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# scipy.sparse.linalg.bicg

scipy.sparse.linalg.bicg(A, b, x0=None, *, tol=<object object>, maxiter=None, M=None, callback=None, atol=0.0, rtol=1e-05)

  Use BIConjugate Gradient iteration to solve `Ax = b`.

  Parameters:
    A{sparse matrix, ndarray, LinearOperator}

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

    b ndarray

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

    x0 ndarray

      Starting guess for the solution.

    rtol, atol float, optional

      Parameters for the convergence test. For convergence, `norm(b - A @ x) <= max(rtol*norm(b), atol)` should be satisfied. The default is `atol=0.` and `rtol=1e-5`.

    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.

    tol float, optional, deprecated

      Deprecated since version 1.12.0: `bicg` keyword argument `tol` is deprecated in favor of `rtol` and will be removed in SciPy 1.14.0.

  Returns:
    x ndarray

      The converged solution.

    info integer
      Provides convergence information:

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

  Examples

      >>> import numpy as np
      >>> from scipy.sparse import csc_matrix
      >>> from scipy.sparse.linalg import bicg
      >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1.]])
      >>> b = np.array([2., 4., -1.])
      >>> x, exitCode = bicg(A, b, atol=1e-5)
      >>> print(exitCode)  # 0 indicates successful convergence
      0
      >>> np.allclose(A.dot(x), b)
      True