200], title="y(200) (=v(4,t))", legend=false)
plot(p5, p6, p7, p8, layout=(2,2), size=(800,600))

Omissions

The following were omitted from the tests due to convergence failures.

#sol = solve(prob,ROCK2()); # Unstable
#sol = solve(prob,ROCK4()); # Unstable

High Tolerances

abstols = 1.0 ./ 10.0 .^ (5:8)
reltols = 1.0 ./ 10.0 .^ (1:4);
setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>NordsieckBDF()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>rodas()),
          Dict(:alg=>radau()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>RadauIIA5()),
          ]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;verbose=SciMLLogging.None(),
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>Kvaerno3()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>KenCarp3()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>lsoda()),
          Dict(:alg=>radau())]
wp = WorkPrecisionSet(prob,abstols,reltols,setups; verbose=SciMLLogging.None(),
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Rosenbrock23()),
          Dict(:alg=>TRBDF2()),
          Dict(:alg=>ImplicitEulerExtrapolation(), :prob_choice=>2),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation(), :prob_choice=>2),
          Dict(:alg=>ImplicitHairerWannerExtrapolation(), :prob_choice=>2),
          Dict(:alg=>ABDF2()),
          Dict(:alg=>FBDF()),
          Dict(:alg=>NordsieckBDF()),
]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=SciMLLogging.None(),
                      save_everystep=false,appxsol=appxsols,maxiters=Int(1e5),numruns=10)
plot(wp)

Low Tolerances

This is the speed at lower tolerances, measuring what's good when accuracy is needed.

abstols = 1.0 ./ 10.0 .^ (7:12)
reltols = 1.0 ./ 10.0 .^ (4:9)

setups = [Dict(:alg=>FBDF()),
          Dict(:alg=>QNDF()),
          Dict(:alg=>NordsieckBDF()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>ddebdf()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>Rodas5P()),
          Dict(:alg=>rodas()),
          Dict(:alg=>radau()),
          Dict(:alg=>lsoda()),
]
wp = WorkPrecisionSet(prob,abstols,reltols,setups;verbose=SciMLLogging.None(),
                      save_everystep=false,appxsol=test_sol,maxiters=Int(1e5),numruns=10)
plot(wp)

setups = [Dict(:alg=>Kvaerno4()),
          Dict(:alg=>Kvaerno5()),
          Dict(:alg=>CVODE_BDF()),
          Dict(:alg=>KenCarp4()),
          Dict(:alg=>KenCarp5()),
          Dict(:alg=>Rodas4()),
          Dict(:alg=>Rodas5P()),
          Dict(:alg=>radau()),
          Dict(:alg=>ImplicitEulerExtrapolation(), :prob_choice=>2),
          Dict(:alg=>ImplicitEulerBarycentricExtrapolation(), :prob_choice=>2),
          Dict(:alg=>ImplicitHairerWannerExtrapolation(), :prob_choice=>2),
          ]
wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=SciMLLogging.None(),
                      save_everystep=false,appxsol=appxsols,maxiters=Int(1e5),numruns=10)
plot(wp)

Multithreading benchmarks with Parallel Extrapolation Methods

#Setting BLAS to one thread to measure gains
LinearAlgebra.BLAS.set_num_threads(1)

abstols = 1.0 ./ 10.0 .^ (10:12)
reltols = 1.0 ./ 10.0 .^ (7:9)

setups = [
            Dict(:alg=>CVODE_BDF()),
            Dict(:alg=>KenCarp4()),
            Dict(:alg=>Rodas4()),
            Dict(:alg=>Rodas5P()),
            Dict(:alg=>QNDF()),
            Dict(:alg=>NordsieckBDF()),
            Dict(:alg=>lsoda()),
            Dict(:alg=>radau()),
            Dict(:alg=>seulex()),
            Dict(:alg=>ImplicitEulerExtrapolation(min_order = 5, init_order = 3,threading = OrdinaryDiffEqCore.PolyesterThreads()), :prob_choice=>2),
            Dict(:alg=>ImplicitEulerExtrapolation(min_order = 5, init_order = 3,threading = false), :prob_choice=>2),
            Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 5, threading = OrdinaryDiffEqCore.PolyesterThreads()), :prob_choice=>2),
            Dict(:alg=>ImplicitEulerBarycentricExtrapolation(min_order = 5, threading = false), :prob_choice=>2),
            Dict(:alg=>ImplicitHairerWannerExtrapolation(threading = OrdinaryDiffEqCore.PolyesterThreads()), :prob_choice=>2),
            Dict(:alg=>ImplicitHairerWannerExtrapolation(threading = false), :prob_choice=>2),
            ]

solnames = ["CVODE_BDF","KenCarp4","Rodas4","Rodas5P","QNDF","NordsieckBDF","lsoda","radau","seulex","ImplEulerExtpl (threaded)", "ImplEulerExtpl (non-threaded)",
            "ImplEulerBaryExtpl (threaded)","ImplEulerBaryExtpl (non-threaded)","ImplHWExtpl (threaded)","ImplHWExtpl (non-threaded)"]

wp = WorkPrecisionSet(probs,abstols,reltols,setups; verbose=SciMLLogging.None(),
                    names = solnames,save_everystep=false,appxsol=appxsols,maxiters=Int(1e5),numruns=10)

plot(wp, title = "Implicit Methods: Medical Akzo Nobel",legend=:outertopleft,size = (1000,500),
     xticks = 10.0 .^ (-15:1:1),
     yticks = 10.0 .^ (-6:0.3:5),
     bottom_margin= 5Plots.mm)

Conclusion

The Medical Akzo Nobel problem is a 400-dimensional banded stiff ODE from oncology. The banded structure (bandwidth 2) makes it well-suited for solvers that exploit sparsity. CVODE_BDF performs very well here across all tolerance ranges. Rodas4 and Rodas5P are good alternatives.

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/StiffODE","MedAkzoNobel.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: 1 default, 0 interactive, 1 GC (on 128 virtual cores)
Environment:
  JULIA_PKG_PRECOMPILE_AUTO = 0

Package Information:

Status `~/sandbox/tmp_20260825_180339_53321/stiffode-master-full/benchmarks/StiffODE/Project.toml`
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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`

And the full manifest:

Status `~/sandbox/tmp_20260825_180339_53321/stiffode-master-full/benchmarks/StiffODE/Manifest.toml`
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  [9b13fd28] IndirectArrays v1.0.0
  [4858937d] InfiniteArrays v0.15.15
  [e1ba4f0e] Infinities v0.1.12
  [d25df0c9] Inflate v0.1.5
  [18e54dd8] IntegerMathUtils v0.1.4
  [a98d9a8b] Interpolations v0.16.3
  [8197267c] IntervalSets v0.7.14
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.6
  [c8e1da08] IterTools v1.10.0
  [82899510] IteratorInterfaceExtensions v1.0.0
  [1019f520] JLFzf v0.1.11
  [692b3bcd] JLLWrappers v1.8.0
⌅ [682c06a0] JSON v0.21.4
  [ae98c720] Jieko v0.2.1
⌃ [ccbc3e58] JumpProcesses v9.29.3
  [ba0b0d4f] Krylov v0.10.9
⌃ [2faa5264] LHLFactorization v2.2.0
  [7f56f5a3] LSODA v1.2.0
  [b964fa9f] LaTeXStrings v1.4.1
  [23fbe1c1] Latexify v0.16.12
  [10f19ff3] LayoutPointers v0.1.17
  [0e77f7df] LazilyInitializedFields v1.3.0
  [5078a376] LazyArrays v2.12.0
⌅ [1d6d02ad] LeftChildRightSiblingTrees v0.2.1
  [87fe0de2] LineSearch v0.1.16
⌃ [7ed4a6bd] LinearSolve v5.13.0
  [2ab3a3ac] LogExpFunctions v1.0.1
  [e6f89c97] LoggingExtras v1.2.0
  [bdcacae8] LoopVectorization v0.12.174
  [1914dd2f] MacroTools v0.5.16
  [d125e4d3] ManualMemory v0.1.8
  [d0879d2d] MarkdownAST v0.1.3
  [a3b82374] MatrixFactorizations v3.1.3
  [bb5d69b7] MaybeInplace v0.1.8
  [739be429] MbedTLS v1.1.10
  [442fdcdd] Measures v0.3.3
⌅ [94925ecb] MethodOfLines v0.11.19
  [e1d29d7a] Missings v1.2.0
⌃ [961ee093] ModelingToolkit v11.39.1
⌃ [7771a370] ModelingToolkitBase v1.68.0
⌃ [6bb917b9] ModelingToolkitTearing v1.20.5
  [2e0e35c7] Moshi v0.3.12
  [46d2c3a1] MuladdMacro v0.2.7
  [102ac46a] MultivariatePolynomials v0.5.19
  [ffc61752] Mustache v1.0.21
  [d8a4904e] MutableArithmetics v1.8.0
  [77ba4419] NaNMath v1.1.4
⌃ [8913a72c] NonlinearSolve v4.28.0
⌃ [be0214bd] NonlinearSolveBase v2.47.0
⌃ [5959db7a] NonlinearSolveFirstOrder v2.4.0
⌃ [9a2c21bd] NonlinearSolveQuasiNewton v1.15.1
⌃ [26075421] NonlinearSolveSpectralMethods v1.8.0
  [54ca160b] ODEInterface v0.5.2
⌅ [09606e27] ODEInterfaceDiffEq v4.1.0
  [6fe1bfb0] OffsetArrays v1.17.0
  [4d8831e6] OpenSSL v1.6.1
⌅ [bac558e1] OrderedCollections v1.8.2
⌃ [1dea7af3] OrdinaryDiffEq v7.7.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.4
⌃ [bbf590c4] OrdinaryDiffEqCore v4.15.0
⌃ [50262376] OrdinaryDiffEqDefault v2.5.0
⌃ [4302a76b] OrdinaryDiffEqDifferentiation v3.10.0
⌃ [e0540318] OrdinaryDiffEqExponentialRK v2.3.0
⌃ [becaefa8] OrdinaryDiffEqExtrapolation v2.5.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.8.0
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.9.0
⌃ [5dd0a6cf] OrdinaryDiffEqPDIRK v2.4.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.7.0
⌃ [b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.1
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.9.0
  [358294b1] OrdinaryDiffEqStabilizedRK v2.6.0
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.3
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.4.0
⌃ [a7812802] PDEBase v0.1.33
  [90014a1f] PDMats v0.11.41
⌃ [65888b18] ParameterizedFunctions v5.26.0
⌅ [d96e819e] Parameters v0.12.3
⌅ [69de0a69] Parsers v2.8.7
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.4
  [91a5bcdd] Plots v1.41.7
  [e409e4f3] PoissonRandom v0.4.13
  [f517fe37] Polyester v0.7.19
  [1d0040c9] PolyesterWeave v0.2.2
  [f27b6e38] Polynomials v4.1.1
⌃ [d236fae5] PreallocationTools v1.6.0
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.5.2
  [08abe8d2] PrettyTables v3.4.8
  [27ebfcd6] Primes v0.5.7
  [132c30aa] ProfileSVG v0.2.2
  [92933f4c] ProgressMeter v1.11.0
  [43287f4e] PtrArrays v1.4.0
  [78ab2635] PureGebal v1.1.0
  [0c0d3e7f] PureKLU v1.4.1
  [1fd47b50] QuadGK v2.11.3
⌅ [a08977f5] QuadratureRules v0.1.10
⌃ [c4ea9172] QuasiArrays v0.13.8
  [c84ed2f1] Ratios v0.4.5
  [988b38a3] ReadOnlyArrays v0.2.0
  [795d4caa] ReadOnlyDicts v1.0.1
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
  [807425ed] RecurrenceRelationships v0.2.0
⌃ [731186ca] RecursiveArrayTools v4.5.0
  [f2c3362d] RecursiveFactorization v0.2.30
  [189a3867] Reexport v1.2.2
  [2792f1a3] RegistryInstances v0.1.0
  [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 v3.0.7
⌅ [fb486d5c] RungeKutta v0.5.23
  [7e49a35a] RuntimeGeneratedFunctions v0.5.25
⌃ [9dfe8606] SCCNonlinearSolve v1.15.0
  [94e857df] SIMDTypes v0.1.0
  [476501e8] SLEEFPirates v0.6.46
  [1bc83da4] SafeTestsets v0.1.0
⌃ [0bca4576] SciMLBase v3.49.2
⌃ [31c91b34] SciMLBenchmarks v0.1.3
⌃ [19f34311] SciMLJacobianOperators v0.1.17
  [a6db7da4] SciMLLogging v2.1.0
⌃ [c0aeaf25] SciMLOperators v1.29.0
  [431bcebd] SciMLPublic v1.3.0
⌃ [53ae85a6] SciMLStructures v1.10.4
  [6c6a2e73] Scratch v1.3.0
  [efcf1570] Setfield v1.1.2
⌃ [992d4aef] Showoff v1.0.3
  [777ac1f9] SimpleBufferStream v1.2.0
⌃ [727e6d20] SimpleNonlinearSolve v2.14.0
⌅ [36b790f5] SimpleSolvers v0.9.2
  [699a6c99] SimpleTraits v0.9.6
  [a2af1166] SortingAlgorithms v1.2.3
  [bd59d7e1] SparseBandedMatrices v1.4.0
  [a57abbd0] SparseColumnPivotedQR v2.1.7
  [0a514795] SparseMatrixColorings v0.4.27
  [276daf66] SpecialFunctions v2.9.0
  [860ef19b] StableRNGs v1.0.4
  [0c0c59c1] StarAlgebras v0.3.0
⌃ [64909d44] StateSelection v1.11.0
  [aedffcd0] Static v1.4.6
  [0d7ed370] StaticArrayInterface v1.10.0
  [90137ffa] StaticArrays v1.9.19
  [1e83bf80] StaticArraysCore v1.4.4
⌃ [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.8.0
  [2913bbd2] StatsBase v0.34.13
  [4c63d2b9] StatsFuns v2.2.1
  [7792a7ef] StrideArraysCore v0.5.9
  [69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.5.0
  [09ab397b] StructArrays v0.7.3
  [c3572dad] Sundials v6.6.0
  [2efcf032] SymbolicIndexingInterface v0.3.55
⌃ [19f23fe9] SymbolicLimits v1.2.0
⌅ [d1185830] SymbolicUtils v4.45.0
⌃ [0c5d862f] Symbolics v7.36.0
  [3783bdb8] TableTraits v1.0.1
  [bd369af6] Tables v1.14.0
  [ed4db957] TaskLocalValues v0.1.3
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [8290d209] ThreadingUtilities v0.5.6
⌅ [a759f4b9] TimerOutputs v0.5.29
  [c751599d] ToeplitzMatrices v0.8.5
  [3bb67fe8] TranscodingStreams v0.11.3
  [d5829a12] TriangularSolve v0.2.6
  [781d530d] TruncatedStacktraces v1.4.0
  [5c2747f8] URIs v1.7.0
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [41fe7b60] Unzip v0.2.0
  [3d5dd08c] VectorizationBase v0.21.74
  [33b4df10] VectorizedRNG v0.2.26
  [81def892] VersionParsing v1.3.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
  [efce3f68] WoodburyMatrices v1.1.0
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [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 v2.8.3+0
⌅ [b22a6f82] FFMPEG_jll v8.1.2+0
  [f5851436] FFTW_jll v3.3.12+0
  [34b6f7d7] FastTransforms_jll v0.6.4+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 v3.4.1+1
⌅ [d2c73de3] GR_jll v0.73.26+0
⌅ [b0724c58] GettextRuntime_jll v0.22.4+0
  [61579ee1] Ghostscript_jll v9.55.1+0
  [020c3dae] Git_LFS_jll v3.7.1+0
  [f8c6e375] Git_jll v2.55.0+0
  [7746bdde] Glib_jll v2.88.3+0
  [3b182d85] Graphite2_jll v1.3.16+0
⌅ [2e76f6c2] HarfBuzz_jll v8.5.1+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 v4.1.0+0
  [1d63c593] LLVMOpenMP_jll v22.1.7+0
  [aae0fff6] LSODA_jll v0.1.2+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
  [c771fb93] ODEInterface_jll v0.0.2+0
  [e7412a2a] Ogg_jll v1.3.6+0
  [656ef2d0] OpenBLAS32_jll v0.3.34+0
  [9bd350c2] OpenSSH_jll v10.5.1+0
⌃ [458c3c95] OpenSSL_jll v3.5.7+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [91d4177d] Opus_jll v1.6.1+0
⌃ [36c8627f] Pango_jll v1.58.0+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
  [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 v5.8.3+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
  [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 v3.14.1+0
⌃ [0ac62f75] libass_jll v0.17.4+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
  [a9144af2] libsodium_jll v1.0.21+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
  [9abbd945] Profile v1.11.0
  [3fa0cd96] REPL v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [1a1011a3] SharedArrays 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
  [781609d7] GMP_jll v6.3.0+0
  [deac9b47] LibCURL_jll v8.6.0+0
  [e37daf67] LibGit2_jll v1.7.2+0
  [29816b5a] LibSSH2_jll v1.11.0+1
  [3a97d323] MPFR_jll v4.2.1+0
  [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`