gcrotmk#
- scipy.sparse.linalg.gcrotmk(A, b, x0=None, *, rtol=1e-05, atol=0.0, maxiter=1000, M=None, callback=None, m=20, k=None, CU=None, discard_C=False, truncate='oldest')[source]#
- Solve - Ax = bwith the flexible GCROT(m,k) algorithm.- Parameters:
- A{sparse array, ndarray, LinearOperator}
- The real or complex N-by-N matrix of the linear system. Alternatively, A can be a linear operator which can produce - Axusing, e.g.,- LinearOperator.
- bndarray
- Right hand side of the linear system. Has shape (N,) or (N,1). 
- x0ndarray
- 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-5and- atol=0.0.
- maxiterint, optional
- Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. The default is - 1000.
- M{sparse array, ndarray, LinearOperator}, optional
- Preconditioner for A. The preconditioner should approximate the inverse of A. gcrotmk is a ‘flexible’ algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. 
- callbackfunction, optional
- User-supplied function to call after each iteration. It is called as - callback(xk), where- xkis the current solution vector.
- mint, optional
- Number of inner FGMRES iterations per each outer iteration. Default: 20 
- kint, optional
- Number of vectors to carry between inner FGMRES iterations. According to [2], good values are around m. Default: m 
- CUlist of tuples, optional
- List of tuples - (c, u)which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The- celements in the tuples can be- None, in which case the vectors are recomputed via- c = A uon start and orthogonalized as described in [3].
- discard_Cbool, optional
- Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems. 
- truncate{‘oldest’, ‘smallest’}, optional
- Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2] for detailed comparison. Default: ‘oldest’ 
 
- Returns:
- xndarray
- The solution found. 
- infoint
- Provides convergence information: - 0 : successful exit 
- >0 : convergence to tolerance not achieved, number of iterations 
 
 
 - References [1]- E. de Sturler, ‘’Truncation strategies for optimal Krylov subspace methods’’, SIAM J. Numer. Anal. 36, 864 (1999). [2] (1,2,3)- J.E. Hicken and D.W. Zingg, ‘’A simplified and flexible variant of GCROT for solving nonsymmetric linear systems’’, SIAM J. Sci. Comput. 32, 172 (2010). [3]- M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ‘’Recycling Krylov subspaces for sequences of linear systems’’, SIAM J. Sci. Comput. 28, 1651 (2006). - Examples - >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import gcrotmk >>> R = np.random.randn(5, 5) >>> A = csc_array(R) >>> b = np.random.randn(5) >>> x, exit_code = gcrotmk(A, b, atol=1e-5) >>> print(exit_code) 0 >>> np.allclose(A.dot(x), b) True