gmres(A, b, x0=None, *, tol=1e-05, atol=0.0, restart=20, maxiter=None, M=None, solve_method='batched')¶
GMRES solves the linear system A x = b for x, given A and b.
A is specified as a function performing A(vi) -> vf = A @ vi, and in principle need not have any particular special properties, such as symmetry. However, convergence is often slow for nearly symmetric operators.
A (ndarray or function) – 2D array or function that calculates the linear map (matrix-vector product)
Axwhen called like
Amust return array(s) with the same structure and shape as its argument.
b (array or tree of arrays) – Right hand side of the linear system representing a single vector. Can be stored as an array or Python container of array(s) with any shape.
x (array or tree of arrays) – The converged solution. Has the same structure as
info (None) – Placeholder for convergence information. In the future, JAX will report the number of iterations when convergence is not achieved, like SciPy.
- Other Parameters
x0 (array, optional) – Starting guess for the solution. Must have the same structure as
b. If this is unspecified, zeroes are used.
tol, atol (float, optional) – Tolerances for convergence,
norm(residual) <= max(tol*norm(b), atol). We do not implement SciPy’s “legacy” behavior, so JAX’s tolerance will differ from SciPy unless you explicitly pass
restart (integer, optional) – Size of the Krylov subspace (“number of iterations”) built between restarts. GMRES works by approximating the true solution x as its projection into a Krylov space of this dimension - this parameter therefore bounds the maximum accuracy achievable from any guess solution. Larger values increase both number of iterations and iteration cost, but may be necessary for convergence. The algorithm terminates early if convergence is achieved before the full subspace is built. Default is 20.
maxiter (integer) – Maximum number of times to rebuild the size-
restartKrylov space starting from the solution found at the last iteration. If GMRES halts or is very slow, decreasing this parameter may help. Default is infinite.
M (ndarray or function) – Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance.
solve_method (‘incremental’ or ‘batched’) – The ‘incremental’ solve method builds a QR decomposition for the Krylov subspace incrementally during the GMRES process using Givens rotations. This improves numerical stability and gives a free estimate of the residual norm that allows for early termination within a single “restart”. In contrast, the ‘batched’ solve method solves the least squares problem from scratch at the end of each GMRES iteration. It does not allow for early termination, but has much less overhead on GPUs.