parameters: (A, b, x0=None, *, tol=1e-5, atol=0.0, restart=20, maxiter=None,
            M=None, solve_method='batched')
return_statement: return x, info
docstring: GMRES solves the linear system :math:`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.
  
      Note:
          In the future, MindSpore will report the number of iterations when convergence
          is not achieved, like SciPy. Currently it is None, as a Placeholder.
  
      Args:
          A (Union[Tensor, function]): 2D Tensor or function that calculates the linear
              map (matrix-vector product) :math:`Ax` when called like :math:`A(x)`.
              As function, `A` must return Tensor with the same structure and shape as its input matrix.
          b (Tensor): Right hand side of the linear system representing a single vector.
              Can be stored as a Tensor.
          x0 (Tensor, optional): Starting guess for the solution. Must have the same structure
              as `b`. If this is unspecified, zeroes are used. Default: None.
          tol (float, optional): Tolerances for convergence,
              :math:`norm(residual) <= max(tol*norm(b), atol)`. We do not implement SciPy's
              "legacy" behavior, so MindSpore's tolerance will differ from SciPy unless you
              explicitly pass `atol` to SciPy's `gmres`. Default: 1e-5.
          atol (float, optional): The same as `tol`. Default: 0.0.
          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: 20.
          maxiter (int): Maximum number of times to rebuild the size-`restart`
              Krylov space starting from the solution found at the last iteration. If GMRES
              halts or is very slow, decreasing this parameter may help.
              Default: None.
          M (Union[Tensor, 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. Default: None.
          solve_method (str): There are two kinds of solve methods,'incremental' or 'batched'. Default: "batched".
  
              - incremental: 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".
              - batched: solve 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.
  
      Returns:
          - Tensor, the converged solution. Has the same structure as `b`.
          - None, placeholder for convergence information.
  
      Supported Platforms:
          ``CPU`` ``GPU``
  
      Examples:
          >>> import numpy as onp
          >>> import mindspore.numpy as mnp
          >>> from mindspore.common import Tensor
          >>> from mindspore.scipy.sparse.linalg import gmres
          >>> A = Tensor(mnp.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=mnp.float32))
          >>> b = Tensor(mnp.array([2, 4, -1], dtype=mnp.float32))
          >>> x, exitCode = gmres(A, b)
          >>> exitCode            # 0 indicates successful convergence
          0
          >>> onp.allclose(mnp.dot(A,x).asnumpy(), b.asnumpy())
          True
docstring_summary: GMRES solves the linear system :math:`A x = b` for x, given A and b.
docstring_tokens: GMRES solves the linear system : math : A x = b for x given A and b .
function: def gmres(A, b, x0=None, *, tol=1e-5, atol=0.0, restart=20, maxiter=None,
            M=None, solve_method='batched'):
      """
      GMRES solves the linear system :math:`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.
  
      Note:
          In the future, MindSpore will report the number of iterations when convergence
          is not achieved, like SciPy. Currently it is None, as a Placeholder.
  
      Args:
          A (Union[Tensor, function]): 2D Tensor or function that calculates the linear
              map (matrix-vector product) :math:`Ax` when called like :math:`A(x)`.
              As function, `A` must return Tensor with the same structure and shape as its input matrix.
          b (Tensor): Right hand side of the linear system representing a single vector.
              Can be stored as a Tensor.
          x0 (Tensor, optional): Starting guess for the solution. Must have the same structure
              as `b`. If this is unspecified, zeroes are used. Default: None.
          tol (float, optional): Tolerances for convergence,
              :math:`norm(residual) <= max(tol*norm(b), atol)`. We do not implement SciPy's
              "legacy" behavior, so MindSpore's tolerance will differ from SciPy unless you
              explicitly pass `atol` to SciPy's `gmres`. Default: 1e-5.
          atol (float, optional): The same as `tol`. Default: 0.0.
          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: 20.
          maxiter (int): Maximum number of times to rebuild the size-`restart`
              Krylov space starting from the solution found at the last iteration. If GMRES
              halts or is very slow, decreasing this parameter may help.
              Default: None.
          M (Union[Tensor, 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. Default: None.
          solve_method (str): There are two kinds of solve methods,'incremental' or 'batched'. Default: "batched".
  
              - incremental: 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".
              - batched: solve 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.
  
      Returns:
          - Tensor, the converged solution. Has the same structure as `b`.
          - None, placeholder for convergence information.
  
      Supported Platforms:
          ``CPU`` ``GPU``
  
      Examples:
          >>> import numpy as onp
          >>> import mindspore.numpy as mnp
          >>> from mindspore.common import Tensor
          >>> from mindspore.scipy.sparse.linalg import gmres
          >>> A = Tensor(mnp.array([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=mnp.float32))
          >>> b = Tensor(mnp.array([2, 4, -1], dtype=mnp.float32))
          >>> x, exitCode = gmres(A, b)
          >>> exitCode            # 0 indicates successful convergence
          0
          >>> onp.allclose(mnp.dot(A,x).asnumpy(), b.asnumpy())
          True
      """
  
      if x0 is None:
          x0 = mnp.zeros_like(b)
      size = b.size
      if maxiter is None:
          maxiter = 10 * size  # copied from scipy
      if restart > size:
          restart = size
  
      if M is None:
          M = lambda x: x
  
      if solve_method == 'incremental':
          x = IterativeGmres(A, M)(b, x0, tol, atol, restart, maxiter)
      elif solve_method == 'batched':
          x = BatchedGmres(A, M)(b, x0, tol, atol, restart, maxiter)
      else:
          _raise_value_error("solve_method should be in ('incremental' or 'batched'), but got {}."
                             .format(solve_method))
      _, x_norm = _safe_normalize(x)
      info = mnp.where(mnp.isnan(x_norm), _INT_NEG_ONE, _INT_ZERO)
      return x, info
function_tokens: def gmres ( A , b , x0 = None , * , tol = 1e-5 , atol = 0.0 , restart = 20 , maxiter = None , M = None , solve_method = 'batched' ) : if x0 is None : x0 = mnp . zeros_like ( b ) size = b . size if maxiter is None : maxiter = 10 * size # copied from scipy if restart > size : restart = size if M is None : M = lambda x : x if solve_method == 'incremental' : x = IterativeGmres ( A , M ) ( b , x0 , tol , atol , restart , maxiter ) elif solve_method == 'batched' : x = BatchedGmres ( A , M ) ( b , x0 , tol , atol , restart , maxiter ) else : _raise_value_error ( "solve_method should be in ('incremental' or 'batched'), but got {}." . format ( solve_method ) ) _ , x_norm = _safe_normalize ( x ) info = mnp . where ( mnp . isnan ( x_norm ) , _INT_NEG_ONE , _INT_ZERO ) return x , info