# scipy.sparse.linalg.cgs

scipy.sparse.linalg.cgs(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None, atol=None)

  Use Conjugate Gradient Squared iteration to solve `Ax = b`.

  Parameters:
    A{sparse matrix, ndarray, LinearOperator}

      The real-valued N-by-N matrix of the linear system. Alternatively, `A` can be a linear operator which can produce `Ax` using, e.g., `scipy.sparse.linalg.LinearOperator`.

    b ndarray

      Right hand side of the linear system. Has shape (N,) or (N,1).

  Returns:
    x ndarray

      The converged solution.

    info integer
      Provides convergence information:

        0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown

  Other Parameters:
    x0 ndarray

      Starting guess for the solution.

    tol, atol float, optional

      Tolerances for convergence, `norm(residual) <= max(tol*norm(b), atol)`. The default for `atol` is `'legacy'`, which emulates a different legacy behavior.

      Warning

      The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.

    maxiter integer

      Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.

    M{sparse matrix, ndarray, LinearOperator}

      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.

    callback function

      User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_matrix
      >>> from scipy.sparse.linalg import cgs
      >>> R = np.array([[4, 2, 0, 1],
      ...               [3, 0, 0, 2],
      ...               [0, 1, 1, 1],
      ...               [0, 2, 1, 0]])
      >>> A = csc_matrix(R)
      >>> b = np.array([-1, -0.5, -1, 2])
      >>> x, exit_code = cgs(A, b)
      >>> print(exit_code)  # 0 indicates successful convergence
      0
      >>> np.allclose(A.dot(x), b)
      True