# 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)[source]#

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, atolfloat, 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.

maxiterint, 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.

callbackfunction, optional

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

showbool, optional

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

tolfloat, 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:
xndarray

The converged solution.

infoint

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
```