code: def _kth_arnoldi_iteration(k, A, M, V, H):
      """
      Performs a single (the k'th) step of the Arnoldi process. Thus,
      adds a new orthonormalized Krylov vector A(M(V[:, k])) to V[:, k+1],
      and that vectors overlaps with the existing Krylov vectors to
      H[k, :]. The tolerance 'tol' sets the threshold at which an invariant
      subspace is declared to have been found, in which case in which case the new
      vector is taken to be the zero vector.
      """
      dtype, _ = dtypes._lattice_result_type(*tree_leaves(V))
      dtype = dtypes.canonicalize_dtype(dtype)
      eps = jnp.finfo(dtype).eps
  
      v = tree_map(lambda x: x[..., k], V)  # Gets V[:, k]
      v = M(A(v))
      _, v_norm_0 = _safe_normalize(v)
      v, h = _iterative_classical_gram_schmidt(
          V, v, v_norm_0, max_iterations=2
      )
  
      tol = eps * v_norm_0
      unit_v, v_norm_1 = _safe_normalize(v, thresh=tol)
      V = tree_map(lambda X, y: X.at[..., k + 1].set(y), V, unit_v)
  
      h = h.at[k + 1].set(v_norm_1.astype(dtype))
      H = H.at[k, :].set(h)
      breakdown = v_norm_1 == 0.0
      return V, H, breakdown
  
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 17 for calling a global function with example usage: _kth_arnoldi_iteration(k, A, M, V, H) and returns: V,  H,  breakdown