&jy? ddlZddlmZddlmZmZmZmZm Z m Z ddl m Z ddl mZdgZ dd Zdd d d dddddd dd dZdS)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_systemgcrotmkFc 0|d}|d}tgd|f\} } } } |g} g}d}tj}|t|z}tjt||f|j}tjd|j}tjd|j}tj|jj}d}t|D]}|r|t|kr ||\}}nm|r!|t|kr||}d}nJ|s5||t|z kr|||t|z z \}}n|| d }d}||||}n| }| |}t|D]2\}}| ||}||||f<| |||j d | }3tj|d z|j}t| D]0\}}| ||}|||<| |||j d | }1| |||d z<tj d d 5d |d z }dddn #1swxYwYtj|r | ||}|d ||zksd}| |||tj|d z|d zf|jd}||d|d zd|d zf<d ||d z|d zf<tj|d z|f|jd} || d|d zddf<t!|| ||ddd\}}t#|d}||ks|rntj|||fst%t'|d|d zd|d zf|d d|d zf\}}!}!}!|ddd|d zf}|||| |||fS)a FGMRES Arnoldi process, with optional projection or augmentation Parameters ---------- matvec : callable Operation A*x v0 : ndarray Initial vector, normalized to nrm2(v0) == 1 m : int Number of GMRES rounds atol : float Absolute tolerance for early exit lpsolve : callable Left preconditioner L rpsolve : callable Right preconditioner R cs : list of (ndarray, ndarray) Columns of matrices C and U in GCROT outer_v : list of ndarrays Augmentation vectors in LGMRES prepend_outer_v : bool, optional Whether augmentation vectors come before or after Krylov iterates Raises ------ LinAlgError If nans encountered Returns ------- Q, R : ndarray QR decomposition of the upper Hessenberg H=QR B : ndarray Projections corresponding to matrix C vs : list of ndarray Columns of matrix V zs : list of ndarray Columns of matrix Z y : ndarray Solution to ||H y - e_1||_2 = min! res : float The final (preconditioned) residual norm Nc|SNrxs RC:\Projects\DSCons\.venv\Lib\site-packages\scipy\sparse\linalg\_isolve\_gcrotmk.pylpsolvez_fgmres..lpsolve@Hc|Srrrs rrpsolvez_fgmres..rpsolveCrraxpydotscalnrm2)dtype)r r )r rFrr ignore)overdivideTFrordercol)which overwrite_qru check_finite)rr )rnpnanlenzerosronesfinfoepsrangecopy enumerateshapeerrstateisfiniteappendrabsrr conj)"matvecv0matolrrcsouter_vprepend_outer_vrrrrvszsyresBQRr2 breakdownjzww_normicalphahcurvQ2R2_s" r_fgmresrWs b      ++J+J+JRERRD#tT B B A &C CLLA #b''1RX...A bh'''A rx(((A (28   CI1XXKK  q3w<<//1:DAqq  c'll!2!2 AAA  Q!c'll*:%:%:1CLL 012DAqq2AA 9q ""AAAabMM / /DAqC1IIEAacFQ171:v..AAx!17+++bMM / /DAqC1IIEDGQ171:v..AADGGQqS [hx 8 8 8  d2hJE                ;u   UAAR3<''I !  ! 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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 ``Ax`` using, e.g., `LinearOperator`. b : ndarray Right hand side of the linear system. Has shape (N,) or (N,1). x0 : ndarray Starting guess for the solution. rtol, atol : float, 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-5`` and ``atol=0.0``. maxiter : int, 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. callback : function, optional User-supplied function to call after each iteration. It is called as ``callback(xk)``, where ``xk`` is the current solution vector. m : int, optional Number of inner FGMRES iterations per each outer iteration. Default: 20 k : int, optional Number of vectors to carry between inner FGMRES iterations. According to [2]_, good values are around `m`. Default: `m` CU : list 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 ``c`` elements in the tuples can be ``None``, in which case the vectors are recomputed via ``c = A u`` on start and orthogonalized as described in [3]_. discard_C : bool, 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 ------- x : ndarray The solution found. info : int 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] 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). 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