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# scipy.sparse.linalg.tfqmr

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

  Use Transpose-Free Quasi-Minimal Residual 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` 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 `rtol=1e-5`, the default for `atol` is `rtol`.

      Warning

      The default value for `atol` will be changed to `0.0` in SciPy 1.14.0.

    maxiter int, optional

      Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. Default is `min(10000, ndofs * 10)`, where `ndofs = A.shape[0]`.

    M{sparse matrix, ndarray, LinearOperator}

      Inverse of the preconditioner of A. M should approximate the inverse of A and be easy to solve for (see Notes). Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. By default, no preconditioner is used.

    callback function, optional

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

    show bool, optional

      Specify `show = True` to show the convergence, `show = False` is to close the output of the convergence. Default is False.

    tol float, optional, deprecated

      Deprecated since version 1.12.0: `tfqmr` 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 int

      Provides convergence information:

      > - 0 : successful exit

      > - >0 : convergence to tolerance not achieved, number of iterations

      > - <0 : illegal input or breakdown

  Notes

  The Transpose-Free QMR algorithm is derived from the CGS algorithm. However, unlike CGS, the convergence curves for the TFQMR method is smoothed by computing a quasi minimization of the residual norm. The implementation supports left preconditioner, and the “residual norm” to compute in convergence criterion is actually an upper bound on the actual residual norm `||b - Axk||`.

  References

  [1]

  R. W. Freund, A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems, SIAM J. Sci. Comput., 14(2), 470-482, 1993.

  [2]

  Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, SIAM, Philadelphia, 2003.

  [3]

  C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, number 16 in Frontiers in Applied Mathematics, SIAM, Philadelphia, 1995.

  Examples

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