quantax.optimizer.lsmr

quantax.optimizer.lsmr(rtol: float = 0.0001, atol: float = 0.0, maxiter: int = 500) → Callable[[...], Array]

Obtain the least-square solver for the linear equation \(Ax=b\) solved by the LSMR method lineax.LSMR.

LSMR is applied to the pure least-square problem \(\min_x \|Ax - b\|^2\) without a diagonal shift. The solution is regularized by early stopping instead: the Krylov space is built from \(b\), so the iteration only explores the dominant singular directions that \(b\) excites and never amplifies the small, noise-dominated directions before the residual tolerance is reached. Working on \(A\) directly rather than \(A^† A\) keeps the effective condition number squared smaller, which is numerically more stable than lstsq_shift_cg, especially in single precision. This solver gives the same solution regardless of whether the problem is over- or under-determined.

Parameters:

• rtol –

  The relative tolerance for terminating the LSMR iteration, default to 1e-4.

  Warning

  For single-precision inputs the internal residual estimate of LSMR is optimistic: the true relative residual \(\|Ax-b\| / \|b\|\) is typically about two orders of magnitude larger than rtol at termination. Choose rtol accordingly, or verify the residual with one extra matrix-vector product after solving.

• atol – The absolute tolerance for terminating the LSMR iteration, default to 0.

• maxiter – The maximum number of LSMR iterations, default to 500.

Returns:

A solver function with two arguments A and b and one output x as the solution of \(A x = b\). It also accepts a keyword argument x0 as the initial guess of the LSMR iteration; by default the iteration starts from zero. Initial guesses taken from previous VMC iterations are nearly orthogonal to the new solution and harm both accuracy and convergence, so they are not recommended.