code: def _gmres_batched(
      A, b, x0, unit_residual, residual_norm, ptol, restart, M
  ):
      """
      Implements a single restart of GMRES. The ``restart``-dimensional Krylov
      subspace
      K(A, x0) = span(A(x0), A@x0, A@A@x0, ..., A^restart @ x0) is built, and the
      projection of the true solution into this subspace is returned.
  
      This implementation solves a dense linear problem instead of building
      a QR factorization during the Arnoldi process.
      """
      del ptol  # unused
      # https://www-users.cs.umn.edu/~saad/Calais/PREC.pdf
      V = tree_map(
          lambda x: jnp.pad(
              x[..., None], ((0, 0),) * x.ndim + ((0, restart),)
          ),
          unit_residual,
      )
      dtype, weak_type = dtypes._lattice_result_type(*tree_leaves(b))
      dtype = dtypes.canonicalize_dtype(dtype)
      H = lax_internal._convert_element_type(
          jnp.eye(restart, restart + 1, dtype=dtype),
          weak_type=weak_type,
      )
  
      def loop_cond(carry):
          _, _, breakdown, k = carry
          return jnp.logical_and(
              k < restart, jnp.logical_not(breakdown)
          )
  
      def arnoldi_process(carry):
          V, H, _, k = carry
          V, H, breakdown = _kth_arnoldi_iteration(k, A, M, V, H)
          return V, H, breakdown, k + 1
  
      carry = (V, H, False, 0)
      V, H, _, _ = lax.while_loop(loop_cond, arnoldi_process, carry)
  
      beta_vec = (
          jnp.zeros_like(H, shape=(restart + 1,))
          .at[0]
          .set(residual_norm.astype(dtype))
      )
      y = _lstsq(H.T, beta_vec)
      dx = tree_map(lambda X: _dot(X[..., :-1], y), V)
  
      x = _add(x0, dx)
  
      residual = M(_sub(b, A(x)))
      unit_residual, residual_norm = _safe_normalize(residual)
      return x, unit_residual, residual_norm
  
imports: from functools import partial
  import operator
  import numpy
  import jax
  import jax.numpy
  from jax import device_put
  from jax import lax
  from jax import scipy
  from jax.tree_util import tree_leaves, tree_map, tree_structure, tree_reduce, Partial
  from jax._src import dtypes
  from jax._src.lax import lax
  from jax._src.util import safe_map
label_desc: Use image node_id 23 for calling a global function with example usage: _gmres_batched(A, b, x0, unit_residual, residual_norm, ptol, restart, M) and returns: x,  unit_residual,  residual_norm, jnp, V,  H,  breakdown,  unknown