Choosing a Solver¤

Circulax separates the linear algebra backend from the simulation algorithm. The backend is selected when calling compile_circuit:

circuit = compile_circuit(net_dict, models_map, backend="klu_split")

Four backends are available. All expose the same interface — you can swap them without changing the rest of your simulation code.

KLU Split (default)¤

circuit = compile_circuit(net_dict, models_map, backend="klu_split")

Separates symbolic analysis (sparsity pattern, done once) from numeric factorisation (done each Newton step). Same AMD-ordered sparse LU as SPICE/Spectre/HSPICE, but avoids repeating the symbolic phase, giving a speedup when Newton needs many iterations.

When to use: Any circuit on CPU. The right default for almost all simulations.

Trade-offs: CPU-only (klujax). GPU falls back to Dense.

KLU¤

backend="klu"

Non-split KLU: performs symbolic + numeric factorisation together on every Newton step. Slightly simpler code path but slower than klu_split for circuits requiring many Newton iterations.

When to use: Fallback if klu_split causes issues. Functionally identical results.

Trade-offs: Same as klu_split, but repeats symbolic analysis unnecessarily.

Dense¤

backend="dense"

Assembles the full \(N \times N\) Jacobian as a dense array and solves with JAX's built-in LU factorisation (jnp.linalg.solve).

When to use:

• Very small circuits (\(N \lesssim 50\) nodes) where sparse overhead dominates.
• GPU execution: dense BLAS routines are highly optimised on GPU.
• Frequency sweeps with Harmonic Balance where jax.vmap over the full solve is desired.

Trade-offs:

Memory scales as \(O(N^2)\) and runtime as \(O(N^3)\), so it becomes impractical quickly. For any real circuit, klu_split will outperform dense.

Sparse¤

backend="sparse"

Iterative BiCGStab in pure JAX. Sparsity pattern is pre-computed at setup; only non-zero values are stored. Runs natively on GPU and TPU.

When to use: Large circuits (\(N \gtrsim 2000\)) on GPU/TPU where Dense runs out of memory.

Trade-offs: Can fail to converge on ill-conditioned systems. Not recommended for DC solves of strongly nonlinear circuits.

Summary¤

Start with klu_split (the default).