Backends and Capability Matrix

torch-sla dispatches each solve() call to one of several backends. Pick a backend explicitly via backend="..." or let backend="auto" choose based on device, dtype, problem size, and which optional dependencies are installed.

The current backend lineup and what each supports:

Capability matrix

Backend	CPU	CUDA	Direct	Iterative	Complex	Batched	Distributed	Autograd
scipy	✔	–	LU / UMFPACK	CG, BiCGStab, GMRES	✔	via batch helpers	–	✔
eigen	✔	–	–	CG, BiCGStab	–	–	–	✔
pytorch	✔	✔	–	CG, BiCGStab, PCG, PBiCGStab	✔	✔	via DSparseTensor	✔
cupy	–	✔	LU (cuSPARSE)	CG, GMRES	✔	via batch helpers	–	✔
cudss	–	✔	LU / Cholesky / LDLT / LDLH	–	✔	–	–	✔
pyamg	✔	✔ (V-cycle only)	–	Ruge-Stuben AMG, Smoothed Aggregation	–	–	–	✔
amgx	–	✔	–	AMG, PCG, PBiCGStab, FGMRES (NVIDIA AmgX)	–	–	–	✔

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Platform availability

Direct-solver backends bind to vendor libraries; the table below records which OS each one builds on today.

Backend	Linux	Windows	macOS	Notes
scipy	✔	✔	✔	Pure SciPy; UMFPACK optional via scikit-umfpack.
eigen	✔	✔	✔	C++ extension, compiled at install time.
pytorch	✔	✔	✔	PyTorch-native; CUDA path active when torch.cuda.is_available().
cupy	✔	✔	–	Requires NVIDIA CUDA. CuPy has no native macOS wheels.
cudss	✔	✔	–	Requires nvmath-python[cu12] + NVIDIA CUDA. macOS is not supported by Nvidia.
pyamg	✔	✔	✔	Setup runs on CPU via the optional pyamg dependency (pip install pyamg); the V-cycle dispatches through torch.sparse so the cycle itself runs on whatever device the matrix lives on. Cross-platform AMG: macOS gets CPU AMG, CUDA boxes get GPU V-cycles.

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When backend="auto" picks what

• CUDA tensors: try cudss (best direct solver) -> cupy (LU) -> pytorch (iterative fallback).

• CPU tensors, small / medium: prefer scipy LU.

• CPU tensors, large or repeated: pytorch CG / BiCGStab keeps the memory footprint flat.

Override via backend="..." whenever you need exact control (e.g. backend="cudss" to force a direct GPU solve for a single ill-conditioned system that iterative methods cannot crack).

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Putting it together

The capability matrix maps directly to the solve() parameters: any combination where the cell is ✔ is supported:

import torch
from torch_sla import solve, PreconditionerConfig

A_csr = ...                          # any accepted matrix format
b = torch.randn(n)

# Direct GPU solve, automatic Cholesky/LDL^H selection
x = solve(A_csr, b, backend="cudss", matrix_type="auto")

# CPU iterative CG with a tuned SSOR preconditioner
x = solve(A_csr, b,
          backend="pytorch", method="cg",
          preconditioner=PreconditionerConfig(kind="ssor", omega=1.2),
          atol=1e-10, maxiter=5_000)

# CPU iterative CG with a real multi-level AMG preconditioner
# (uses PyAMG when installed, falls back to the lightweight
# 2-level stub otherwise). Reduces the iteration count by 10-100x
# on ill-conditioned PDE problems.
x = solve(A_csr, b,
          backend="pytorch", method="cg",
          preconditioner="amg",  # or PreconditionerConfig(kind="amg", ...)
          atol=1e-10, maxiter=200)

# Diagnostic return -- iteration count + residual
x, info = solve(A_csr, b, return_info=True)
print(info.iter_count, info.residual, info.converged)

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Future backends (roadmap)

The next wave of backends will extend the table with cross-platform AMG preconditioning and high-end GPU AMG:

Backend	Status	Capability	Notes
pyamg	available (this release)	CPU AMG setup + cross-device V-cycle	Already shipping. See above. Standalone solver + PyAMGHierarchy for preconditioner re-use.
amgx	available (this release)	CUDA AMG + Krylov (Nvidia AmgX)	Linux + Windows only. NVIDIA GPU required. Install via pip install torch-sla[amgx] (pulls torch-amgx wheels).
petsc	investigating	CPU/GPU direct + iterative, distributed (PETSc/hypre BoomerAMG)	Linux + macOS easy; Windows via WSL.