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Two-Bit Adding Unit DAE Work-Precision Diagrams

This is a benchmark of the Two-Bit Adding Unit, an index-1 DAE of dimension 350 from the IVP Test Set (Bär & Söhnlein 1998, tba.f).

The system models a CMOS digital circuit that computes the sum of two 2-bit binary numbers plus a carry-in:

\[A_1 \cdot 2 + A_0 + B_1 \cdot 2 + B_0 + C_{in} = C \cdot 4 + S_1 \cdot 2 + S_0\]

The circuit consists of ten logical subcircuits (3 NOR gates, 5 ANDOI gates, 1 NAND gate, 1 ORANI gate) plus three additional enhancement transistors in series. The transistor model is the Shichman–Hodges MOSFET model with depletion- and enhancement-type devices.

Variables (350 state variables):

  • 175 charge variables (differential): $Q_1, \ldots, Q_{175}$
  • 175 node potentials (algebraic): $V_1, \ldots, V_{175}$

System structure: The charge-driven formulation gives $\dot{Q} = f(t, V), \qquad 0 = Q - g(V)$ where $f$ contains the static currents (resistive + MOSFET drain/source/bulk) and $g$ is the nonlinear charge function. The Jacobian $\partial g/\partial V$ corresponds to the nodal capacitance matrix augmented by voltage-dependent bulk capacitances. In the operating regime of the circuit, this matrix remains nonsingular, ensuring index-1 structure.

Physically relevant outputs (3 signals):

  • \[V_{49}\]

    = $S_0$ (sum bit 0)

  • \[V_{130}\]

    = $S_1$ (sum bit 1)

  • \[V_{148}\]

    = $C$ (carry out)

Input signals: Five periodic pulse waveforms drive the circuit with periods 20, 40, 80, 160, and 320 (scaled time), producing derivative discontinuities at $t = 0, 5, 10, \ldots, 320$.

Reference: Bär, M. and Söhnlein, K.: Test Set for IVP Solvers, http://www.dm.uniba.it/~testset/

using OrdinaryDiffEq, DiffEqDevTools, Sundials, ModelingToolkit, Plots
using OrdinaryDiffEqBDF, OrdinaryDiffEqFIRK, OrdinaryDiffEqRosenbrock
using DASSL, DASKR
using LinearAlgebra, SparseArrays

Physical Constants and MOS Parameters

All parameters are from tba.f. Time is scaled by CTIME = 10⁴ and stiffness by STIFF = 5.

# --- Scaling ---
const CTIME = 1.0e4
const STIFF = 5.0

# --- MOS Parameters ---
const RGS   = 0.4e2 / (CTIME * STIFF)
const RGD   = 0.4e2 / (CTIME * STIFF)
const RBS   = 0.1e3 / (CTIME * STIFF)
const RBD   = 0.1e3 / (CTIME * STIFF)
const CGS   = 0.6e-4 * CTIME
const CGD   = 0.6e-4 * CTIME
const CBD   = 2.4e-5 * CTIME
const CBS   = 2.4e-5 * CTIME
const DELTA = 0.2e-1
const CURIS = 1.0e-15 * CTIME * STIFF
const VTH   = 25.85
const VDD   = 5.0
const VBB   = -2.5
const CLOAD = 0.0
const COUT  = 2.0e-4 * CTIME - CLOAD
2.0

MOSFET Model Functions (Shichman–Hodges)

"""Voltage-dependent bulk capacitance."""
function CBDBS(V)
    PHIB = 0.87
    if V <= 0.0
        return CBD / sqrt(1.0 - V / PHIB)
    else
        return CBD * (1.0 + V / (2.0 * PHIB))
    end
end

"""pn-junction current: bulk-source."""
function IBS_func(VBS)
    if VBS <= 0.0
        return -CURIS * (exp(VBS / VTH) - 1.0)
    else
        return 0.0
    end
end

"""pn-junction current: bulk-drain."""
function IBD_func(VBD)
    if VBD <= 0.0
        return -CURIS * (exp(VBD / VTH) - 1.0)
    else
        return 0.0
    end
end

"""Get MOS parameters by type: 0=depletion, 1=enhancement, 2=2x series, 3=3x series."""
function mos_params(NED)
    if NED == 0
        return -2.43, 0.2, 1.28, 53.5e-6 * CTIME * STIFF
    elseif NED == 1
        return 0.2, 0.035, 1.01, 4 * 43.7e-6 * CTIME * STIFF
    elseif NED == 2
        return 0.2, 0.035, 1.01, 8 * 43.7e-6 * CTIME * STIFF
    else
        return 0.2, 0.035, 1.01, 12 * 43.7e-6 * CTIME * STIFF
    end
end

"""Drain function for VDS > 0."""
function GDSP(NED, VDS, VGS, VBS)
    VT0, CGAMMA, PHI, BETA = mos_params(NED)
    if PHI - VBS < 0.0 || PHI < 0.0
        return 0.0, true  # error flag
    end
    VTE = VT0 + CGAMMA * (sqrt(PHI - VBS) - sqrt(PHI))
    if VGS - VTE <= 0.0
        return 0.0, false
    elseif VGS - VTE <= VDS
        return -BETA * (VGS - VTE)^2 * (1.0 + DELTA * VDS), false
    else
        return -BETA * VDS * (2.0 * (VGS - VTE) - VDS) * (1.0 + DELTA * VDS), false
    end
end

"""Drain function for VDS < 0."""
function GDSM(NED, VDS, VGD, VBD)
    VT0, CGAMMA, PHI, BETA = mos_params(NED)
    if PHI - VBD < 0.0 || PHI < 0.0
        return 0.0, true  # error flag
    end
    VTE = VT0 + CGAMMA * (sqrt(PHI - VBD) - sqrt(PHI))
    if VGD - VTE <= 0.0
        return 0.0, false
    elseif VGD - VTE <= -VDS
        return BETA * (VGD - VTE)^2 * (1.0 - DELTA * VDS), false
    else
        return -BETA * VDS * (2.0 * (VGD - VTE) + VDS) * (1.0 - DELTA * VDS), false
    end
end

"""Drain current (Shichman–Hodges model)."""
function IDS_func(NED, VDS, VGS, VBS, VGD, VBD)
    if VDS > 0.0
        return GDSP(NED, VDS, VGS, VBS)
    elseif VDS == 0.0
        return 0.0, false
    else
        return GDSM(NED, VDS, VGD, VBD)
    end
end
Main.var"##WeaveSandBox#232".IDS_func

Input Signal (Periodic Pulse)

"""
Evaluate periodic pulse signal at time X.

Signal structure: LOW → ramp up (T1) → HIGH (T2) → ramp down (T3) → LOW
"""
function pulse(X, LOW, HIGH, DELAY, T1, T2, T3, PERIOD)
    TIME = mod(X, PERIOD)
    if TIME > DELAY + T1 + T2 + T3
        return LOW, 0.0
    elseif TIME > DELAY + T1 + T2
        VIN = ((HIGH - LOW) / T3) * (DELAY + T1 + T2 + T3 - TIME) + LOW
        return VIN, -(HIGH - LOW) / T3
    elseif TIME > DELAY + T1
        return HIGH, 0.0
    elseif TIME > DELAY
        VIN = ((HIGH - LOW) / T1) * (TIME - DELAY) + LOW
        return VIN, (HIGH - LOW) / T1
    else
        return LOW, 0.0
    end
end
Main.var"##WeaveSandBox#232".pulse

Logic Gate Subcircuit Functions

Each gate computes the static current (right-hand side) for its nodes.

"""NOR gate: NOT(U1 OR U2). Nodes I..I+12."""
function nor_gate!(F, Y, I, U1, U2, U1D, U2D)
    ids_dep, _ = IDS_func(0, Y[I+1]-Y[I], Y[I+4]-Y[I], Y[I+2]-Y[I+4], Y[I+4]-Y[I+1], Y[I+3]-VDD)
    F[I]   = -(Y[I]-Y[I+4])/RGS - ids_dep
    F[I+1] = -(Y[I+1]-VDD)/RGD + ids_dep
    F[I+2] = -(Y[I+2]-VBB)/RBS + IBS_func(Y[I+2]-Y[I+4])
    F[I+3] = -(Y[I+3]-VBB)/RBD + IBD_func(Y[I+3]-VDD)
    # Result node I+4
    F[I+4] = -(Y[I+4]-Y[I])/RGS - IBS_func(Y[I+2]-Y[I+4]) -
             (Y[I+4]-Y[I+6])/RGD - IBD_func(Y[I+8]-Y[I+4]) -
             (Y[I+4]-Y[I+10])/RGD - IBD_func(Y[I+12]-Y[I+4])
    ids_val, _ = IDS_func(1, Y[I+6]-Y[I+5], U1-Y[I+5], Y[I+7], U1-Y[I+6], Y[I+8]-Y[I+4])
    F[I+5] = CGS*U1D - Y[I+5]/RGS - ids_val
    F[I+6] = CGD*U1D - (Y[I+6]-Y[I+4])/RGD + ids_val
    F[I+7] = -(Y[I+7]-VBB)/RBS + IBS_func(Y[I+7])
    F[I+8] = -(Y[I+8]-VBB)/RBD + IBD_func(Y[I+8]-Y[I+4])
    ids_val, _ = IDS_func(1, Y[I+10]-Y[I+9], U2-Y[I+9], Y[I+11], U2-Y[I+10], Y[I+12]-Y[I+4])
    F[I+9]  = CGS*U2D - Y[I+9]/RGS - ids_val
    F[I+10] = CGD*U2D - (Y[I+10]-Y[I+4])/RGD + ids_val
    F[I+11] = -(Y[I+11]-VBB)/RBS + IBS_func(Y[I+11])
    F[I+12] = -(Y[I+12]-VBB)/RBD + IBD_func(Y[I+12]-Y[I+4])
end

"""ANDOI gate: NOT(U1 OR (U2 AND U3)). Nodes I..I+17."""
function andoi_gate!(F, Y, I, U1, U2, U3, U1D, U2D, U3D)
    ids_val, _ = IDS_func(0, Y[I+1]-Y[I], Y[I+4]-Y[I], Y[I+2]-Y[I+4], Y[I+4]-Y[I+1], Y[I+3]-VDD)
    F[I]   = -(Y[I]-Y[I+4])/RGS - ids_val
    F[I+1] = -(Y[I+1]-VDD)/RGD + ids_val
    F[I+2] = -(Y[I+2]-VBB)/RBS + IBS_func(Y[I+2]-Y[I+4])
    F[I+3] = -(Y[I+3]-VBB)/RBD + IBD_func(Y[I+3]-VDD)
    # Result node I+4
    F[I+4] = -(Y[I+4]-Y[I])/RGS - IBS_func(Y[I+2]-Y[I+4]) -
             (Y[I+4]-Y[I+6])/RGD - IBD_func(Y[I+8]-Y[I+4]) -
             (Y[I+4]-Y[I+10])/RGD - IBD_func(Y[I+12]-Y[I+4])
    ids_val, _ = IDS_func(1, Y[I+6]-Y[I+5], U1-Y[I+5], Y[I+7], U1-Y[I+6], Y[I+8]-Y[I+4])
    F[I+5] = CGS*U1D - Y[I+5]/RGS - ids_val
    F[I+6] = CGD*U1D - (Y[I+6]-Y[I+4])/RGD + ids_val
    F[I+7] = -(Y[I+7]-VBB)/RBS + IBS_func(Y[I+7])
    F[I+8] = -(Y[I+8]-VBB)/RBD + IBD_func(Y[I+8]-Y[I+4])
    ids_val, _ = IDS_func(2, Y[I+10]-Y[I+9], U2-Y[I+9], Y[I+11]-Y[I+13], U2-Y[I+10], Y[I+12]-Y[I+4])
    F[I+9]  = CGS*U2D - (Y[I+9]-Y[I+13])/RGS - ids_val
    F[I+10] = CGD*U2D - (Y[I+10]-Y[I+4])/RGD + ids_val
    F[I+11] = -(Y[I+11]-VBB)/RBS + IBS_func(Y[I+11]-Y[I+13])
    F[I+12] = -(Y[I+12]-VBB)/RBD + IBD_func(Y[I+12]-Y[I+4])
    # Coupling node I+13
    F[I+13] = -(Y[I+13]-Y[I+9])/RGS - IBS_func(Y[I+11]-Y[I+13]) -
              (Y[I+13]-Y[I+15])/RGD - IBD_func(Y[I+17]-Y[I+13])
    ids_val, _ = IDS_func(2, Y[I+15]-Y[I+14], U3-Y[I+14], Y[I+16], U3-Y[I+15], Y[I+17]-Y[I+13])
    F[I+14] = CGS*U3D - Y[I+14]/RGS - ids_val
    F[I+15] = CGD*U3D - (Y[I+15]-Y[I+13])/RGD + ids_val
    F[I+16] = -(Y[I+16]-VBB)/RBS + IBS_func(Y[I+16])
    F[I+17] = -(Y[I+17]-VBB)/RBD + IBD_func(Y[I+17]-Y[I+13])
end

"""ANDOI gate with capacitive coupling at result node (for TBA output node 148).
   Nodes I..I+17, with extra coupling to nodes 163, 165."""
function andoip_gate!(F, Y, I, U1, U2, U3, U1D, U2D, U3D)
    ids_val, _ = IDS_func(0, Y[I+1]-Y[I], Y[I+4]-Y[I], Y[I+2]-Y[I+4], Y[I+4]-Y[I+1], Y[I+3]-VDD)
    F[I]   = -(Y[I]-Y[I+4])/RGS - ids_val
    F[I+1] = -(Y[I+1]-VDD)/RGD + ids_val
    F[I+2] = -(Y[I+2]-VBB)/RBS + IBS_func(Y[I+2]-Y[I+4])
    F[I+3] = -(Y[I+3]-VBB)/RBD + IBD_func(Y[I+3]-VDD)
    # Result node I+4, extra coupling to nodes 163, 165
    F[I+4] = -(Y[I+4]-Y[I])/RGS - IBS_func(Y[I+2]-Y[I+4]) -
             (Y[I+4]-Y[I+6])/RGD - IBD_func(Y[I+8]-Y[I+4]) -
             (Y[I+4]-Y[I+10])/RGD - IBD_func(Y[I+12]-Y[I+4]) -
             (Y[I+4]-Y[163])/RGD - IBD_func(Y[165]-Y[I+4])
    ids_val, _ = IDS_func(1, Y[I+6]-Y[I+5], U1-Y[I+5], Y[I+7], U1-Y[I+6], Y[I+8]-Y[I+4])
    F[I+5] = CGS*U1D - Y[I+5]/RGS - ids_val
    F[I+6] = CGD*U1D - (Y[I+6]-Y[I+4])/RGD + ids_val
    F[I+7] = -(Y[I+7]-VBB)/RBS + IBS_func(Y[I+7])
    F[I+8] = -(Y[I+8]-VBB)/RBD + IBD_func(Y[I+8]-Y[I+4])
    ids_val, _ = IDS_func(2, Y[I+10]-Y[I+9], U2-Y[I+9], Y[I+11]-Y[I+13], U2-Y[I+10], Y[I+12]-Y[I+4])
    F[I+9]  = CGS*U2D - (Y[I+9]-Y[I+13])/RGS - ids_val
    F[I+10] = CGD*U2D - (Y[I+10]-Y[I+4])/RGD + ids_val
    F[I+11] = -(Y[I+11]-VBB)/RBS + IBS_func(Y[I+11]-Y[I+13])
    F[I+12] = -(Y[I+12]-VBB)/RBD + IBD_func(Y[I+12]-Y[I+4])
    # Coupling node I+13
    F[I+13] = -(Y[I+13]-Y[I+9])/RGS - IBS_func(Y[I+11]-Y[I+13]) -
              (Y[I+13]-Y[I+15])/RGD - IBD_func(Y[I+17]-Y[I+13])
    ids_val, _ = IDS_func(2, Y[I+15]-Y[I+14], U3-Y[I+14], Y[I+16], U3-Y[I+15], Y[I+17]-Y[I+13])
    F[I+14] = CGS*U3D - Y[I+14]/RGS - ids_val
    F[I+15] = CGD*U3D - (Y[I+15]-Y[I+13])/RGD + ids_val
    F[I+16] = -(Y[I+16]-VBB)/RBS + IBS_func(Y[I+16])
    F[I+17] = -(Y[I+17]-VBB)/RBD + IBD_func(Y[I+17]-Y[I+13])
end

"""NAND gate: NOT(U1 AND U2). Nodes I..I+13."""
function nand_gate!(F, Y, I, U1, U2, U1D, U2D)
    ids_val, _ = IDS_func(0, Y[I+1]-Y[I], Y[I+4]-Y[I], Y[I+2]-Y[I+4], Y[I+4]-Y[I+1], Y[I+3]-VDD)
    F[I]   = -(Y[I]-Y[I+4])/RGS - ids_val
    F[I+1] = -(Y[I+1]-VDD)/RGD + ids_val
    F[I+2] = -(Y[I+2]-VBB)/RBS + IBS_func(Y[I+2]-Y[I+4])
    F[I+3] = -(Y[I+3]-VBB)/RBD + IBD_func(Y[I+3]-VDD)
    # Result node I+4
    F[I+4] = -(Y[I+4]-Y[I])/RGS - IBS_func(Y[I+2]-Y[I+4]) -
             (Y[I+4]-Y[I+6])/RGD - IBD_func(Y[I+8]-Y[I+4])
    ids_val, _ = IDS_func(2, Y[I+6]-Y[I+5], U1-Y[I+5], Y[I+7]-Y[I+9], U1-Y[I+6], Y[I+8]-Y[I+4])
    F[I+5] = CGS*U1D - (Y[I+5]-Y[I+9])/RGS - ids_val
    F[I+6] = CGD*U1D - (Y[I+6]-Y[I+4])/RGD + ids_val
    F[I+7] = -(Y[I+7]-VBB)/RBS + IBS_func(Y[I+7]-Y[I+9])
    F[I+8] = -(Y[I+8]-VBB)/RBD + IBD_func(Y[I+8]-Y[I+4])
    # Coupling node I+9
    F[I+9]  = -(Y[I+9]-Y[I+5])/RGS - IBS_func(Y[I+7]-Y[I+9]) -
              (Y[I+9]-Y[I+11])/RGD - IBD_func(Y[I+13]-Y[I+9])
    ids_val, _ = IDS_func(2, Y[I+11]-Y[I+10], U2-Y[I+10], Y[I+12], U2-Y[I+11], Y[I+13]-Y[I+9])
    F[I+10] = CGS*U2D - Y[I+10]/RGS - ids_val
    F[I+11] = CGD*U2D - (Y[I+11]-Y[I+9])/RGD + ids_val
    F[I+12] = -(Y[I+12]-VBB)/RBS + IBS_func(Y[I+12])
    F[I+13] = -(Y[I+13]-VBB)/RBD + IBD_func(Y[I+13]-Y[I+9])
end

"""ORANI gate: NOT(U1 AND (U2 OR U3)). Nodes I..I+17."""
function orani_gate!(F, Y, I, U1, U2, U3, U1D, U2D, U3D)
    ids_val, _ = IDS_func(0, Y[I+1]-Y[I], Y[I+4]-Y[I], Y[I+2]-Y[I+4], Y[I+4]-Y[I+1], Y[I+3]-VDD)
    F[I]   = -(Y[I]-Y[I+4])/RGS - ids_val
    F[I+1] = -(Y[I+1]-VDD)/RGD + ids_val
    F[I+2] = -(Y[I+2]-VBB)/RBS + IBS_func(Y[I+2]-Y[I+4])
    F[I+3] = -(Y[I+3]-VBB)/RBD + IBD_func(Y[I+3]-VDD)
    # Result node I+4
    F[I+4] = -(Y[I+4]-Y[I])/RGS - IBS_func(Y[I+2]-Y[I+4]) -
             (Y[I+4]-Y[I+6])/RGD - IBD_func(Y[I+8]-Y[I+4])
    ids_val, _ = IDS_func(2, Y[I+6]-Y[I+5], U1-Y[I+5], Y[I+7]-Y[I+9], U1-Y[I+6], Y[I+8]-Y[I+4])
    F[I+5] = CGS*U1D - (Y[I+5]-Y[I+9])/RGS - ids_val
    F[I+6] = CGD*U1D - (Y[I+6]-Y[I+4])/RGD + ids_val
    F[I+7] = -(Y[I+7]-VBB)/RBS + IBS_func(Y[I+7]-Y[I+9])
    F[I+8] = -(Y[I+8]-VBB)/RBD + IBD_func(Y[I+8]-Y[I+4])
    # Coupling node I+9
    F[I+9]  = -(Y[I+9]-Y[I+5])/RGS - IBS_func(Y[I+7]-Y[I+9]) -
              (Y[I+9]-Y[I+11])/RGD - IBD_func(Y[I+13]-Y[I+9]) -
              (Y[I+9]-Y[I+15])/RGD - IBD_func(Y[I+17]-Y[I+9])
    ids_val, _ = IDS_func(2, Y[I+11]-Y[I+10], U2-Y[I+10], Y[I+12], U2-Y[I+11], Y[I+13]-Y[I+9])
    F[I+10] = CGS*U2D - Y[I+10]/RGS - ids_val
    F[I+11] = CGD*U2D - (Y[I+11]-Y[I+9])/RGD + ids_val
    F[I+12] = -(Y[I+12]-VBB)/RBS + IBS_func(Y[I+12])
    F[I+13] = -(Y[I+13]-VBB)/RBD + IBD_func(Y[I+13]-Y[I+9])
    ids_val, _ = IDS_func(2, Y[I+15]-Y[I+14], U3-Y[I+14], Y[I+16], U3-Y[I+15], Y[I+17]-Y[I+9])
    F[I+14] = CGS*U3D - Y[I+14]/RGS - ids_val
    F[I+15] = CGD*U3D - (Y[I+15]-Y[I+9])/RGD + ids_val
    F[I+16] = -(Y[I+16]-VBB)/RBS + IBS_func(Y[I+16])
    F[I+17] = -(Y[I+17]-VBB)/RBD + IBD_func(Y[I+17]-Y[I+9])
end
Main.var"##WeaveSandBox#232".orani_gate!

Static Current Function FCN (Right-Hand Side)

The full right-hand side assembles all ten subcircuits plus three additional enhancement transistors.

"""Compute static currents F[1:175] from node potentials Y[1:175] at time X."""
function FCN!(F, X, Y)
    # Input signals
    V1,  V1D  = pulse(X, 0.0, 5.0, 0.0,   5.0, 5.0,  5.0, 20.0)
    V2,  V2D  = pulse(X, 0.0, 5.0, 10.0,  5.0, 15.0, 5.0, 40.0)
    V3,  V3D  = pulse(X, 0.0, 5.0, 30.0,  5.0, 35.0, 5.0, 80.0)
    V4,  V4D  = pulse(X, 0.0, 5.0, 70.0,  5.0, 75.0, 5.0, 160.0)
    CIN, CIND = pulse(X, 0.0, 5.0, 150.0, 5.0, 155.0, 5.0, 320.0)

    # NOR-gate 1: nodes 1–13
    nor_gate!(F, Y, 1, V1, V2, V1D, V2D)
    # ANDOI-gate 1: nodes 14–31
    andoi_gate!(F, Y, 14, Y[5], V2, V1, 0.0, V2D, V1D)
    # NOR-gate 2: nodes 32–44
    nor_gate!(F, Y, 32, Y[18], CIN, 0.0, CIND)
    # ANDOI-gate 2: nodes 45–62
    andoi_gate!(F, Y, 45, Y[36], CIN, Y[18], 0.0, CIND, 0.0)
    # ANDOI-gate 3: nodes 63–80
    andoi_gate!(F, Y, 63, Y[5], CIN, Y[18], 0.0, CIND, 0.0)
    # NOR-gate 3: nodes 81–93
    nor_gate!(F, Y, 81, V3, V4, V3D, V4D)
    # ANDOI-gate 4: nodes 94–111
    andoi_gate!(F, Y, 94, Y[85], V4, V3, 0.0, V4D, V3D)
    # NAND-gate: nodes 112–125
    nand_gate!(F, Y, 112, Y[67], Y[98], 0.0, 0.0)
    # ORANI-gate 1: nodes 126–143
    orani_gate!(F, Y, 126, Y[116], Y[67], Y[98], 0.0, 0.0, 0.0)
    # ANDOI-gate 5 (capacitive coupling): nodes 144–161
    andoip_gate!(F, Y, 144, Y[85], Y[5], Y[98], 0.0, 0.0, 0.0)

    # Three additional enhancement transistors in series (nodes 162–175)
    ids_val, _ = IDS_func(3, Y[163]-Y[162], Y[98]-Y[162], Y[164]-Y[166], Y[98]-Y[163], Y[165]-Y[148])
    F[162] = -(Y[162]-Y[166])/RGS - ids_val
    F[163] = -(Y[163]-Y[148])/RGD + ids_val
    F[164] = -(Y[164]-VBB)/RBS + IBS_func(Y[164]-Y[166])
    F[165] = -(Y[165]-VBB)/RBD + IBD_func(Y[165]-Y[148])
    F[166] = -IBS_func(Y[164]-Y[166]) - (Y[166]-Y[162])/RGS -
              IBD_func(Y[170]-Y[166]) - (Y[166]-Y[168])/RGD

    ids_val, _ = IDS_func(3, Y[168]-Y[167], Y[18]-Y[167], Y[169]-Y[171], Y[18]-Y[168], Y[170]-Y[166])
    F[167] = -(Y[167]-Y[171])/RGS - ids_val
    F[168] = -(Y[168]-Y[166])/RGD + ids_val
    F[169] = -(Y[169]-VBB)/RBS + IBS_func(Y[169]-Y[171])
    F[170] = -(Y[170]-VBB)/RBD + IBD_func(Y[170]-Y[166])
    F[171] = -IBS_func(Y[169]-Y[171]) - (Y[171]-Y[167])/RGS -
              IBD_func(Y[175]-Y[171]) - (Y[171]-Y[173])/RGD

    ids_val, _ = IDS_func(3, Y[173]-Y[172], CIN-Y[172], Y[174], CIN-Y[173], Y[175]-Y[171])
    F[172] = CGS*CIND - Y[172]/RGS - ids_val
    F[173] = CGD*CIND - (Y[173]-Y[171])/RGD + ids_val
    F[174] = -(Y[174]-VBB)/RBS + IBS_func(Y[174])
    F[175] = -(Y[175]-VBB)/RBD + IBD_func(Y[175]-Y[171])
end
Main.var"##WeaveSandBox#232".FCN!

Charge Function GCN

"""Charge function for NOR gate. Nodes I..I+12."""
function dnor!(G, U, I)
    G[I]    += CGS*(U[I]-U[I+4])
    G[I+1]  += CGD*(U[I+1]-U[I+4])
    G[I+2]  += CBDBS(U[I+2]-U[I+4])*(U[I+2]-U[I+4])
    G[I+3]  += CBDBS(U[I+3]-VDD)*U[I+3]
    G[I+4]  += CGS*(U[I+4]-U[I]) + CGD*(U[I+4]-U[I+1]) +
               CBDBS(U[I+2]-U[I+4])*(U[I+4]-U[I+2]) +
               CBDBS(U[I+8]-U[I+4])*(U[I+4]-U[I+8]) +
               CBDBS(U[I+12]-U[I+4])*(U[I+4]-U[I+12]) +
               CLOAD*U[I+4]
    G[I+5]  += CGS*U[I+5]
    G[I+6]  += CGD*U[I+6]
    G[I+7]  += CBDBS(U[I+7])*U[I+7]
    G[I+8]  += CBDBS(U[I+8]-U[I+4])*(U[I+8]-U[I+4])
    G[I+9]  += CGS*U[I+9]
    G[I+10] += CGD*U[I+10]
    G[I+11] += CBDBS(U[I+11])*U[I+11]
    G[I+12] += CBDBS(U[I+12]-U[I+4])*(U[I+12]-U[I+4])
end

"""Charge function for ANDOI gate. Nodes I..I+17."""
function dandoi!(G, U, I)
    G[I]     += CGS*(U[I]-U[I+4])
    G[I+1]   += CGD*(U[I+1]-U[I+4])
    G[I+2]   += CBDBS(U[I+2]-U[I+4])*(U[I+2]-U[I+4])
    G[I+3]   += CBDBS(U[I+3]-VDD)*U[I+3]
    G[I+4]   += CGS*(U[I+4]-U[I]) + CGD*(U[I+4]-U[I+1]) +
                CBDBS(U[I+2]-U[I+4])*(U[I+4]-U[I+2]) +
                CBDBS(U[I+8]-U[I+4])*(U[I+4]-U[I+8]) +
                CBDBS(U[I+12]-U[I+4])*(U[I+4]-U[I+12]) +
                CLOAD*U[I+4]
    G[I+5]   += CGS*U[I+5]
    G[I+6]   += CGD*U[I+6]
    G[I+7]   += CBDBS(U[I+7])*U[I+7]
    G[I+8]   += CBDBS(U[I+8]-U[I+4])*(U[I+8]-U[I+4])
    G[I+9]   += CGS*U[I+9]
    G[I+10]  += CGD*U[I+10]
    G[I+11]  += CBDBS(U[I+11]-U[I+13])*(U[I+11]-U[I+13])
    G[I+12]  += CBDBS(U[I+12]-U[I+4])*(U[I+12]-U[I+4])
    G[I+13]  += CBDBS(U[I+11]-U[I+13])*(U[I+13]-U[I+11]) +
                CBDBS(U[I+17]-U[I+13])*(U[I+13]-U[I+17]) +
                CLOAD*U[I+13]
    G[I+14]  += CGS*U[I+14]
    G[I+15]  += CGD*U[I+15]
    G[I+16]  += CBDBS(U[I+16])*U[I+16]
    G[I+17]  += CBDBS(U[I+17]-U[I+13])*(U[I+17]-U[I+13])
end

"""Charge function for NAND gate. Nodes I..I+13."""
function dnand!(G, U, I)
    G[I]     += CGS*(U[I]-U[I+4])
    G[I+1]   += CGD*(U[I+1]-U[I+4])
    G[I+2]   += CBDBS(U[I+2]-U[I+4])*(U[I+2]-U[I+4])
    G[I+3]   += CBDBS(U[I+3]-VDD)*U[I+3]
    G[I+4]   += CGS*(U[I+4]-U[I]) + CGD*(U[I+4]-U[I+1]) +
                CBDBS(U[I+2]-U[I+4])*(U[I+4]-U[I+2]) +
                CBDBS(U[I+8]-U[I+4])*(U[I+4]-U[I+8]) +
                CLOAD*U[I+4]
    G[I+5]   += CGS*U[I+5]
    G[I+6]   += CGD*U[I+6]
    G[I+7]   += CBDBS(U[I+7]-U[I+9])*(U[I+7]-U[I+9])
    G[I+8]   += CBDBS(U[I+8]-U[I+4])*(U[I+8]-U[I+4])
    G[I+9]   += CBDBS(U[I+7]-U[I+9])*(U[I+9]-U[I+7]) +
                CBDBS(U[I+13]-U[I+9])*(U[I+9]-U[I+13]) +
                CLOAD*U[I+9]
    G[I+10]  += CGS*U[I+10]
    G[I+11]  += CGD*U[I+11]
    G[I+12]  += CBDBS(U[I+12])*U[I+12]
    G[I+13]  += CBDBS(U[I+13]-U[I+9])*(U[I+13]-U[I+9])
end

"""Charge function for ORANI gate. Nodes I..I+17."""
function dorani!(G, U, I)
    G[I]     += CGS*(U[I]-U[I+4])
    G[I+1]   += CGD*(U[I+1]-U[I+4])
    G[I+2]   += CBDBS(U[I+2]-U[I+4])*(U[I+2]-U[I+4])
    G[I+3]   += CBDBS(U[I+3]-VDD)*U[I+3]
    G[I+4]   += CGS*(U[I+4]-U[I]) + CGD*(U[I+4]-U[I+1]) +
                CBDBS(U[I+2]-U[I+4])*(U[I+4]-U[I+2]) +
                CBDBS(U[I+8]-U[I+4])*(U[I+4]-U[I+8]) +
                CLOAD*U[I+4]
    G[I+5]   += CGS*U[I+5]
    G[I+6]   += CGD*U[I+6]
    G[I+7]   += CBDBS(U[I+7]-U[I+9])*(U[I+7]-U[I+9])
    G[I+8]   += CBDBS(U[I+8]-U[I+4])*(U[I+8]-U[I+4])
    G[I+9]   += CBDBS(U[I+7]-U[I+9])*(U[I+9]-U[I+7]) +
                CBDBS(U[I+13]-U[I+9])*(U[I+9]-U[I+13]) +
                CBDBS(U[I+17]-U[I+9])*(U[I+9]-U[I+17]) +
                CLOAD*U[I+9]
    G[I+10]  += CGS*U[I+10]
    G[I+11]  += CGD*U[I+11]
    G[I+12]  += CBDBS(U[I+12])*U[I+12]
    G[I+13]  += CBDBS(U[I+13]-U[I+9])*(U[I+13]-U[I+9])
    G[I+14]  += CGS*U[I+14]
    G[I+15]  += CGD*U[I+15]
    G[I+16]  += CBDBS(U[I+16])*U[I+16]
    G[I+17]  += CBDBS(U[I+17]-U[I+9])*(U[I+17]-U[I+9])
end

"""Full charge function G[1:175] from node potentials U[1:175]."""
function GCN!(G, U)
    G .= 0.0

    # Ten logical subcircuits
    dnor!(G, U, 1)
    dandoi!(G, U, 14)
    dnor!(G, U, 32)
    dandoi!(G, U, 45)
    dandoi!(G, U, 63)
    dnor!(G, U, 81)
    dandoi!(G, U, 94)
    dnand!(G, U, 112)
    dorani!(G, U, 126)
    dandoi!(G, U, 144)

    # Capacitive coupling: result node NOR-gate 1 (node 5)
    G[5]  += CGS*(U[5]-U[19]) + CGD*(U[5]-U[20]) +
             CGS*(U[5]-U[68]) + CGD*(U[5]-U[69]) +
             CGS*(U[5]-U[153]) + CGD*(U[5]-U[154])
    G[19]  -= CGS*U[5]
    G[20]  -= CGD*U[5]
    G[68]  -= CGS*U[5]
    G[69]  -= CGD*U[5]
    G[153] -= CGS*U[5]
    G[154] -= CGD*U[5]

    # Capacitive coupling: result node ANDOI-gate 1 (node 18)
    G[18] += CGS*(U[18]-U[37]) + CGD*(U[18]-U[38]) +
             CGS*(U[18]-U[59]) + CGD*(U[18]-U[60]) +
             CGS*(U[18]-U[77]) + CGD*(U[18]-U[78]) +
             CGS*(U[18]-U[167]) + CGD*(U[18]-U[168])
    G[37]  -= CGS*U[18]
    G[38]  -= CGD*U[18]
    G[59]  -= CGS*U[18]
    G[60]  -= CGD*U[18]
    G[77]  -= CGS*U[18]
    G[78]  -= CGD*U[18]

    # Capacitive coupling: result node NOR-gate 2 (node 36)
    G[36] += CGS*(U[36]-U[50]) + CGD*(U[36]-U[51])
    G[50]  -= CGS*U[36]
    G[51]  -= CGD*U[36]

    # Capacitive coupling: result node ANDOI-gate 2 = S0 (node 49)
    G[49] += COUT*U[49]

    # Capacitive coupling: result node ANDOI-gate 3 (node 67)
    G[67] += CGS*(U[67]-U[117]) + CGD*(U[67]-U[118]) +
             CGS*(U[67]-U[136]) + CGD*(U[67]-U[137])
    G[117] -= CGS*U[67]
    G[118] -= CGD*U[67]
    G[136] -= CGS*U[67]
    G[137] -= CGD*U[67]

    # Capacitive coupling: result node NOR-gate 3 (node 85)
    G[85] += CGS*(U[85]-U[99]) + CGD*(U[85]-U[100]) +
             CGS*(U[85]-U[149]) + CGD*(U[85]-U[150])
    G[99]  -= CGS*U[85]
    G[100] -= CGD*U[85]
    G[149] -= CGS*U[85]
    G[150] -= CGD*U[85]

    # Capacitive coupling: result node ANDOI-gate 4 (node 98)
    G[98] += CGS*(U[98]-U[122]) + CGD*(U[98]-U[123]) +
             CGS*(U[98]-U[140]) + CGD*(U[98]-U[141]) +
             CGS*(U[98]-U[158]) + CGD*(U[98]-U[159]) +
             CGS*(U[98]-U[162]) + CGD*(U[98]-U[163])
    G[122] -= CGS*U[98]
    G[123] -= CGD*U[98]
    G[140] -= CGS*U[98]
    G[141] -= CGD*U[98]
    G[158] -= CGS*U[98]
    G[159] -= CGD*U[98]

    # Capacitive coupling: result node NAND-gate (node 116)
    G[116] += CGS*(U[116]-U[131]) + CGD*(U[116]-U[132])
    G[131] -= CGS*U[116]
    G[132] -= CGD*U[116]

    # Capacitive coupling: result node ORANI-gate = S1 (node 130)
    G[130] += COUT*U[130]

    # Capacitive coupling: result ANDOI-gate 5 = C_inverse (node 148)
    G[148] += CBDBS(U[165]-U[148])*(U[148]-U[165]) + COUT*U[148]

    # Three additional transistors
    G[162] += CGS*(U[162]-U[98])
    G[163] += CGD*(U[163]-U[98])
    G[164] += CBDBS(U[164]-U[166])*(U[164]-U[166])
    G[165] += CBDBS(U[165]-U[148])*(U[165]-U[148])
    G[166] += CBDBS(U[164]-U[166])*(U[166]-U[164]) +
              CBDBS(U[170]-U[166])*(U[166]-U[170]) +
              CLOAD*U[166]
    G[167] += CGS*(U[167]-U[18])
    G[168] += CGD*(U[168]-U[18])
    G[169] += CBDBS(U[169]-U[171])*(U[169]-U[171])
    G[170] += CBDBS(U[170]-U[166])*(U[170]-U[166])
    G[171] += CBDBS(U[169]-U[171])*(U[171]-U[169]) +
              CBDBS(U[175]-U[171])*(U[171]-U[175]) +
              CLOAD*U[171]
    G[172] += CGS*U[172]
    G[173] += CGD*U[173]
    G[174] += CBDBS(U[174])*U[174]
    G[175] += CBDBS(U[175]-U[171])*(U[175]-U[171])
end
Main.var"##WeaveSandBox#232".GCN!

Consistent Initial Conditions

From init subroutine in tba.f. The initial voltage vector U(1:175) is provided, then y[1:175] = GCN(U) (charges) and y[176:350] = U (potentials).

function tba_initial_conditions()
    U = zeros(175)
    U[1]   =  4.999999999996544;  U[2]   =  4.999999999999970
    U[3]   = -2.499999999999975;  U[4]   = -2.499999999999975
    U[5]   =  4.999999999996514;  U[6]   =  0.000000000000000
    U[7]   =  4.999999999996514;  U[8]   = -2.499999999999991
    U[9]   = -2.499999999999975;  U[10]  =  0.000000000000000
    U[11]  =  4.999999999996514;  U[12]  = -2.499999999999991
    U[13]  = -2.499999999999975;  U[14]  =  0.215858486765796
    U[15]  =  4.988182208251953;  U[16]  = -2.499999999999990
    U[17]  = -2.499999999999975;  U[18]  =  0.204040695017748
    U[19]  =  0.011817791748026;  U[20]  =  0.192222903269723
    U[21]  = -2.499999999999991;  U[22]  = -2.499999999999990
    U[23]  = -0.228160951881239;  U[24]  =  0.204040695017748
    U[25]  = -2.499999999999992;  U[26]  = -2.499999999999990
    U[27]  = -0.228160951881241;  U[28]  =  0.000000000000000
    U[29]  = -0.228160951881239;  U[30]  = -2.499999999999991
    U[31]  = -2.499999999999992;  U[32]  =  4.999999999996547
    U[33]  =  4.999999999999970;  U[34]  = -2.499999999999975
    U[35]  = -2.499999999999975;  U[36]  =  4.999999999996517
    U[37]  =  0.000000000000000;  U[38]  =  4.999999999996517
    U[39]  = -2.499999999999991;  U[40]  = -2.499999999999975
    U[41]  =  0.000000000000000;  U[42]  =  4.999999999996517
    U[43]  = -2.499999999999991;  U[44]  = -2.499999999999975
    U[45]  =  0.215858484247529;  U[46]  =  4.988182208251953
    U[47]  = -2.499999999999990;  U[48]  = -2.499999999999975
    U[49]  =  0.204040692499482;  U[50]  =  0.011817791748035
    U[51]  =  0.192222900751447;  U[52]  = -2.499999999999991
    U[53]  = -2.499999999999990;  U[54]  = -0.026041071738432
    U[55]  =  0.204040692499482;  U[56]  = -2.499999999999992
    U[57]  = -2.499999999999990;  U[58]  = -0.026041071738434
    U[59]  =  0.000000000000000;  U[60]  = -0.026041071738432
    U[61]  = -2.499999999999991;  U[62]  = -2.499999999999992
    U[63]  =  0.215858484880918;  U[64]  =  4.988182208251953
    U[65]  = -2.499999999999990;  U[66]  = -2.499999999999975
    U[67]  =  0.204040693132870;  U[68]  =  0.011817791748026
    U[69]  =  0.192222901384845;  U[70]  = -2.499999999999991
    U[71]  = -2.499999999999990;  U[72]  = -0.026041071737961
    U[73]  =  0.204040693132870;  U[74]  = -2.499999999999992
    U[75]  = -2.499999999999990;  U[76]  = -0.026041071737963
    U[77]  =  0.000000000000000;  U[78]  = -0.026041071737961
    U[79]  = -2.499999999999991;  U[80]  = -2.499999999999992
    U[81]  =  4.999999999996546;  U[82]  =  4.999999999999970
    U[83]  = -2.499999999999975;  U[84]  = -2.499999999999975
    U[85]  =  4.999999999996516;  U[86]  =  0.000000000000000
    U[87]  =  4.999999999996516;  U[88]  = -2.499999999999991
    U[89]  = -2.499999999999975;  U[90]  =  0.000000000000000
    U[91]  =  4.999999999996516;  U[92]  = -2.499999999999991
    U[93]  = -2.499999999999975;  U[94]  =  0.215858481060569
    U[95]  =  4.988182208251953;  U[96]  = -2.499999999999990
    U[97]  = -2.499999999999975;  U[98]  =  0.204040689312522
    U[99]  =  0.011817791748023;  U[100] =  0.192222897564498
    U[101] = -2.499999999999991;  U[102] = -2.499999999999990
    U[103] =  4.734672533390068;  U[104] =  0.204040689312522
    U[105] = -2.499999999999977;  U[106] = -2.499999999999990
    U[107] =  4.734672533390062;  U[108] =  0.000000000000000
    U[109] =  4.734672533390068;  U[110] = -2.499999999999991
    U[111] = -2.499999999999977;  U[112] =  4.999999999996870
    U[113] =  4.999999999999972;  U[114] = -2.499999999999975
    U[115] = -2.499999999999975;  U[116] =  4.999999999996843
    U[117] = -0.025968303070038;  U[118] =  4.999999999996843
    U[119] = -2.499999999999992;  U[120] = -2.499999999999975
    U[121] = -0.025968303070040;  U[122] =  0.000000000000000
    U[123] = -0.025968303070038;  U[124] = -2.499999999999991
    U[125] = -2.499999999999992;  U[126] =  4.999999999997699
    U[127] =  4.999999999999980;  U[128] = -2.499999999999975
    U[129] = -2.499999999999975;  U[130] =  4.999999999997678
    U[131] =  4.744923533081106;  U[132] =  4.999999999997678
    U[133] = -2.499999999999977;  U[134] = -2.499999999999975
    U[135] =  4.744923533081098;  U[136] =  0.000000000000000
    U[137] =  4.744923533081106;  U[138] = -2.499999999999991
    U[139] = -2.499999999999977;  U[140] =  0.000000000000000
    U[141] =  4.744923533081106;  U[142] = -2.499999999999991
    U[143] = -2.499999999999977;  U[144] =  0.215858484844162
    U[145] =  4.988182208251953;  U[146] = -2.499999999999990
    U[147] = -2.499999999999975;  U[148] =  0.204040693096114
    U[149] =  0.011817791748023;  U[150] =  0.192222901348091
    U[151] = -2.499999999999991;  U[152] = -2.499999999999990
    U[153] =  0.204040693096045;  U[154] =  0.204040693096107
    U[155] = -2.499999999999990;  U[156] = -2.499999999999990
    U[157] =  0.204040693096037;  U[158] =  0.000000000000000
    U[159] =  0.204040693096037;  U[160] = -2.499999999999991
    U[161] = -2.499999999999990;  U[162] = -0.026017361873565
    U[163] =  0.204040693096114;  U[164] = -2.499999999999992
    U[165] = -2.499999999999990;  U[166] = -0.026017361873568
    U[167] = -0.026017590106916;  U[168] = -0.026017361873565
    U[169] = -2.499999999999992;  U[170] = -2.499999999999992
    U[171] = -0.026017590106918;  U[172] =  0.000000000000000
    U[173] = -0.026017590106916;  U[174] = -2.499999999999991
    U[175] = -2.499999999999992

    # Compute initial charges
    G = zeros(175)
    GCN!(G, U)

    # Full state: y[1:175] = charges, y[176:350] = potentials
    y0 = zeros(350)
    y0[1:175]   .= G
    y0[176:350] .= U
    return y0
end

y0 = tba_initial_conditions()
println("Initial condition: 350 variables (175 charges + 175 potentials)")
println("  S0  (V₄₉)  = ", y0[224])
println("  S1  (V₁₃₀) = ", y0[305])
println("  C   (V₁₄₈) = ", y0[323])
Initial condition: 350 variables (175 charges + 175 potentials)
  S0  (V₄₉)  = 0.204040692499482
  S1  (V₁₃₀) = 4.999999999997678
  C   (V₁₄₈) = 0.204040693096114

Derivative Discontinuity Times

The five input pulse signals create derivative discontinuities every 5 time units (at t = 5, 10, …, 315). The Fortran reference (tba.f) specifies ndisc = 63 with interaction points at multiples of 5. Communicating these to the solver via tstops prevents wasteful step rejections at transitions.

disc_times = collect(5.0:5.0:315.0)
63-element Vector{Float64}:
   5.0
  10.0
  15.0
  20.0
  25.0
  30.0
  35.0
  40.0
  45.0
  50.0
   ⋮
 275.0
 280.0
 285.0
 290.0
 295.0
 300.0
 305.0
 310.0
 315.0

Step 1: Mass-Matrix ODE Formulation — Build and Verify

The DAE is written as M·dy/dt = f(t, y) where M = diag(I₁₇₅, 0₁₇₅). The first 175 equations are differential (charge conservation), and the remaining 175 are algebraic (charge = g(V)).

function tba_rhs!(dy, y, p, t)
    @views begin
        x = y[176:350]
        # Differential equations: dQ/dt = f(t, V)
        FCN!(dy, t, x)              # writes dy[1:175]
        # Algebraic equations: 0 = Q - g(V)
        GCN!(dy[176:350], x)        # writes dy[176:350] = g(V)
        @. dy[176:350] = y[1:175] - dy[176:350]  # Q - g(V)
    end
end

# Mass matrix: identity for charges, zero for potentials (sparse)
M_tba = spdiagm(0 => [ones(175); zeros(175)])

tspan = (0.0, 320.0)
mmf = ODEFunction(tba_rhs!, mass_matrix = M_tba)
prob_mm = ODEProblem(mmf, y0, tspan;
                     isoutofdomain = (u, p, t) -> any(isnan, u))

println("=== Step 1: Mass-Matrix ODE Verification ===")
mm_test = solve(prob_mm, Rodas5P(), reltol = 1e-4, abstol = 1e-4,
                maxiters = 1_000_000, tstops = disc_times)
println("Rodas5P: retcode = $(mm_test.retcode), steps = $(length(mm_test.t)), ",
        "t_final = $(mm_test.t[end])")
println("  S₀ (V₄₉)  = $(mm_test[224, end])")
println("  S₁ (V₁₃₀) = $(mm_test[305, end])")
println("  C  (V₁₄₈) = $(mm_test[323, end])")
=== Step 1: Mass-Matrix ODE Verification ===
Rodas5P: retcode = Success, steps = 2056, t_final = 320.0
  S₀ (V₄₉)  = 0.20404191280755501
  S₁ (V₁₃₀) = 4.997246133507289
  C  (V₁₄₈) = 0.20389855467018875

Step 2: DAE Residual Form — Build and Verify

The same system as a DAE residual F(du, u, t) = M·du − f(u, t) = 0, for testing DAE-specific solvers (IDA, DASKR, DASSL).

function tba_dae!(res, du, u, p, t)
    tba_rhs!(res, u, p, t)
    @views begin
        @. res[1:175]   = du[1:175] - res[1:175]     # M*du - f (differential)
        @. res[176:350] = -res[176:350]               # 0 - (Q - g(V)) (algebraic)
    end
end

du0 = zeros(350)
tba_rhs!(du0, y0, nothing, 0.0)
# For consistent ICs: du0[176:350] should be zero (algebraic)
# and du0[1:175] = f(0, V0)
du0_dae = copy(du0)
du0_dae[176:350] .= 0.0

differential_vars = [trues(175); falses(175)]
prob_dae = DAEProblem(tba_dae!, du0_dae, y0, tspan,
                      differential_vars = differential_vars)
println("DAE problem: 350 variables, 175 differential + 175 algebraic")

# Verify DAE residual at initial conditions
res_check = similar(y0)
tba_dae!(res_check, du0_dae, y0, nothing, 0.0)
println("=== Step 2: DAE Residual Verification ===")
println("DAE residual norm at IC: ", norm(res_check))

println("\nTesting IDA (Sundials):")
try
    dae_test = solve(prob_dae, IDA(), reltol = 1e-4, abstol = 1e-4,
                     maxiters = 1_000_000, tstops = disc_times)
    println("  IDA: retcode = $(dae_test.retcode), steps = $(length(dae_test.t)), ",
            "t_final = $(dae_test.t[end])")
catch e
    println("  IDA failed: ", sprint(showerror, e)[1:min(end, 200)])
end

println("\nTesting DASKR:")
try
    daskr_test = solve(prob_dae, DASKR.daskr(), reltol = 1e-4, abstol = 1e-4,
                       maxiters = 1_000_000, tstops = disc_times)
    println("  DASKR: retcode = $(daskr_test.retcode), steps = $(length(daskr_test.t)), ",
            "t_final = $(daskr_test.t[end])")
catch e
    println("  DASKR failed: ", sprint(showerror, e)[1:min(end, 200)])
end

# Note: DASSL may fail with "stepsize too small" on this problem due to
# the stiff MOSFET switching combined with derivative discontinuities.
println("\nTesting DASSL:")
try
    dassl_test = solve(prob_dae, DASSL.dassl(), reltol = 1e-4, abstol = 1e-4,
                       maxiters = 1_000_000, tstops = disc_times)
    println("  DASSL: retcode = $(dassl_test.retcode), steps = $(length(dassl_test.t)), ",
            "t_final = $(dassl_test.t[end])")
catch e
    println("  DASSL failed: ", sprint(showerror, e)[1:min(end, 200)])
end
DAE problem: 350 variables, 175 differential + 175 algebraic
=== Step 2: DAE Residual Verification ===
DAE residual norm at IC: 0.0

Testing IDA (Sundials):
  IDA: retcode = Success, steps = 5309, t_final = 320.0

Testing DASKR:
  DASKR: retcode = Success, steps = 5482, t_final = 320.0

Testing DASSL:
  DASSL: retcode = Success, steps = 6934, t_final = 320.0

Step 3: MTK Symbolic Formulation — Attempt

ModelingToolkit can automatically index-reduce DAEs via structural_simplify. We attempt modelingtoolkitize on the mass-matrix ODE to check whether symbolic tracing is possible for this large piecewise-switching circuit.

println("=== Step 3: MTK Symbolic Formulation ===")
println("Attempting modelingtoolkitize on mass-matrix ODE form...")
mtk_success = false
try
    global sys_mtk = modelingtoolkitize(prob_mm)
    println("  Tracing succeeded. Attempting structural_simplify...")
    sys_simplified = structural_simplify(sys_mtk)
    global prob_mtk = ODEProblem(sys_simplified, [], tspan)
    println("  MTK problem built: $(length(ModelingToolkit.unknowns(sys_simplified))) states")
    mtk_test = solve(prob_mtk, Rodas5P(), reltol = 1e-4, abstol = 1e-4,
                     maxiters = 1_000_000, tstops = disc_times)
    println("  Rodas5P (MTK): retcode = $(mtk_test.retcode), t_final = $(mtk_test.t[end])")
    global mtk_success = true
catch e
    println("  MTK failed: ", sprint(showerror, e)[1:min(end, 300)])
    println("\n  Expected: the Shichman–Hodges MOSFET model and pulse generator use")
    println("  piecewise if/else branching that cannot be symbolically traced.")
    println("  A full MTK formulation would require rewriting all transistor equations")
    println("  with symbolic-friendly ifelse(), prohibitive for 350 variables.")
end
=== Step 3: MTK Symbolic Formulation ===
Attempting modelingtoolkitize on mass-matrix ODE form...
  MTK failed: TypeError: non-boolean (Symbolics.Num) used in boolean contex
t
A symbolic expression appeared in a Boolean context. This error arises in s
ituations where Julia expects a Bool, like
if boolean_condition		 use ifelse(boolean_condition, then branch, else bran
ch)
x && y				 use x & y
boolean_condition

  Expected: the Shichman–Hodges MOSFET model and pulse generator use
  piecewise if/else branching that cannot be symbolically traced.
  A full MTK formulation would require rewriting all transistor equations
  with symbolic-friendly ifelse(), prohibitive for 350 variables.

Reference Solution

We compute a high-accuracy reference using Rodas5P on the mass-matrix form at tight tolerance. The derivative discontinuities at pulse transitions are communicated via tstops for efficiency. The IVP Test Set reference was computed with RADAU5 at rtol = atol = 1e-5. We use 1e-8 rather than 1e-10 because tighter tolerances are unnecessary for this stiff, discontinuous circuit and can trigger instabilities on some BLAS backends.

ts = collect(0.0:0.5:320.0)

ref_sol = solve(prob_mm, Rodas5P(), reltol = 1e-8, abstol = 1e-8,
                maxiters = 10_000_000, tstops = disc_times)
println("Reference solution: retcode = $(ref_sol.retcode), ",
        "npoints = $(length(ref_sol.t)), t_final = $(ref_sol.t[end])")
if ref_sol.retcode != ReturnCode.Success
    error("Reference solution failed with retcode = $(ref_sol.retcode)")
end

if mtk_success
    global mtk_ref = solve(prob_mtk, Rodas5P(), reltol = 1e-8, abstol = 1e-8,
                           maxiters = 10_000_000, tstops = disc_times)
    println("MTK reference: retcode = $(mtk_ref.retcode), ",
            "npoints = $(length(mtk_ref.t)), t_final = $(mtk_ref.t[end])")
end
Reference solution: retcode = Success, npoints = 16499, t_final = 320.0

Output Signal Plots

The three physically relevant output signals are the digital outputs S0, S1, and C. These correspond to node potentials V₄₉, V₁₃₀, V₁₄₈ (indices 224, 305, 323 in the full state vector).

plot(ref_sol, idxs = [224], title = "S₀ Output Signal (V₄₉)",
     xlabel = "Time", ylabel = "Voltage", lw = 1.5, legend = false)

plot(ref_sol, idxs = [305], title = "S₁ Output Signal (V₁₃₀)",
     xlabel = "Time", ylabel = "Voltage", lw = 1.5, legend = false)

plot(ref_sol, idxs = [323], title = "Carry Output Signal C (V₁₄₈)",
     xlabel = "Time", ylabel = "Voltage", lw = 1.5, legend = false)

plot(ref_sol, idxs = [224, 305, 323],
     title = "Two-Bit Adder: All Output Signals",
     xlabel = "Time", ylabel = "Voltage", lw = 1.5,
     label = ["S₀ (V₄₉)" "S₁ (V₁₃₀)" "C (V₁₄₈)"])

Verification against IVP Test Set Reference

The solut subroutine provides reference values at t = 320. We verify the three output components.

archimede_y224 = 0.2040419147264534   # x(49)  = y(224)
archimede_y305 = 0.4997238455712048e1 # x(130) = y(305)
archimede_y323 = 0.2038985905095614   # x(148) = y(323)

sol_final = ref_sol.u[end]
println("=== Verification at t = 320 ===")
println("Variable | ARCHIMEDE Reference       | Our Solution              | Rel Error")
println("-"^80)
for (name, idx, ref_val) in [("y(224) S₀", 224, archimede_y224),
                              ("y(305) S₁", 305, archimede_y305),
                              ("y(323) C ", 323, archimede_y323)]
    our_val = sol_final[idx]
    relerr = abs(ref_val) > 0 ? abs((our_val - ref_val) / ref_val) : abs(our_val)
    status = relerr < 1e-6 ? "✓" : (relerr < 1e-3 ? "~" : "✗")
    println("$(rpad(name, 12))| $(lpad(string(ref_val), 26)) | $(lpad(string(round(our_val, sigdigits=12)), 26)) | $(relerr) $status")
end
=== Verification at t = 320 ===
Variable | ARCHIMEDE Reference       | Our Solution              | Rel Erro
r
---------------------------------------------------------------------------
-----
y(224) S₀   |         0.2040419147264534 |             0.204041912817 | 9.3
5739694431443e-9 ✓
y(305) S₁   |          4.997238455712048 |              4.99724244497 | 7.9
82928766272776e-7 ✓
y(323) C    |         0.2038985905095614 |              0.20389857591 | 7.1
59991334559498e-8 ✓

Work-Precision Diagrams

We benchmark up to three formulations: DAE residual (IDA, DASKR, DASSL), mass-matrix ODE (Rosenbrock-W, implicit RK, BDF), and — if MTK tracing succeeded — the symbolic MTK form. Derivative discontinuity times from the input pulses are communicated via tstops.

probs = [prob_dae, prob_mm]
refs  = [ref_sol, ref_sol]

if mtk_success
    push!(probs, prob_mtk)
    push!(refs, mtk_ref)
end

High Tolerances

abstols = 1.0 ./ 10.0 .^ (3:7)
reltols = 1.0 ./ 10.0 .^ (1:5)
setups = [
    Dict(:prob_choice => 1, :alg => IDA()),
    Dict(:prob_choice => 2, :alg => Rodas5P()),
    Dict(:prob_choice => 2, :alg => Rodas4P()),
    Dict(:prob_choice => 2, :alg => FBDF()),
    Dict(:prob_choice => 2, :alg => QNDF()),
    Dict(:prob_choice => 2, :alg => NordsieckBDF()),
    Dict(:prob_choice => 2, :alg => RadauIIA5()),
]
labels = ["IDA (DAE)", "Rodas5P (MM)", "Rodas4P (MM)", "FBDF (MM)", "QNDF (MM)", "NordsieckBDF (MM)", "RadauIIA5 (MM)"]

if mtk_success
    push!(setups, Dict(:prob_choice => 3, :alg => Rodas5P()))
    push!(labels, "Rodas5P (MTK)")
end

wp = WorkPrecisionSet(probs, abstols, reltols, setups;
    names = reshape(labels, 1, :), appxsol = refs, save_everystep = false,
    maxiters = Int(1e6), numruns = 1, tstops = disc_times)
plot(wp, title = "Two-Bit Adder: High Tolerances")

Medium Tolerances

abstols = 1.0 ./ 10.0 .^ (5:7)
reltols = 1.0 ./ 10.0 .^ (3:5)
setups = [
    Dict(:prob_choice => 1, :alg => IDA()),
    Dict(:prob_choice => 1, :alg => DASKR.daskr()),
    Dict(:prob_choice => 2, :alg => Rodas5P()),
    Dict(:prob_choice => 2, :alg => Rodas4P()),
    Dict(:prob_choice => 2, :alg => FBDF()),
    Dict(:prob_choice => 2, :alg => NordsieckBDF()),
    Dict(:prob_choice => 2, :alg => RadauIIA5()),
]
labels = ["IDA (DAE)", "DASKR (DAE)", "Rodas5P (MM)", "Rodas4P (MM)", "FBDF (MM)", "NordsieckBDF (MM)", "RadauIIA5 (MM)"]

if mtk_success
    push!(setups, Dict(:prob_choice => 3, :alg => Rodas5P()))
    push!(setups, Dict(:prob_choice => 3, :alg => Rodas4P()))
    push!(labels, "Rodas5P (MTK)")
    push!(labels, "Rodas4P (MTK)")
end

wp = WorkPrecisionSet(probs, abstols, reltols, setups;
    names = reshape(labels, 1, :), appxsol = refs, save_everystep = false,
    maxiters = Int(1e6), numruns = 1, tstops = disc_times)
plot(wp, title = "Two-Bit Adder: Medium Tolerances")

Timeseries Errors (L2)

Tolerances start at 1e-4/1e-2 rather than 1e-3/1e-1 because very loose tolerances on this stiff switching circuit amplify phase errors at input transitions, making L2 comparisons noisy.

abstols = 1.0 ./ 10.0 .^ (4:7)
reltols = 1.0 ./ 10.0 .^ (2:5)
setups = [
    Dict(:prob_choice => 1, :alg => IDA()),
    Dict(:prob_choice => 2, :alg => Rodas5P()),
    Dict(:prob_choice => 2, :alg => Rodas4P()),
    Dict(:prob_choice => 2, :alg => FBDF()),
    Dict(:prob_choice => 2, :alg => NordsieckBDF()),
    Dict(:prob_choice => 2, :alg => RadauIIA5()),
]
labels = ["IDA (DAE)", "Rodas5P (MM)", "Rodas4P (MM)", "FBDF (MM)", "NordsieckBDF (MM)", "RadauIIA5 (MM)"]

if mtk_success
    push!(setups, Dict(:prob_choice => 3, :alg => Rodas5P()))
    push!(labels, "Rodas5P (MTK)")
end

wp = WorkPrecisionSet(probs, abstols, reltols, setups;
    names = reshape(labels, 1, :), appxsol = refs,
    saveat = ts, maxiters = Int(1e6), numruns = 1,
    error_estimate = :l2, tstops = disc_times)
plot(wp, title = "Two-Bit Adder: Timeseries Error (L2)")

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/DAE","two_bit_adder.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_PKG_PRECOMPILE_AUTO = 0
  JULIA_NUM_THREADS = auto

Package Information:

Status `~/sandbox/tmp_20260825_180339_53321/dae-pr1670-validate/benchmarks/DAE/Project.toml`
⌃ [165a45c3] DASKR v3.1.5
⌃ [e993076c] DASSL v3.1.0
⌃ [f3b72e0c] DiffEqDevTools v3.2.0
⌃ [961ee093] ModelingToolkit v11.39.0
⌅ [09606e27] ODEInterfaceDiffEq v4.1.0
⌃ [1dea7af3] OrdinaryDiffEq v7.6.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.2
⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.6.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.6.5
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.8.2
⌃ [91a5bcdd] Plots v1.41.6
⌃ [31c91b34] SciMLBenchmarks v0.1.3
⌃ [90137ffa] StaticArrays v1.9.18
⌃ [10745b16] Statistics v1.11.1
⌃ [c3572dad] Sundials v6.5.1
⌃ [0c5d862f] Symbolics v7.36.0
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/dae-pr1670-validate/benchmarks/DAE/Manifest.toml`
⌃ [47edcb42] ADTypes v1.23.0
  [14f7f29c] AMD v0.5.3
  [6e696c72] AbstractPlutoDingetjes v1.4.0
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  [79e6a3ab] Adapt v4.7.0
  [66dad0bd] AliasTables v1.1.3
  [ec485272] ArnoldiMethod v0.4.0
⌃ [4fba245c] ArrayInterface v7.28.1
  [4c555306] ArrayLayouts v1.12.2
⌃ [aae01518] BandedMatrices v1.11.0
  [e2ed5e7c] Bijections v0.2.2
⌃ [b2a6c25c] BinaryHeaps v1.0.4
⌃ [caf10ac8] BipartiteGraphs v0.1.11
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  [8e7c35d0] BlockArrays v1.10.0
⌃ [70df07ce] BracketingNonlinearSolve v1.12.5
  [fa961155] CEnum v0.5.0
  [2a0fbf3d] CPUSummary v0.2.7
  [fb6a15b2] CloseOpenIntervals v0.1.13
⌃ [944b1d66] CodecZlib v0.7.8
  [35d6a980] ColorSchemes v3.31.0
  [3da002f7] ColorTypes v0.12.1
  [c3611d14] ColorVectorSpace v0.11.0
  [5ae59095] Colors v0.13.1
⌅ [861a8166] Combinatorics v1.0.2
⌃ [38540f10] CommonSolve v0.2.13
  [bbf7d656] CommonSubexpressions v0.3.1
⌃ [f70d9fcc] CommonWorldInvalidations v1.1.2
  [34da2185] Compat v4.18.1
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⌃ [2569d6c7] ConcreteStructs v0.2.7
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  [8f4d0f93] Conda v1.10.3
  [187b0558] ConstructionBase v1.6.0
  [d38c429a] Contour v0.6.3
  [adafc99b] CpuId v0.3.1
  [a8cc5b0e] Crayons v4.2.0
⌃ [165a45c3] DASKR v3.1.5
⌃ [e993076c] DASSL v3.1.0
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⌃ [7034ab61] FastBroadcast v1.3.6
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⌃ [a4df4552] FastPower v1.4.1
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⌃ [28b8d3ca] GR v0.73.26
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⌅ [cd3eb016] HTTP v1.11.0
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⌃ [ccbc3e58] JumpProcesses v9.29.2
  [ba0b0d4f] Krylov v0.10.9
⌃ [b964fa9f] LaTeXStrings v1.4.0
⌃ [23fbe1c1] Latexify v0.16.11
  [10f19ff3] LayoutPointers v0.1.17
⌃ [87fe0de2] LineSearch v0.1.14
⌃ [7ed4a6bd] LinearSolve v5.10.0
  [2ab3a3ac] LogExpFunctions v1.0.1
  [e6f89c97] LoggingExtras v1.2.0
  [1914dd2f] MacroTools v0.5.16
  [d125e4d3] ManualMemory v0.1.8
⌃ [bb5d69b7] MaybeInplace v0.1.7
  [739be429] MbedTLS v1.1.10
  [442fdcdd] Measures v0.3.3
  [e1d29d7a] Missings v1.2.0
⌃ [961ee093] ModelingToolkit v11.39.0
⌃ [7771a370] ModelingToolkitBase v1.65.0
⌃ [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 v4.26.1
⌃ [be0214bd] NonlinearSolveBase v2.43.0
⌃ [5959db7a] NonlinearSolveFirstOrder v2.3.2
⌃ [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 v7.6.0
⌃ [6ad6398a] OrdinaryDiffEqBDF v2.4.2
⌃ [bbf590c4] OrdinaryDiffEqCore v4.14.3
⌃ [50262376] OrdinaryDiffEqDefault v2.4.4
⌃ [4302a76b] OrdinaryDiffEqDifferentiation v3.9.0
⌃ [5960d6e9] OrdinaryDiffEqFIRK v2.6.0
⌃ [127b3ac7] OrdinaryDiffEqNonlinearSolve v2.8.0
⌃ [43230ef6] OrdinaryDiffEqRosenbrock v2.6.5
⌃ [b4bd8bb3] OrdinaryDiffEqRosenbrockTableaus v2.4.1
⌃ [2d112036] OrdinaryDiffEqSDIRK v2.8.2
⌃ [b1df2697] OrdinaryDiffEqTsit5 v2.1.3
⌃ [79d7bb75] OrdinaryDiffEqVerner v2.2.2
  [90014a1f] PDMats v0.11.41
⌅ [69de0a69] Parsers v2.8.7
  [ccf2f8ad] PlotThemes v3.3.0
  [995b91a9] PlotUtils v1.4.4
⌃ [91a5bcdd] Plots v1.41.6
  [e409e4f3] PoissonRandom v0.4.13
  [f517fe37] Polyester v0.7.19
  [1d0040c9] PolyesterWeave v0.2.2
⌃ [d236fae5] PreallocationTools v1.5.0
⌅ [aea7be01] PrecompileTools v1.2.1
  [21216c6a] Preferences v1.5.2
⌃ [08abe8d2] PrettyTables v3.4.6
  [27ebfcd6] Primes v0.5.7
  [43287f4e] PtrArrays v1.4.0
  [0c0d3e7f] PureKLU v1.4.1
  [1fd47b50] QuadGK v2.11.3
  [988b38a3] ReadOnlyArrays v0.2.0
  [795d4caa] ReadOnlyDicts v1.0.1
  [3cdcf5f2] RecipesBase v1.3.4
  [01d81517] RecipesPipeline v0.6.12
⌃ [731186ca] RecursiveArrayTools v4.4.0
  [189a3867] Reexport v1.2.2
  [05181044] RelocatableFolders v1.0.1
  [ae029012] Requires v1.3.1
⌃ [ae5879a3] ResettableStacks v1.3.0
⌃ [9fe22ead] RespecializeParams v1.2.0
  [79098fc4] Rmath v0.9.0
⌃ [47965b36] RootedTrees v2.25.4
⌃ [f2b01f46] Roots v3.0.6
⌃ [7e49a35a] RuntimeGeneratedFunctions v0.5.24
⌃ [9dfe8606] SCCNonlinearSolve v1.14.1
  [94e857df] SIMDTypes v0.1.0
⌅ [0bca4576] SciMLBase v3.46.1
⌃ [31c91b34] SciMLBenchmarks v0.1.3
⌃ [19f34311] SciMLJacobianOperators v0.1.17
⌃ [a6db7da4] SciMLLogging v2.0.4
⌃ [c0aeaf25] SciMLOperators v1.26.1
⌃ [431bcebd] SciMLPublic v1.2.4
⌃ [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
  [699a6c99] SimpleTraits v0.9.6
  [a2af1166] SortingAlgorithms v1.2.3
⌃ [a57abbd0] SparseColumnPivotedQR v2.1.6
  [0a514795] SparseMatrixColorings v0.4.27
⌃ [276daf66] SpecialFunctions v2.8.3
  [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 v1.9.18
  [1e83bf80] StaticArraysCore v1.4.4
⌃ [10745b16] Statistics v1.11.1
  [82ae8749] StatsAPI v1.8.0
⌃ [2913bbd2] StatsBase v0.34.12
  [4c63d2b9] StatsFuns v2.2.1
  [7792a7ef] StrideArraysCore v0.5.9
  [69024149] StringEncodings v0.3.7
⌅ [892a3eda] StringManipulation v0.4.7
  [09ab397b] StructArrays v0.7.3
⌃ [c3572dad] Sundials v6.5.1
⌃ [2efcf032] SymbolicIndexingInterface v0.3.54
⌃ [19f23fe9] SymbolicLimits v1.1.5
⌅ [d1185830] SymbolicUtils v4.45.0
⌃ [0c5d862f] Symbolics v7.36.0
  [3783bdb8] TableTraits v1.0.1
⌃ [bd369af6] Tables v1.13.0
  [ed4db957] TaskLocalValues v0.1.3
  [62fd8b95] TensorCore v0.1.1
  [8ea1fca8] TermInterface v2.0.0
  [8290d209] ThreadingUtilities v0.5.6
  [a759f4b9] TimerOutputs v1.2.0
  [3bb67fe8] TranscodingStreams v0.11.3
  [781d530d] TruncatedStacktraces v1.4.0
⌃ [5c2747f8] URIs v1.6.3
  [3a884ed6] UnPack v1.0.2
  [1cfade01] UnicodeFun v0.4.1
  [41fe7b60] Unzip v0.2.0
  [81def892] VersionParsing v1.3.0
  [d30d5f5c] WeakCacheSets v0.1.0
  [44d3d7a6] Weave v0.10.12
  [ddb6d928] YAML v0.4.16
  [c2297ded] ZMQ v1.5.1
  [6e34b625] Bzip2_jll v1.0.9+0
  [83423d85] Cairo_jll v1.18.7+0
  [655fdf9c] DASKR_jll v1.0.1+0
  [ee1fde0b] Dbus_jll v1.16.2+0
  [2702e6a9] EpollShim_jll v0.0.20230411+1
⌃ [2e619515] Expat_jll v2.8.2+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 v3.4.1+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
⌅ [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 v10.4.1+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
  [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`