scipy.sparse.linalg.qmr¶
-
scipy.sparse.linalg.
qmr
(A, b, x0=None, tol=1e-05, maxiter=None, M1=None, M2=None, callback=None, atol=None)¶ Use Quasi-Minimal Residual iteration to solve
Ax = b
.- Parameters
- A{sparse matrix, dense matrix, LinearOperator}
The real-valued N-by-N matrix of the linear system. It is required that the linear operator can produce
Ax
andA^T x
.- b{array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
- Returns
- x{array, matrix}
The converged solution.
- infointeger
- Provides convergence information:
0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown
- Other Parameters
- x0{array, matrix}
Starting guess for the solution.
- tol, atolfloat, optional
Tolerances for convergence,
norm(residual) <= max(tol*norm(b), atol)
. The default foratol
is'legacy'
, which emulates a different legacy behavior.Warning
The default value for atol will be changed in a future release. For future compatibility, specify atol explicitly.
- maxiterinteger
Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved.
- M1{sparse matrix, dense matrix, LinearOperator}
Left preconditioner for A.
- M2{sparse matrix, dense matrix, LinearOperator}
Right preconditioner for A. Used together with the left preconditioner M1. The matrix M1*A*M2 should have better conditioned than A alone.
- callbackfunction
User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector.
See also
Examples
>>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import qmr >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = qmr(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True