tfqmr(A, b, x0=None, tol=1e-05, maxiter=None, M=None, callback=None, atol=None, show=False)

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|| .

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 :None:None:`tol` is 1.0e-5. The default for atol is tol * norm(b-Ax0) .

warning

The default value for :None:None:`atol` will be changed in a future release. For future compatibility, specify :None:None:`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 :None:None:`callback(xk)`, where :None:None:`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 :None:None:`False`.

Returns

x : ndarray

The converged solution.

info : int

Provides convergence information:

Use Transpose-Free Quasi-Minimal Residual iteration to solve Ax = b .

Examples

>>> 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

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.sparse.linalg._isolve.tfqmr.tfqmr

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