code: def custom_linear_solve(
      matvec,
      b,
      solve,
      transpose_solve=None,
      symmetric=False,
      has_aux=False,
  ):
      """Perform a matrix-free linear solve with implicitly defined gradients.
  
      This function allows for overriding or defining gradients for a linear
      solve directly via implicit differentiation at the solution, rather than by
      differentiating *through* the solve operation. This can sometimes be much faster
      or more numerically stable, or differentiating through the solve operation
      may not even be implemented (e.g., if ``solve`` uses ``lax.while_loop``).
  
      Required invariant::
  
          x = solve(matvec, b)  # solve the linear equation
          assert matvec(x) == b  # not checked
  
      Args:
        matvec: linear function to invert. Must be differentiable.
        b: constant right handle side of the equation. May be any nested structure
          of arrays.
        solve: higher level function that solves for solution to the linear
          equation, i.e., ``solve(matvec, x) == x`` for all ``x`` of the same form
          as ``b``. This function need not be differentiable.
        transpose_solve: higher level function for solving the transpose linear
          equation, i.e., ``transpose_solve(vecmat, x) == x``, where ``vecmat`` is
          the transpose of the linear map ``matvec`` (computed automatically with
          autodiff). Required for backwards mode automatic differentiation, unless
          ``symmetric=True``, in which case ``solve`` provides the default value.
        symmetric: bool indicating if it is safe to assume the linear map
          corresponds to a symmetric matrix, i.e., ``matvec == vecmat``.
        has_aux: bool indicating whether the ``solve`` and ``transpose_solve`` functions
          return auxiliary data like solver diagnostics as a second argument.
  
      Returns:
        Result of ``solve(matvec, b)``, with gradients defined assuming that the
          solution ``x`` satisfies the linear equation ``matvec(x) == b``.
      """
      if transpose_solve is None and symmetric:
          transpose_solve = solve
  
      b_flat, in_args_tree = tree_flatten((b,))
      b_avals = tuple(_map(_abstractify, b_flat))
  
      (tree,) = treedef_children(in_args_tree)
  
      def _shape_checked(fun, name, has_aux):
          def f(x):
              y = fun(x)
              _check_shapes(name, "b", y, b_flat)
              return y
  
          def f_aux(x):
              y, aux = fun(x)
              _check_shapes(name, "b", y, b_flat)
              return y, aux
  
          return f_aux if has_aux else f
  
      # no auxiliary data assumed for matvec
      matvec_jaxpr, matvec_consts, out_tree = _initial_style_jaxpr(
          _shape_checked(matvec, "matvec", False),
          in_args_tree,
          b_avals,
          "custom_linear_solve",
      )
      _check_tree("matvec", "b", out_tree, tree, False)
  
      solve_jaxpr, solve_consts, out_tree = _initial_style_jaxpr(
          _shape_checked(partial(solve, matvec), "solve", has_aux),
          in_args_tree,
          b_avals,
          "custom_linear_solve",
      )
      _check_tree("solve", "b", out_tree, tree, has_aux)
  
      if transpose_solve is None:
          vecmat_jaxpr = tr_solve_jaxpr = None
          vecmat_consts = tr_solve_consts = []
      else:
          if symmetric:
              vecmat = matvec
              vecmat_jaxpr = matvec_jaxpr
              vecmat_consts = matvec_consts
          else:
              vecmat = _transpose_one_output(matvec, b)
              (
                  vecmat_jaxpr,
                  vecmat_consts,
                  out_tree,
              ) = _initial_style_jaxpr(
                  vecmat, in_args_tree, b_avals, "custom_linear_solve"
              )
              assert out_tree == tree
  
          (
              tr_solve_jaxpr,
              tr_solve_consts,
              out_tree,
          ) = _initial_style_jaxpr(
              _shape_checked(
                  partial(transpose_solve, vecmat),
                  "transpose_solve",
                  has_aux,
              ),
              in_args_tree,
              b_avals,
              "custom_linear_solve",
          )
          _check_tree("transpose_solve", "b", out_tree, tree, has_aux)
  
      all_consts = [
          matvec_consts,
          vecmat_consts,
          solve_consts,
          tr_solve_consts,
      ]
      const_lengths = _LinearSolveTuple(*_map(len, all_consts))
      jaxprs = _LinearSolveTuple(
          matvec_jaxpr, vecmat_jaxpr, solve_jaxpr, tr_solve_jaxpr
      )
  
      out_flat = linear_solve_p.bind(
          *(_flatten(all_consts) + b_flat),
          const_lengths=const_lengths,
          jaxprs=jaxprs,
      )
  
      return tree_unflatten(out_tree, out_flat)
  
imports: import collections
  from functools import partial
  import operator
  import jax
  from jax.tree_util import tree_flatten, treedef_children, tree_leaves, tree_unflatten, treedef_tuple
  from jax._src import ad_util
  from jax._src import core
  from jax._src import linear_util
  from jax._src.core import raise_to_shaped
  from jax._src.interpreters import ad
  from jax._src.interpreters import batching
  from jax._src.interpreters import mlir
  from jax._src.interpreters import xla
  from jax._src.traceback_util import api_boundary
  from jax._src.util import split_list, safe_map
  import numpy
  from jax._src.lax.control_flow.common import _abstractify, _check_tree, _initial_style_jaxpr
label_desc: Use image node_id 9 for calling a global function with example usage: custom_linear_solve(matvec, b, solve, transpose_solve, symmetric, has_aux) and returns: tree_unflatten, unknown, y, y,  aux