qmr#
- scipy.sparse.linalg.qmr(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=None, M1=None, M2=None, callback=None)[source]#
- Use Quasi-Minimal Residual iteration to solve - Ax = b.- Parameters:
- A{sparse array, ndarray, LinearOperator}
- The real-valued N-by-N matrix of the linear system. Alternatively, - Acan be a linear operator which can produce- Axand- A^T xusing, e.g.,- scipy.sparse.linalg.LinearOperator.
- bndarray
- Right hand side of the linear system. Has shape (N,) or (N,1). 
- x0ndarray
- Starting guess for the solution. 
- atol, rtolfloat, optional
- Parameters for the convergence test. For convergence, - norm(b - A @ x) <= max(rtol*norm(b), atol)should be satisfied. The default is- atol=0.and- rtol=1e-5.
- maxiterinteger
- Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. 
- M1{sparse array, ndarray, LinearOperator}
- Left preconditioner for A. 
- M2{sparse array, ndarray, 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. 
 
- Returns:
- xndarray
- The converged solution. 
- infointeger
- Provides convergence information:
- 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : parameter breakdown 
 
 
 - See also - Examples - >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import qmr >>> A = csc_array([[3., 2., 0.], [1., -1., 0.], [0., 5., 1.]]) >>> b = np.array([2., 4., -1.]) >>> x, exitCode = qmr(A, b, atol=1e-5) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True