Single Pedulum Comparison

Solving single pendulums by DifferentialEquations.jl

In this notebook, we shall solve the single pendulum equation:

\[\ddot q = -\sin q,\]

where $q$ means the angle.

Hamiltonian:

\[H(q,p) = \frac{1}{2}p^2 - \cos q + 1.\]

Canonical equation:

\[\dot q = p, \quad \dot p = - \sin q.\]

Initial condition:

\[q(0) = 0, \quad p(0) = 2k.\]

Exact solution:

\[q(t) = 2\arcsin(k\,\mathrm{sn}(t,k)).\]

Maximum of $q(t)$:

\[\sin(q_{\max}/2) = k, \quad q_{\max} = \max\{q(t)\}.\]

Define $y(t)$ by

\[y(t) = \sin(q(t)/2) = k\,\mathrm{sn}(t,k), \quad y_{\max} = k.\]

# Single pendulums shall be solved numerically.
#
using OrdinaryDiffEq, Elliptic, Printf, DiffEqPhysics, Statistics

sol2q(sol) = [sol.u[i][j] for i in 1:length(sol.u), j in 1:length(sol.u[1])÷2]
sol2p(sol) = [sol.u[i][j] for i in 1:length(sol.u), j in length(sol.u[1])÷2+1:length(sol.u[1])]
sol2tqp(sol) = (sol.t, sol2q(sol), sol2p(sol))

# The exact solutions of single pendulums can be expressed by the Jacobian elliptic functions.
#
sn(u, k) = Jacobi.sn(u, k^2) # the Jacobian sn function

# Use PyPlot.
#
using PyPlot

colorlist = [
    "#1f77b4", "#ff7f0e", "#2ca02c", "#d62728", "#9467bd",
    "#8c564b", "#e377c2", "#7f7f7f", "#bcbd22", "#17becf",
]
cc(k) = colorlist[mod1(k, length(colorlist))]

# plot the sulution of a Hamiltonian problem
#
function plotsol(sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local d = size(q)[2]
    for j in 1:d
        j_str = d > 1 ? "[$j]" : ""
        plot(t, q[:,j], color=cc(2j-1), label="q$(j_str)", lw=1)
        plot(t, p[:,j], color=cc(2j),   label="p$(j_str)", lw=1, ls="--")
    end
    grid(ls=":")
    xlabel("t")
    legend()
end

# plot the solution of a Hamiltonian problem on the 2D phase space
#
function plotsol2(sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local d = size(q)[2]
    for j in 1:d
        j_str = d > 1 ? "[$j]" : ""
        plot(q[:,j], p[:,j], color=cc(j), label="(q$(j_str),p$(j_str))", lw=1)
    end
    grid(ls=":")
    xlabel("q")
    ylabel("p")
    legend()
end

# plot the energy of a Hamiltonian problem
#
function plotenergy(H, sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local energy = [H(q[i,:], p[i,:], nothing) for i in 1:size(q)[1]]
    plot(t, energy, label="energy", color="red", lw=1)
    grid(ls=":")
    xlabel("t")
    legend()
    local stdenergy_str = @sprintf("%.3e", std(energy))
    title("                    std(energy) = $stdenergy_str", fontsize=10)
end

# plot the numerical and exact solutions of a single pendulum
#
# Warning: Assume q(0) = 0, p(0) = 2k.   (for the sake of laziness)
#
function plotcomparison(k, sol::ODESolution)
    local t, q, p
    t, q, p = sol2tqp(sol)
    local y = sin.(q/2)
    local y_exact = k*sn.(t, k) # the exact solution

    plot(t, y,       label="numerical", lw=1)
    plot(t, y_exact, label="exact",     lw=1, ls="--")
    grid(ls=":")
    xlabel("t")
    ylabel("y = sin(q(t)/2)")
    legend()
    local error_str = @sprintf("%.3e", maximum(abs.(y - y_exact)))
    title("maximum(abs(numerical - exact)) = $error_str", fontsize=10)
end

# plot solution and energy
#
function plotsolenergy(H, integrator, Δt, sol::ODESolution)
    local integrator_str = replace("$integrator", r"^[^.]*\." => "")

    figure(figsize=(10,8))

    subplot2grid((21,20), ( 1, 0), rowspan=10, colspan=10)
    plotsol(sol)

    subplot2grid((21,20), ( 1,10), rowspan=10, colspan=10)
    plotsol2(sol)

    subplot2grid((21,20), (11, 0), rowspan=10, colspan=10)
    plotenergy(H, sol)

    suptitle("=====    $integrator_str,   Δt = $Δt    =====")
end

# Solve a single pendulum
#
function singlependulum(k, integrator, Δt; t0 = 0.0, t1 = 100.0)
    local H(p,q,params) = p[1]^2/2 - cos(q[1]) + 1
    local q0 = [0.0]
    local p0 = [2k]
    local prob = HamiltonianProblem(H, p0, q0, (t0, t1))

    local integrator_str = replace("$integrator", r"^[^.]*\." => "")
    @printf("%-25s", "$integrator_str:")
    sol = solve(prob, integrator, dt=Δt)
    @time local sol = solve(prob, integrator, dt=Δt)

    sleep(0.1)
    figure(figsize=(10,8))

    subplot2grid((21,20), ( 1, 0), rowspan=10, colspan=10)
    plotsol(sol)

    subplot2grid((21,20), ( 1,10), rowspan=10, colspan=10)
    plotsol2(sol)

    subplot2grid((21,20), (11, 0), rowspan=10, colspan=10)
    plotenergy(H, sol)

    subplot2grid((21,20), (11,10), rowspan=10, colspan=10)
    plotcomparison(k, sol)

    suptitle("=====    $integrator_str,   Δt = $Δt    =====")
end
Collecting package metadata (current_repodata.json): ...working... done
Solving environment: ...working... failed with initial frozen solve. Retryi
ng with flexible solve.
Solving environment: ...working... failed with repodata from current_repoda
ta.json, will retry with next repodata source.
Collecting package metadata (repodata.json): ...working... done
Solving environment: ...working... done

## Package Plan ##

  environment location: /cache/julia-buildkite-plugin/depots/5b300254-1738-
4989-ae0a-f4d2d937f953/conda/3

  added / updated specs:
    - matplotlib


The following packages will be downloaded:

    package                    |            build
    ---------------------------|-----------------
    alsa-lib-1.2.6.1           |       h7f98852_0         578 KB  conda-for
ge
    attr-2.5.1                 |       h166bdaf_1          69 KB  conda-for
ge
    brotli-1.0.9               |       h166bdaf_7          18 KB  conda-for
ge
    brotli-bin-1.0.9           |       h166bdaf_7          19 KB  conda-for
ge
    cycler-0.11.0              |     pyhd8ed1ab_0          10 KB  conda-for
ge
    dbus-1.13.6                |       h5008d03_3         604 KB  conda-for
ge
    expat-2.4.8                |       h27087fc_0         187 KB  conda-for
ge
    fftw-3.3.10                |nompi_ha7695d1_103         6.5 MB  conda-fo
rge
    font-ttf-dejavu-sans-mono-2.37|       hab24e00_0         388 KB  conda-
forge
    font-ttf-inconsolata-3.000 |       h77eed37_0          94 KB  conda-for
ge
    font-ttf-source-code-pro-2.038|       h77eed37_0         684 KB  conda-
forge
    font-ttf-ubuntu-0.83       |       hab24e00_0         1.9 MB  conda-for
ge
    fontconfig-2.14.0          |       h8e229c2_0         305 KB  conda-for
ge
    fonts-conda-ecosystem-1    |                0           4 KB  conda-for
ge
    fonts-conda-forge-1        |                0           4 KB  conda-for
ge
    fonttools-4.34.4           |   py39hb9d737c_0         1.6 MB  conda-for
ge
    freetype-2.10.4            |       hca18f0e_2         913 KB  conda-for
ge
    gettext-0.19.8.1           |    h73d1719_1008         3.6 MB  conda-for
ge
    glib-2.72.1                |       h6239696_0         443 KB  conda-for
ge
    glib-tools-2.72.1          |       h6239696_0         107 KB  conda-for
ge
    gst-plugins-base-1.20.3    |       hf6a322e_0         2.8 MB  conda-for
ge
    gstreamer-1.20.3           |       hd4edc92_0         2.0 MB  conda-for
ge
    icu-70.1                   |       h27087fc_0        13.5 MB  conda-for
ge
    jack-1.9.18                |    h8c3723f_1002         643 KB  conda-for
ge
    jpeg-9e                    |       h166bdaf_2         269 KB  conda-for
ge
    keyutils-1.6.1             |       h166bdaf_0         115 KB  conda-for
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    kiwisolver-1.4.4           |   py39hf939315_0          76 KB  conda-for
ge
    krb5-1.19.3                |       h3790be6_0         1.4 MB  conda-for
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    lcms2-2.12                 |       hddcbb42_0         443 KB  conda-for
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    lerc-4.0.0                 |       h27087fc_0         275 KB  conda-for
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    libbrotlicommon-1.0.9      |       h166bdaf_7          65 KB  conda-for
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    libbrotlidec-1.0.9         |       h166bdaf_7          33 KB  conda-for
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    libbrotlienc-1.0.9         |       h166bdaf_7         287 KB  conda-for
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    libcap-2.64                |       ha37c62d_0          96 KB  conda-for
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    libclang-14.0.6            |default_h2e3cab8_0         127 KB  conda-fo
rge
    libclang13-14.0.6          |default_h3a83d3e_0        10.6 MB  conda-fo
rge
    libcups-2.3.3              |       h3e49a29_2         4.5 MB  conda-for
ge
    libdb-6.2.32               |       h9c3ff4c_0        23.3 MB  conda-for
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    libdeflate-1.13            |       h166bdaf_0          79 KB  conda-for
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    libedit-3.1.20191231       |       he28a2e2_2         121 KB  conda-for
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    libevent-2.1.10            |       h9b69904_4         1.1 MB  conda-for
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    libflac-1.3.4              |       h27087fc_0         474 KB  conda-for
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    libglib-2.72.1             |       h2d90d5f_0         3.1 MB  conda-for
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    libiconv-1.16              |       h516909a_0         1.4 MB  conda-for
ge
    libllvm14-14.0.6           |       he0ac6c6_0        35.2 MB  conda-for
ge
    libogg-1.3.4               |       h7f98852_1         206 KB  conda-for
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    libopus-1.3.1              |       h7f98852_1         255 KB  conda-for
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    libpng-1.6.37              |       h753d276_4         371 KB  conda-for
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    libpq-14.5                 |       hd77ab85_0         3.0 MB  conda-for
ge
    libsndfile-1.0.31          |       h9c3ff4c_1         602 KB  conda-for
ge
    libtiff-4.4.0              |       h0e0dad5_3         642 KB  conda-for
ge
    libtool-2.4.6              |    h9c3ff4c_1008         511 KB  conda-for
ge
    libudev1-249               |       h166bdaf_4         109 KB  conda-for
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    libvorbis-1.3.7            |       h9c3ff4c_0         280 KB  conda-for
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    libwebp-base-1.2.4         |       h166bdaf_0         404 KB  conda-for
ge
    libxcb-1.13                |    h7f98852_1004         391 KB  conda-for
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    libxkbcommon-1.0.3         |       he3ba5ed_0         581 KB  conda-for
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    libxml2-2.9.14             |       h22db469_4         771 KB  conda-for
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    lz4-c-1.9.3                |       h9c3ff4c_1         179 KB  conda-for
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    matplotlib-3.5.2           |   py39hf3d152e_1           7 KB  conda-for
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    matplotlib-base-3.5.2      |   py39h700656a_1         7.4 MB  conda-for
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    munkres-1.1.4              |     pyh9f0ad1d_0          12 KB  conda-for
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    mysql-common-8.0.30        |       haf5c9bc_0         1.9 MB  conda-for
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    mysql-libs-8.0.30          |       h28c427c_0         1.9 MB  conda-for
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    nspr-4.32                  |       h9c3ff4c_1         233 KB  conda-for
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    nss-3.78                   |       h2350873_0         2.1 MB  conda-for
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    openjpeg-2.5.0             |       h7d73246_1         533 KB  conda-for
ge
    packaging-21.3             |     pyhd8ed1ab_0          36 KB  conda-for
ge
    pcre-8.45                  |       h9c3ff4c_0         253 KB  conda-for
ge
    pillow-9.2.0               |   py39hd5dbb17_2        44.9 MB  conda-for
ge
    ply-3.11                   |             py_1          44 KB  conda-for
ge
    portaudio-19.6.0           |       h57a0ea0_5         131 KB  conda-for
ge
    pthread-stubs-0.4          |    h36c2ea0_1001           5 KB  conda-for
ge
    pulseaudio-14.0            |       h7f54b18_8         1.7 MB  conda-for
ge
    pyparsing-3.0.9            |     pyhd8ed1ab_0          79 KB  conda-for
ge
    pyqt-5.15.7                |   py39h18e9c17_0         6.2 MB  conda-for
ge
    pyqt5-sip-12.11.0          |   py39h5a03fae_0          86 KB  conda-for
ge
    python-dateutil-2.8.2      |     pyhd8ed1ab_0         240 KB  conda-for
ge
    qt-main-5.15.4             |       ha5833f6_2        61.5 MB  conda-for
ge
    sip-6.6.2                  |   py39h5a03fae_0         495 KB  conda-for
ge
    toml-0.10.2                |     pyhd8ed1ab_0          18 KB  conda-for
ge
    tornado-6.2                |   py39hb9d737c_0         658 KB  conda-for
ge
    unicodedata2-14.0.0        |   py39hb9d737c_1         498 KB  conda-for
ge
    xcb-util-0.4.0             |       h166bdaf_0          20 KB  conda-for
ge
    xcb-util-image-0.4.0       |       h166bdaf_0          24 KB  conda-for
ge
    xcb-util-keysyms-0.4.0     |       h166bdaf_0          12 KB  conda-for
ge
    xcb-util-renderutil-0.3.9  |       h166bdaf_0          15 KB  conda-for
ge
    xcb-util-wm-0.4.1          |       h166bdaf_0          55 KB  conda-for
ge
    xorg-libxau-1.0.9          |       h7f98852_0          13 KB  conda-for
ge
    xorg-libxdmcp-1.1.3        |       h7f98852_0          19 KB  conda-for
ge
    zstd-1.5.2                 |       h8a70e8d_4         448 KB  conda-for
ge
    ------------------------------------------------------------
                                           Total:       259.5 MB

The following NEW packages will be INSTALLED:

  alsa-lib           conda-forge/linux-64::alsa-lib-1.2.6.1-h7f98852_0
  attr               conda-forge/linux-64::attr-2.5.1-h166bdaf_1
  brotli             conda-forge/linux-64::brotli-1.0.9-h166bdaf_7
  brotli-bin         conda-forge/linux-64::brotli-bin-1.0.9-h166bdaf_7
  cycler             conda-forge/noarch::cycler-0.11.0-pyhd8ed1ab_0
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  expat              conda-forge/linux-64::expat-2.4.8-h27087fc_0
  fftw               conda-forge/linux-64::fftw-3.3.10-nompi_ha7695d1_103
  font-ttf-dejavu-s~ conda-forge/noarch::font-ttf-dejavu-sans-mono-2.37-hab
24e00_0
  font-ttf-inconsol~ conda-forge/noarch::font-ttf-inconsolata-3.000-h77eed3
7_0
  font-ttf-source-c~ conda-forge/noarch::font-ttf-source-code-pro-2.038-h77
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  font-ttf-ubuntu    conda-forge/noarch::font-ttf-ubuntu-0.83-hab24e00_0
  fontconfig         conda-forge/linux-64::fontconfig-2.14.0-h8e229c2_0
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  gettext            conda-forge/linux-64::gettext-0.19.8.1-h73d1719_1008
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  lerc               conda-forge/linux-64::lerc-4.0.0-h27087fc_0
  libbrotlicommon    conda-forge/linux-64::libbrotlicommon-1.0.9-h166bdaf_7
  libbrotlidec       conda-forge/linux-64::libbrotlidec-1.0.9-h166bdaf_7
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  libclang13         conda-forge/linux-64::libclang13-14.0.6-default_h3a83d
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_1
  xcb-util           conda-forge/linux-64::xcb-util-0.4.0-h166bdaf_0
  xcb-util-image     conda-forge/linux-64::xcb-util-image-0.4.0-h166bdaf_0
  xcb-util-keysyms   conda-forge/linux-64::xcb-util-keysyms-0.4.0-h166bdaf_
0
  xcb-util-renderut~ conda-forge/linux-64::xcb-util-renderutil-0.3.9-h166bd
af_0
  xcb-util-wm        conda-forge/linux-64::xcb-util-wm-0.4.1-h166bdaf_0
  xorg-libxau        conda-forge/linux-64::xorg-libxau-1.0.9-h7f98852_0
  xorg-libxdmcp      conda-forge/linux-64::xorg-libxdmcp-1.1.3-h7f98852_0
  zstd               conda-forge/linux-64::zstd-1.5.2-h8a70e8d_4


Preparing transaction: ...working... done
Verifying transaction: ...working... done
Executing transaction: ...working... done
singlependulum (generic function with 1 method)

Tests

# Single pendulum

k = rand()
integrator = VelocityVerlet()
Δt = 0.1
singlependulum(k, integrator, Δt, t0=-20.0, t1=20.0)
VelocityVerlet():          0.000257 seconds (2.90 k allocations: 205.859 Ki
B)
PyObject Text(0.5, 0.98, '=====    VelocityVerlet(),   Δt = 0.1    =====')
# Two single pendulums

H(q,p,param) = sum(p.^2/2 .- cos.(q) .+ 1)
q0 = pi*rand(2)
p0 = zeros(2)
t0, t1 = -20.0, 20.0
prob = HamiltonianProblem(H, q0, p0, (t0, t1))

integrator = McAte4()
Δt = 0.1
sol = solve(prob, integrator, dt=Δt)
@time sol = solve(prob, integrator, dt=Δt)

sleep(0.1)
plotsolenergy(H, integrator, Δt, sol)
0.000821 seconds (13.70 k allocations: 1.295 MiB)
PyObject Text(0.5, 0.98, '=====    McAte4(),   Δt = 0.1    =====')

Comparison of symplectic Integrators

SymplecticIntegrators = [
    SymplecticEuler(),
    VelocityVerlet(),
    VerletLeapfrog(),
    PseudoVerletLeapfrog(),
    McAte2(),
    Ruth3(),
    McAte3(),
    CandyRoz4(),
    McAte4(),
    CalvoSanz4(),
    McAte42(),
    McAte5(),
    Yoshida6(),
    KahanLi6(),
    McAte8(),
    KahanLi8(),
    SofSpa10(),
]

k = 0.999
Δt = 0.1
for integrator in SymplecticIntegrators
    singlependulum(k, integrator, Δt)
end
SymplecticEuler():         0.000496 seconds (7.09 k allocations: 505.797 Ki
B)
VelocityVerlet():          0.000419 seconds (7.10 k allocations: 505.859 Ki
B)
VerletLeapfrog():          0.000582 seconds (7.09 k allocations: 505.984 Ki
B)
PseudoVerletLeapfrog():    0.000633 seconds (7.09 k allocations: 505.984 Ki
B)
McAte2():                  0.000602 seconds (7.09 k allocations: 505.984 Ki
B)
Ruth3():                   0.000660 seconds (7.11 k allocations: 506.406 Ki
B)
McAte3():                  0.000626 seconds (7.09 k allocations: 506.094 Ki
B)
CandyRoz4():               0.000677 seconds (7.10 k allocations: 506.438 Ki
B)
McAte4():                  0.000704 seconds (7.09 k allocations: 506.250 Ki
B)
CalvoSanz4():              0.000709 seconds (7.09 k allocations: 506.391 Ki
B)
McAte42():                 0.000721 seconds (7.09 k allocations: 506.391 Ki
B)
McAte5():                  0.000766 seconds (7.09 k allocations: 506.609 Ki
B)
Yoshida6():                0.000900 seconds (7.09 k allocations: 506.672 Ki
B)
KahanLi6():                0.000996 seconds (7.09 k allocations: 507.000 Ki
B)
McAte8():                  0.001281 seconds (7.09 k allocations: 507.797 Ki
B)
KahanLi8():                0.001375 seconds (7.09 k allocations: 508.047 Ki
B)
SofSpa10():                0.002165 seconds (7.09 k allocations: 510.297 Ki
B)
k = 0.999
Δt = 0.01
for integrator in SymplecticIntegrators[1:4]
    singlependulum(k, integrator, Δt)
end
SymplecticEuler():         0.004166 seconds (70.09 k allocations: 4.888 MiB
)
VelocityVerlet():          0.004395 seconds (70.10 k allocations: 4.889 MiB
)
VerletLeapfrog():          0.005147 seconds (70.09 k allocations: 4.889 MiB
)
PseudoVerletLeapfrog():    0.005602 seconds (70.09 k allocations: 4.889 MiB
)
k = 0.999
Δt = 0.001
singlependulum(k, SymplecticEuler(), Δt)
SymplecticEuler():         0.051536 seconds (700.09 k allocations: 48.834 M
iB)
PyObject Text(0.5, 0.98, '=====    SymplecticEuler(),   Δt = 0.001    =====
')
k = 0.999
Δt = 0.0001
singlependulum(k, SymplecticEuler(), Δt)
SymplecticEuler():         0.518554 seconds (7.00 M allocations: 488.287 Mi
B)
PyObject Text(0.5, 0.98, '=====    SymplecticEuler(),   Δt = 0.0001    ====
=')

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/DynamicalODE","single_pendulums.jmd")

Computer Information:

Julia Version 1.7.3
Commit 742b9abb4d (2022-05-06 12:58 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
  CPU: AMD EPYC 7502 32-Core Processor
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-12.0.1 (ORCJIT, znver2)
Environment:
  JULIA_CPU_THREADS = 128
  BUILDKITE_PLUGIN_JULIA_CACHE_DIR = /cache/julia-buildkite-plugin
  JULIA_DEPOT_PATH = /cache/julia-buildkite-plugin/depots/5b300254-1738-4989-ae0a-f4d2d937f953

Package Information:

      Status `/cache/build/exclusive-amdci1-0/julialang/scimlbenchmarks-dot-jl/benchmarks/DynamicalODE/Project.toml`
  [459566f4] DiffEqCallbacks v2.23.1
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  [92b13dbe] TaylorIntegration v0.8.11
  [37e2e46d] LinearAlgebra
  [de0858da] Printf
  [10745b16] Statistics

And the full manifest:

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  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools v1.1.1
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads v1.6.0
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [b27032c2] LibCURL v0.6.3
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions v1.2.0
  [44cfe95a] Pkg v1.8.0
  [de0858da] Printf
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML v1.0.0
  [a4e569a6] Tar v1.10.0
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll v0.5.2+0
  [deac9b47] LibCURL_jll v7.81.0+0
  [29816b5a] LibSSH2_jll v1.10.2+0
  [c8ffd9c3] MbedTLS_jll v2.28.0+0
  [14a3606d] MozillaCACerts_jll v2022.2.1
  [4536629a] OpenBLAS_jll v0.3.20+0
  [05823500] OpenLibm_jll v0.8.1+0
  [efcefdf7] PCRE2_jll v10.40.0+0
  [bea87d4a] SuiteSparse_jll v5.10.1+0
  [83775a58] Zlib_jll v1.2.12+3
  [8e850b90] libblastrampoline_jll v5.1.0+0
  [8e850ede] nghttp2_jll v1.41.0+1
  [3f19e933] p7zip_jll v17.4.0+0