tfqmr#
- scipy.sparse.linalg.tfqmr(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M=None, callback=None, 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 isrtol=1e-5
, the default foratol
is0.0
.- 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)
, wherendofs = 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)
, wherexk
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.
- 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, atol=0.0) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True