# Implement saturation of DAEs output using a combination of continuous and discrete callbacks

**URL:** <https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238>\
**Category:** General Usage\
**Tags:** differentialequation\
**Created:** [October 12, 2024, 10:41pm UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238 "2024-10-12T22:41:48Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![Mouhamed](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@Mouhamed](https://discourse.julialang.org/u/Mouhamed)\
**Post date:** [October 12, 2024, 10:41pm UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/1 "2024-10-12T22:41:48Z")

</div>

Hello,

In a nutshell what I do is the following :

1- First I monitor the evolution of one output through a continuous callback and when it hits a limit value I saturate the derivative output by multiplying it by a boolean set to ` `false ` `.  
2- Through a discrete callback that becomes active aftetwards, I compute(via interpolation) at each integration step the expected value at ` `t+dt ` ` without saturation. If that value is below limit I deactivate the saturation.

My problem is that the value extrapolated when the saturation is deactivated does not evolve while it should. So I want to know if the integrator’s extrapolation is affected by the modification of the parameters it holds during a callback. Or what am I missing ?

There goes a reproducible example :

```julia
using DifferentialEquations
using Plots
#params
H = 0.0371
ΔPe = 0.05
saturated = [false]
observer = [false]
p = (H, ΔPe, saturated)
#DAE definition
function system!(du, u, p, t)
    H, ΔPe, saturated = p
    du[1] = -5.0u[1] - 1.0u[5]
    du[2] = 0.0228532u[1] - 0.0263158u[2]
    du[3] = 0.789474u[1] + 6.0u[2] -2u[3]
    du[4] = (-0.263158u[1] -2.0u[2] +1.0u[3] - 1.0u[4] ) *!saturated[1]
    du[5] = (u[4] -ΔPe)/(2*H)
end
#Continuous Callback
function condition(u, t, integrator)
    u[4] - 0.06
end
function affect!(integrator)
    observer[1] = true
    integrator.p[3] .= true
    integrator.u[4] = 0.06
end
cb1 = ContinuousCallback(condition, affect!, nothing);
#Discrete Callback
function condition_(u, t, integrator)
    Bool(observer[1])
end
function affect_!(integrator)#Check if the value is increasing without any limitation; if yes, continue to saturate
    integrator.p[3] .= false
    s_t = integrator.sol(integrator.t)
    s_tplus = integrator.sol(integrator.t+integrator.dt)#evaluate next value without saturation
    @show s_t[4], s_tplus[4]
    #
    if s_tplus[4] < 0.06 #if under limit, deactivate saturation
        observer[1] = false
    elseif s_tplus[4] >= 0.06#if still above limit, maintain saturation
        integrator.p[3] .= true
    end
end
cb2 = DiscreteCallback(condition_, affect_!)
cb = CallbackSet(cb1, cb2);
#DAE Solving
M = [1.0 0 0 0 0
     0 1.0 0 0 0
     0 0 1.0 0 0
     0 0 0 0 0
     0 0 0 0 1.0]
u0 = [0.0, 0.0, 0.0, 0.0, 0.0]
tspan = (0.0,30.0)
#with_logger(TerminalLogger()) do
prob = ODEProblem(ODEFunction(system!, mass_matrix = M),u0,tspan, p)
sol = solve(prob,Rodas5(), reltol = 1e-8, abstol = 1e-8, callback = cb)

```

This is the output from @show :

```julia
(s_t[4], s_tplus[4]) = (0.06, 0.06)
(s_t[4], s_tplus[4]) = (0.06, 0.06)
(s_t[4], s_tplus[4]) = (0.06, 0.06)
(s_t[4], s_tplus[4]) = (0.06, 0.06)

```

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [October 13, 2024, 9:14am UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/2 "2024-10-13T09:14:29Z")

</div>

> [@Mouhamed](#):
>
> So I want to know if the integrator’s extrapolation is affected by the modification of the parameters it holds during a callback.

It is not.

> [@Mouhamed](#):
>
> Or what am I missing ?

Did you mean to interpolate between tprev and t?

---

<div class="post-metadata">

**Author:** ![Mouhamed](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@Mouhamed](https://discourse.julialang.org/u/Mouhamed)\
**Post date:** [October 13, 2024, 11:13am UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/3 "2024-10-13T11:13:01Z")

</div>

> [@ChrisRackauckas](#):
>
> Did you mean to interpolate between tprev and t?

Actually I am trying to extrapolate the value at `t+dt` (i.e. beyond the current integration interval). However I wanted to do it without the saturation being activated. That would require modifying the parameters inside `integration.p` before the interpolation

---

<div class="post-metadata">

**Author:** ![Mouhamed](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@Mouhamed](https://discourse.julialang.org/u/Mouhamed)\
**Post date:** [October 13, 2024, 11:21am UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/4 "2024-10-13T11:21:44Z")

</div>

But what’s more strange is this : `du[5] = (u[4] - ΔPe)/(2*H)` and `u[4]` is constant and different from `ΔPe`. Consequently `u[5]` should follow a linear trend.  
But the results show this 👇  
 ![image](https://global.discourse-cdn.com/julialang/original/3X/a/2/a2f1732be13586d5f949d337ad65fbb09adbeeac.png)

My feeling is that somehow my actions are freezing the integrator…

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [October 21, 2024, 9:28am UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/5 "2024-10-21T09:28:15Z")

</div>

Can you open an issue?

---

<div class="post-metadata">

**Author:** ![Mouhamed](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@Mouhamed](https://discourse.julialang.org/u/Mouhamed)\
**Post date:** [October 29, 2024, 12:07pm UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/6 "2024-10-29T12:07:11Z")

</div>

Sure !!!  
Another information that could be of help (maybe…) is that when I remove the discrete callback and only consider the continuous one, I get the following error :

```julia
Warning: At t=1.722792175267728, dt was forced below floating point epsilon 2.220446049250313e-16, and step error estimate = 4041.2061747989337. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).

```

The solver seems to consider the discontinuity as an error while in the documentation it was said to be safe to modify the problem via the callbacks. Strange enough, chaining the continuous callback with the discrete one make the error vanish but the results are not what’s expected.

---

<div class="post-metadata">

**Author:** ![Mouhamed](https://avatars.discourse-cdn.com/v4/letter/m/c2a13f/32.png) [@Mouhamed](https://discourse.julialang.org/u/Mouhamed)\
**Post date:** [October 29, 2024, 1:05pm UTC](https://discourse.julialang.org/t/implement-saturation-of-daes-output-using-a-combination-of-continuous-and-discrete-callbacks/121238/7 "2024-10-29T13:05:15Z")

</div>

> [@ChrisRackauckas](#):
>
> Can you open an issue?

> <https://github.com/SciML/DifferentialEquations.jl/issues/1061>
>
> \*\*Describe the bug 🐞\*\*
> 
> Simply put, I am trying to implement a saturation feat…ure on ODEs' output by detecting a threshhold value via a continuous callback. 
> Then via a discrete callback I can check if yes or no I can deactivate the saturation. The problem is that once the continuous callback triggers and the problem is modified (i.e. activate the saturation by flipping a boolean state) two strange behaviours happen : 
> 
> \- If the discrete callback is activated, the integrator outputs erroneous results different from what the expected output of the ODE should be
> \- If the discrete callback is deactivated (it was done in order to get a hint at the problem's origin), I realize that an error is thrown and the solver seems to struggle to handle the discontinuity.
> 
> \*\*Expected behavior\*\*
> 
> \- If the discrete callback is to be activated, I would expect from the integrator correct results. Because it is easy in this example to know how the solution should evolve (detailed in Error and Stacktrace section). 
> \- If the discrete callback is to be deactivated, I would expect the discontinuity to be handled when it is caused in an \`affect\` function of the continuous callback. However an error is thrown when the problem is modified.
> 
> \*\*Minimal Reproducible Example 👇\*\*
> 
> \`\`\`julia
> using DifferentialEquations
> using Plots
> \#params
> H = 0.0371
> ΔPe = 0.05
> saturated = \[false\]
> observer = \[false\]
> p = (H, ΔPe, saturated)
> \#DAE definition
> function system!(du, u, p, t)
> H, ΔPe, saturated = p
> du\[1\] = -5.0u\[1\] - 1.0u\[5\]
> du\[2\] = 0.0228532u\[1\] - 0.0263158u\[2\]
> du\[3\] = 0.789474u\[1\] + 6.0u\[2\] -2u\[3\]
> du\[4\] = (-0.263158u\[1\] -2.0u\[2\] +1.0u\[3\] - 1.0u\[4\] ) \*!saturated\[1\]
> du\[5\] = (u\[4\] -ΔPe)/(2\*H)
> end
> \#Continuous Callback
> function condition(u, t, integrator)
> u\[4\] - 0.06
> end
> function affect!(integrator)
> observer\[1\] = true
> integrator.p\[3\] .= true
> integrator.u\[4\] = 0.06
> end
> cb1 = ContinuousCallback(condition, affect!, nothing);
> \#Discrete Callback
> function condition\_(u, t, integrator)
> Bool(observer\[1\])
> end
> function affect\_!(integrator)#Check if the value is increasing without any limitation; if yes, continue to saturate
> integrator.p\[3\] .= false
> s\_t = integrator.sol(integrator.t)
> s\_tplus = integrator.sol(integrator.t+integrator.dt)#evaluate next value without saturation
> @show s\_t\[4\], s\_tplus\[4\]
> #
> if s\_tplus\[4\] \< 0.06 #if under limit, deactivate saturation
> observer\[1\] = false
> elseif s\_tplus\[4\] \>= 0.06#if still above limit, maintain saturation
> integrator.p\[3\] .= true
> end
> end
> cb2 = DiscreteCallback(condition\_, affect\_!)
> cb = CallbackSet(cb1, cb2);
> \#DAE Solving
> M = \[1.0 0 0 0 0
> 0 1.0 0 0 0
> 0 0 1.0 0 0
> 0 0 0 0 0
> 0 0 0 0 1.0\]
> u0 = \[0.0, 0.0, 0.0, 0.0, 0.0\]
> tspan = (0.0,30.0)
> \#with\_logger(TerminalLogger()) do
> prob = ODEProblem(ODEFunction(system!, mass\_matrix = M),u0,tspan, p)
> sol = solve(prob,Rodas5(), reltol = 1e-8, abstol = 1e-8, callback = cb)
> 
> \`\`\`
> 
> \*\*Error & Stacktrace ⚠️\*\*
> 
> \- Stacktrace when discrete callback activated
> \`\`\`julia
> No error thrown but results are erroneous.
> The equation du\[5\] = (u\[4\] -ΔPe)/(2\*H) should have a linear increase because in this example u\[4\] = 0.06 after saturation through callback and ΔPe = 0.05 . However the results are static; as if the integrator "froze" on the its last value.
> \`\`\`
> 
> \- Stacktrace when discrete callback deactivated
> \`\`\`julia
> Warning: At t=1.722792175267728, dt was forced below floating point epsilon 2.220446049250313e-16, and step error estimate = 4041.2061747989337. Aborting. There is either an error in your model specification or the true solution is unstable (or the true solution can not be represented in the precision of Float64).
> \`\`\`
> 
> \*\*Environment :\*\*
> 
> - Output of \`using Pkg; Pkg.status()\`
> 
> \`\`\`julia
> Status \`C:\\Users\\lenovo\\.julia\\environments\\v1.10\\Project.toml\`
> \[0a1fb500\] BlockDiagonals v0.1.42
> ⌃ \[336ed68f\] CSV v0.10.14
> ⌃ \[717857b8\] DSP v0.7.9
> ⌅ \[a93c6f00\] DataFrames v1.6.1
> \[864edb3b\] DataStructures v0.18.20
> ⌃ \[0c46a032\] DifferentialEquations v7.13.0
> ⌃ \[31c24e10\] Distributions v0.25.110
> \[7a1cc6ca\] FFTW v1.8.0
> ⌃ \[2e9cd046\] Gurobi v1.3.0
> ⌃ \[87dc4568\] HiGHS v1.9.2
> \[7073ff75\] IJulia v1.25.0
> ⌃ \[b6b21f68\] Ipopt v1.6.6
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> ⌃ \[1dea7af3\] OrdinaryDiffEq v6.84.0
> \[14b8a8f1\] PkgTemplates v0.7.52
> ⌃ \[91a5bcdd\] Plots v1.40.4
> ⌃ \[c3e4b0f8\] Pluto v0.19.46
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> ⌃ \[bcd98974\] PowerSystems v4.3.1
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> 
> \`\`\`
> 
> - Output of \`using Pkg; Pkg.status(; mode = PKGMODE\_MANIFEST)\`
> 
> \`\`\`julia
> Status \`C:\\Users\\lenovo\\.julia\\environments\\v1.10\\Manifest.toml\`
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> ⌃ \[e6f89c97\] LoggingExtras v1.0.3
> ⌃ \[bdcacae8\] LoopVectorization v0.12.170
> \[6c6e2e6c\] MIMEs v0.1.4
> \[3da0fdf6\] MPIPreferences v0.1.11
> \[1914dd2f\] MacroTools v0.5.13
> ⌃ \[36869731\] Malt v1.1.1
> \[d125e4d3\] ManualMemory v0.1.8
> ⌃ \[b8f27783\] MathOptInterface v1.31.2
> ⌃ \[a3b82374\] MatrixFactorizations v3.0.0
> ⌃ \[bb5d69b7\] MaybeInplace v0.1.3
> \[739be429\] MbedTLS v1.1.9
> \[442fdcdd\] Measures v0.3.2
> \[e1d29d7a\] Missings v1.2.0
> ⌅ \[78c3b35d\] Mocking v0.7.9
> \[99f44e22\] MsgPack v1.2.1
> \[46d2c3a1\] MuladdMacro v0.2.4
> \[6f286f6a\] MultivariateStats v0.10.3
> \[ffc61752\] Mustache v1.0.20
> ⌃ \[d8a4904e\] MutableArithmetics v1.4.6
> \[d41bc354\] NLSolversBase v7.8.3
> \[2774e3e8\] NLsolve v4.5.1
> \[77ba4419\] NaNMath v1.0.2
> \[b8a86587\] NearestNeighbors v0.4.20
> \[2bd173c7\] NodeJS v2.0.0
> ⌃ \[8913a72c\] NonlinearSolve v3.13.1
> \[510215fc\] Observables v0.5.5
> ⌃ \[6fe1bfb0\] OffsetArrays v1.14.0
> \[4d8831e6\] OpenSSL v1.4.3
> \[429524aa\] Optim v1.9.4
> \[bac558e1\] OrderedCollections v1.6.3
> ⌃ \[1dea7af3\] OrdinaryDiffEq v6.84.0
> \[90014a1f\] PDMats v0.11.31
> \[65ce6f38\] PackageExtensionCompat v1.0.2
> \[d96e819e\] Parameters v0.12.3
> \[69de0a69\] Parsers v2.8.1
> \[b98c9c47\] Pipe v1.3.0
> \[14b8a8f1\] PkgTemplates v0.7.52
> \[ccf2f8ad\] PlotThemes v3.2.0
> ⌃ \[995b91a9\] PlotUtils v1.4.1
> ⌃ \[91a5bcdd\] Plots v1.40.4
> ⌃ \[c3e4b0f8\] Pluto v0.19.46
> ⌅ \[72656b73\] PlutoDependencyExplorer v1.0.4
> \[e409e4f3\] PoissonRandom v0.4.4
> ⌃ \[f517fe37\] Polyester v0.7.15
> ⌃ \[1d0040c9\] PolyesterWeave v0.2.1
> \[f27b6e38\] Polynomials v4.0.11
> \[2dfb63ee\] PooledArrays v1.4.3
> \[85a6dd25\] PositiveFactorizations v0.2.4
> \[dd99e9e3\] PowerFlowData v1.5.0
> ⌃ \[f00506e0\] PowerSystemCaseBuilder v1.3.6
> ⌃ \[bcd98974\] PowerSystems v4.3.1
> ⌃ \[d236fae5\] PreallocationTools v0.4.22
> \[91cefc8d\] PrecompileSignatures v3.0.3
> \[aea7be01\] PrecompileTools v1.2.1
> \[21216c6a\] Preferences v1.4.3
> ⌃ \[08abe8d2\] PrettyTables v2.3.2
> \[33c8b6b6\] ProgressLogging v0.1.4
> ⌃ \[43287f4e\] PtrArrays v1.2.0
> ⌃ \[1fd47b50\] QuadGK v2.10.1
> \[74087812\] Random123 v1.7.0
> ⌃ \[e6cf234a\] RandomNumbers v1.5.3
> \[c84ed2f1\] Ratios v0.4.5
> \[3cdcf5f2\] RecipesBase v1.3.4
> \[01d81517\] RecipesPipeline v0.6.12
> ⌃ \[731186ca\] RecursiveArrayTools v3.23.1
> \[f2c3362d\] RecursiveFactorization v0.2.23
> \[189a3867\] Reexport v1.2.2
> \[2792f1a3\] RegistryInstances v0.1.0
> \[05181044\] RelocatableFolders v1.0.1
> \[ae029012\] Requires v1.3.0
> \[ae5879a3\] ResettableStacks v1.1.1
> \[708f8203\] Richardson v1.4.2
> ⌅ \[79098fc4\] Rmath v0.7.1
> \[7e49a35a\] RuntimeGeneratedFunctions v0.5.13
> \[94e857df\] SIMDTypes v0.1.0
> ⌃ \[476501e8\] SLEEFPirates v0.6.42
> \[0aa819cd\] SQLite v1.6.1
> ⌃ \[0bca4576\] SciMLBase v2.41.3
> ⌃ \[c0aeaf25\] SciMLOperators v0.3.8
> ⌃ \[53ae85a6\] SciMLStructures v1.3.0
> \[30cb0354\] SciMLTutorials v1.0.0 \`\[https://github.com/SciML/SciMLTutorials.jl.git#master\`\](https://github.com/SciML/SciMLTutorials.jl.git#master%60)
> \[6c6a2e73\] Scratch v1.2.1
> ⌃ \[91c51154\] SentinelArrays v1.4.5
> \[efcf1570\] Setfield v1.1.1
> \[992d4aef\] Showoff v1.0.3
> ⌃ \[777ac1f9\] SimpleBufferStream v1.1.0
> ⌃ \[727e6d20\] SimpleNonlinearSolve v1.10.1
> \[699a6c99\] SimpleTraits v0.9.4
> \[ce78b400\] SimpleUnPack v1.1.0
> \[b85f4697\] SoftGlobalScope v1.1.0
> \[a2af1166\] SortingAlgorithms v1.2.1
> ⌃ \[47a9eef4\] SparseDiffTools v2.19.0
> ⌅ \[0a514795\] SparseMatrixColorings v0.3.3
> \[e56a9233\] Sparspak v0.3.9
> \[276daf66\] SpecialFunctions v2.4.0
> \[3756410d\] StableDistributions v1.0.1
> ⌅ \[aedffcd0\] Static v0.8.10
> ⌃ \[0d7ed370\] StaticArrayInterface v1.5.1
> ⌃ \[90137ffa\] StaticArrays v1.9.5
> \[1e83bf80\] StaticArraysCore v1.4.3
> \[82ae8749\] StatsAPI v1.7.0
> \[2913bbd2\] StatsBase v0.34.3
> ⌃ \[4c63d2b9\] StatsFuns v1.3.1
> \[f3b207a7\] StatsPlots v0.15.7
> ⌃ \[9672c7b4\] SteadyStateDiffEq v2.2.0
> ⌃ \[789caeaf\] StochasticDiffEq v6.65.1
> ⌃ \[7792a7ef\] StrideArraysCore v0.5.6
> \[69024149\] StringEncodings v0.3.7
> ⌅ \[892a3eda\] StringManipulation v0.3.4
> ⌃ \[856f2bd8\] StructTypes v1.10.0
> ⌃ \[c3572dad\] Sundials v4.24.0
> ⌃ \[2efcf032\] SymbolicIndexingInterface v0.3.22
> \[ab02a1b2\] TableOperations v1.2.0
> \[3783bdb8\] TableTraits v1.0.1
> \[382cd787\] TableTraitsUtils v1.0.2
> ⌃ \[bd369af6\] Tables v1.11.1
> \[62fd8b95\] TensorCore v0.1.1
> \[5d786b92\] TerminalLoggers v0.1.7
> \[8290d209\] ThreadingUtilities v0.5.2
> \[9e3dc215\] TimeSeries v0.24.2
> ⌃ \[a759f4b9\] TimerOutputs v0.5.24
> ⌅ \[3bb67fe8\] TranscodingStreams v0.10.10
> \[592b5752\] Trapz v2.0.3
> ⌃ \[d5829a12\] TriangularSolve v0.2.0
> ⌃ \[410a4b4d\] Tricks v0.1.8
> \[781d530d\] TruncatedStacktraces v1.4.0
> \[30578b45\] URIParser v0.4.1
> \[5c2747f8\] URIs v1.5.1
> \[3a884ed6\] UnPack v1.0.2
> \[1cfade01\] UnicodeFun v0.4.1
> ⌃ \[1986cc42\] Unitful v1.20.0
> ⌃ \[45397f5d\] UnitfulLatexify v1.6.3
> \[41fe7b60\] Unzip v0.2.0
> ⌃ \[3d5dd08c\] VectorizationBase v0.21.69
> \[239c3e63\] Vega v2.7.0
> \[112f6efa\] VegaLite v3.3.0
> \[81def892\] VersionParsing v1.3.0
> \[19fa3120\] VertexSafeGraphs v0.2.0
> \[ea10d353\] WeakRefStrings v1.4.2
> \[44d3d7a6\] Weave v0.10.12
> \[cc8bc4a8\] Widgets v0.6.6
> \[efce3f68\] WoodburyMatrices v1.0.0
> \[76eceee3\] WorkerUtilities v1.6.1
> ⌃ \[fdbf4ff8\] XLSX v0.10.1
> \[ddb6d928\] YAML v0.4.12
> ⌃ \[c2297ded\] ZMQ v1.2.6
> \[a5390f91\] ZipFile v0.10.1
> \[ae81ac8f\] ASL\_jll v0.1.3+0
> ⌅ \[68821587\] Arpack\_jll v3.5.1+1
> ⌃ \[0b7ba130\] Blosc\_jll v1.21.5+0
> ⌃ \[6e34b625\] Bzip2\_jll v1.0.8+1
> ⌃ \[83423d85\] Cairo\_jll v1.18.0+2
> \[2702e6a9\] EpollShim\_jll v0.0.20230411+0
> \[2e619515\] Expat\_jll v2.6.2+0
> ⌅ \[b22a6f82\] FFMPEG\_jll v4.4.4+1
> ⌃ \[f5851436\] FFTW\_jll v3.3.10+0
> \[a3f928ae\] Fontconfig\_jll v2.13.96+0
> \[d7e528f0\] FreeType2\_jll v2.13.2+0
> \[559328eb\] FriBidi\_jll v1.0.14+0
> ⌃ \[0656b61e\] GLFW\_jll v3.3.9+0
> ⌅ \[d2c73de3\] GR\_jll v0.73.6+0
> \[78b55507\] Gettext\_jll v0.21.0+0
> ⌃ \[7746bdde\] Glib\_jll v2.80.2+0
> \[3b182d85\] Graphite2\_jll v1.3.14+0
> ⌃ \[c018c7e6\] Gurobi\_jll v11.0.2+2
> \[0234f1f7\] HDF5\_jll v1.14.3+3
> ⌅ \[2e76f6c2\] HarfBuzz\_jll v2.8.1+1
> ⌅ \[8fd58aa0\] HiGHS\_jll v1.7.2+0
> ⌃ \[e33a78d0\] Hwloc\_jll v2.11.1+0
> ⌃ \[1d5cc7b8\] IntelOpenMP\_jll v2024.1.0+0
> \[9cc047cb\] Ipopt\_jll v300.1400.1600+0
> ⌃ \[aacddb02\] JpegTurbo\_jll v3.0.3+0
> \[c1c5ebd0\] LAME\_jll v3.100.2+0
> ⌅ \[88015f11\] LERC\_jll v3.0.0+1
> ⌃ \[1d63c593\] LLVMOpenMP\_jll v15.0.7+0
> ⌃ \[dd4b983a\] LZO\_jll v2.10.2+0
> ⌅ \[e9f186c6\] Libffi\_jll v3.2.2+1
> \[d4300ac3\] Libgcrypt\_jll v1.8.11+0
> \[7e76a0d4\] Libglvnd\_jll v1.6.0+0
> ⌃ \[7add5ba3\] Libgpg\_error\_jll v1.49.0+0
> ⌃ \[94ce4f54\] Libiconv\_jll v1.17.0+0
> \[4b2f31a3\] Libmount\_jll v2.40.1+0
> ⌅ \[89763e89\] Libtiff\_jll v4.5.1+1
> \[38a345b3\] Libuuid\_jll v2.40.1+0
> ⌃ \[5ced341a\] Lz4\_jll v1.10.0+0
> \[d00139f3\] METIS\_jll v5.1.2+0
> ⌃ \[856f044c\] MKL\_jll v2024.1.0+0
> ⌃ \[7cb0a576\] MPICH\_jll v4.2.2+0
> ⌃ \[f1f71cc9\] MPItrampoline\_jll v5.4.0+0
> ⌅ \[d7ed1dd3\] MUMPS\_seq\_jll v500.700.300+0
> \[9237b28f\] MicrosoftMPI\_jll v10.1.4+2
> \[e7412a2a\] Ogg\_jll v1.3.5+1
> ⌅ \[656ef2d0\] OpenBLAS32\_jll v0.3.24+0
> ⌅ \[fe0851c0\] OpenMPI\_jll v4.1.6+0
> ⌃ \[458c3c95\] OpenSSL\_jll v3.0.14+0
> \[efe28fd5\] OpenSpecFun\_jll v0.5.5+0
> ⌃ \[91d4177d\] Opus\_jll v1.3.2+0
> \[30392449\] Pixman\_jll v0.43.4+0
> \[c0090381\] Qt6Base\_jll v6.7.1+1
> ⌅ \[f50d1b31\] Rmath\_jll v0.4.2+0
> \[319450e9\] SPRAL\_jll v2024.5.8+0
> \[76ed43ae\] SQLite\_jll v3.45.3+0
> ⌅ \[fb77eaff\] Sundials\_jll v5.2.2+0
> \[a44049a8\] Vulkan\_Loader\_jll v1.3.243+0
> \[a2964d1f\] Wayland\_jll v1.21.0+1
> \[2381bf8a\] Wayland\_protocols\_jll v1.31.0+0
> ⌃ \[02c8fc9c\] XML2\_jll v2.12.7+0
> ⌃ \[aed1982a\] XSLT\_jll v1.1.34+0
> ⌃ \[ffd25f8a\] XZ\_jll v5.4.6+0
> \[f67eecfb\] Xorg\_libICE\_jll v1.1.1+0
> \[c834827a\] Xorg\_libSM\_jll v1.2.4+0
> \[4f6342f7\] Xorg\_libX11\_jll v1.8.6+0
> \[0c0b7dd1\] Xorg\_libXau\_jll v1.0.11+0
> \[935fb764\] Xorg\_libXcursor\_jll v1.2.0+4
> \[a3789734\] Xorg\_libXdmcp\_jll v1.1.4+0
> \[1082639a\] Xorg\_libXext\_jll v1.3.6+0
> \[d091e8ba\] Xorg\_libXfixes\_jll v5.0.3+4
> \[a51aa0fd\] Xorg\_libXi\_jll v1.7.10+4
> \[d1454406\] Xorg\_libXinerama\_jll v1.1.4+4
> \[ec84b674\] Xorg\_libXrandr\_jll v1.5.2+4
> \[ea2f1a96\] Xorg\_libXrender\_jll v0.9.11+0
> \[14d82f49\] Xorg\_libpthread\_stubs\_jll v0.1.1+0
> \[c7cfdc94\] Xorg\_libxcb\_jll v1.17.0+0
> \[cc61e674\] Xorg\_libxkbfile\_jll v1.1.2+0
> \[e920d4aa\] Xorg\_xcb\_util\_cursor\_jll v0.1.4+0
> \[12413925\] Xorg\_xcb\_util\_image\_jll v0.4.0+1
> \[2def613f\] Xorg\_xcb\_util\_jll v0.4.0+1
> \[975044d2\] Xorg\_xcb\_util\_keysyms\_jll v0.4.0+1
> \[0d47668e\] Xorg\_xcb\_util\_renderutil\_jll v0.3.9+1
> \[c22f9ab0\] Xorg\_xcb\_util\_wm\_jll v0.4.1+1
> \[35661453\] Xorg\_xkbcomp\_jll v1.4.6+0
> \[33bec58e\] Xorg\_xkeyboard\_config\_jll v2.39.0+0
> \[c5fb5394\] Xorg\_xtrans\_jll v1.5.0+0
> ⌃ \[8f1865be\] ZeroMQ\_jll v4.3.5+0
> ⌃ \[3161d3a3\] Zstd\_jll v1.5.6+0
> \[35ca27e7\] eudev\_jll v3.2.9+0
> ⌅ \[214eeab7\] fzf\_jll v0.43.0+0
> \[1a1c6b14\] gperf\_jll v3.1.1+0
> \[477f73a3\] libaec\_jll v1.1.2+0
> \[a4ae2306\] libaom\_jll v3.9.0+0
> ⌃ \[0ac62f75\] libass\_jll v0.15.1+0
> \[2db6ffa8\] libevdev\_jll v1.11.0+0
> ⌃ \[f638f0a6\] libfdk\_aac\_jll v2.0.2+0
> \[36db933b\] libinput\_jll v1.18.0+0
> ⌃ \[b53b4c65\] libpng\_jll v1.6.43+1
> ⌃ \[a9144af2\] libsodium\_jll v1.0.20+0
> ⌃ \[f27f6e37\] libvorbis\_jll v1.3.7+1
> \[009596ad\] mtdev\_jll v1.1.6+0
> \[1317d2d5\] oneTBB\_jll v2021.12.0+0
> ⌅ \[1270edf5\] x264\_jll v2021.5.5+0
> ⌅ \[dfaa095f\] x265\_jll v3.5.0+0
> \[d8fb68d0\] xkbcommon\_jll v1.4.1+1
> \[0dad84c5\] ArgTools v1.1.1
> \[56f22d72\] Artifacts
> \[2a0f44e3\] Base64
> \[ade2ca70\] Dates
> \[8ba89e20\] Distributed
> \[f43a241f\] Downloads v1.6.0
> \[7b1f6079\] FileWatching
> \[9fa8497b\] Future
> \[b77e0a4c\] InteractiveUtils
> \[4af54fe1\] LazyArtifacts
> \[b27032c2\] LibCURL v0.6.4
> \[76f85450\] LibGit2
> \[8f399da3\] Libdl
> \[37e2e46d\] LinearAlgebra
> \[56ddb016\] Logging
> \[d6f4376e\] Markdown
> \[a63ad114\] Mmap
> \[ca575930\] NetworkOptions v1.2.0
> \[44cfe95a\] Pkg v1.10.0
> \[de0858da\] Printf
> \[9abbd945\] Profile
> \[3fa0cd96\] REPL
> \[9a3f8284\] Random
> \[ea8e919c\] SHA v0.7.0
> \[9e88b42a\] Serialization
> \[1a1011a3\] SharedArrays
> \[6462fe0b\] Sockets
> \[2f01184e\] SparseArrays v1.10.0
> \[10745b16\] Statistics v1.10.0
> \[4607b0f0\] SuiteSparse
> \[fa267f1f\] TOML v1.0.3
> \[a4e569a6\] Tar v1.10.0
> \[8dfed614\] Test
> \[cf7118a7\] UUIDs
> \[4ec0a83e\] Unicode
> \[e66e0078\] CompilerSupportLibraries\_jll v1.1.1+0
> \[deac9b47\] LibCURL\_jll v8.4.0+0
> \[e37daf67\] LibGit2\_jll v1.6.4+0
> \[29816b5a\] LibSSH2\_jll v1.11.0+1
> \[c8ffd9c3\] MbedTLS\_jll v2.28.2+1
> \[14a3606d\] MozillaCACerts\_jll v2023.1.10
> \[4536629a\] OpenBLAS\_jll v0.3.23+4
> \[05823500\] OpenLibm\_jll v0.8.1+2
> \[efcefdf7\] PCRE2\_jll v10.42.0+1
> \[bea87d4a\] SuiteSparse\_jll v7.2.1+1
> \[83775a58\] Zlib\_jll v1.2.13+1
> \[8e850b90\] libblastrampoline\_jll v5.8.0+1
> \[8e850ede\] nghttp2\_jll v1.52.0+1
> \[3f19e933\] p7zip\_jll v17.4.0+2
> 
> \`\`\`
> 
> - Output of \`versioninfo()\`
> 
> \`\`\`julia
> Julia Version 1.10.4
> Commit 48d4fd4843 (2024-06-04 10:41 UTC)
> Build Info:
> Official https://julialang.org/ release
> Platform Info:
> OS: Windows (x86\_64-w64-mingw32)
> CPU: 8 × 11th Gen Intel(R) Core(TM) i7-1185G7 @ 3.00GHz
> WORD\_SIZE: 64
> LIBM: libopenlibm
> LLVM: libLLVM-15.0.7 (ORCJIT, tigerlake)
> Threads: 1 default, 0 interactive, 1 GC (on 8 virtual cores)
> 
> \`\`\`