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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.
tol, atol : float, optional
Tolerances for convergence, ``norm(residual) <= max(tol*norm(b-Ax0), atol)``.
The default for `tol` is 1.0e-5.
The default for `atol` is ``tol * norm(b-Ax0)``.
.. warning::
The default value for `atol` will be changed in a future release.
For future compatibility, specify `atol` explicitly.
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`.
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
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