xfp>lddlZddlZddlmZddlmZmZmZm Z m Z m Z ddl m Z dgZ ddZ d dZy) N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq) make_systemgcrotmkc |d}|d}tgd|f\} } } } |g} g}d}tj}|t|z}tjt||f|j }tj d|j }tjd|j }tj|j j}d}t|D]^}|r|t|kr ||\}}nS|r|t|k(r ||}d}n8|s)||t|z k\r|||t|z z \}}n || d }d}||||}n|j}| |}t|D].\}}| ||}||||f<| |||jd | }0tj|d z|j }t| D],\}}| ||}|||<| |||jd | }.| ||d z<tjd d 5d |d z }dddtjr | ||}|d ||zkDsd}| j||j|tj|d z|d zf|j d}||d|d zd|d zf<d ||d z|d zf<tj|d z|f|j d} || d|d zddf<t!|| ||ddd\}}t#|d}||ks|s_ntj||fs t%t'|d|d zd|d zf|d d|d zfj)\}}!}!}!|ddd|d zf}|||| |||fS#1swYvxYw)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|SNxs F/usr/lib/python3/dist-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolvez_fgmres..lpsolve@Hc|Srrrs rrpsolvez_fgmres..rpsolveCrraxpydotscalnrm2)dtype)r)rrFrrignore)overdivideTFrordercol)which overwrite_qru check_finite)rr)rnpnanlenzerosronesfinfoepsrangecopy enumerateshapeerrstateisfiniteappendrabsrr conj)"matvecv0matolrrcsouter_vprepend_outer_vrrrrvszsyresBQRr1 breakdownjzww_normicalphahcurvQ2R2_s" r_fgmresrVsb  ++JRERD#tT B B A &&C CLA #b'1RXX.A bhh'A rxx(A ((288  CI1XK q3w</1:DAq c'l!2 AA Q!c'l*:%:1CL 012DAq2AA 9q "AAabM /DAq1IEAacFQ1771:v.A / xx!177+bM /DAq1IEDGQ1771:v.A /GQqS [[hx 8 d2hJE  ;;u UAAR3<'I !  ! 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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. tol, atol : float, optional Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``. The default for ``atol`` is `tol`. .. warning:: The default value for `atol` will be changed in a future release. For future compatibility, specify `atol` explicitly. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse matrix, 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 Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import gcrotmk >>> R = np.random.randn(5, 5) >>> A = csc_matrix(R) >>> b = np.random.randn(5) >>> x, exit_code = gcrotmk(A, b) >>> print(exit_code) 0 >>> np.allclose(A.dot(x), b) True 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). z$RHS must contain only finite numbers)oldestsmallestzInvalid value for 'truncate': Nzscipy.sparse.linalg.gcrotmk called without specifying `atol`. The default value will change in the future. To preserve current behavior, set ``atol=tol``.r )category stacklevel)NNNrrc|dduS)Nrr)cus rzgcrotmk..Dsr!uD0r)keyr$r%rTeconomic) overwrite_amodepivotingg-q=)rrg?rrr)rr>r?rXrY)!r r+r7all ValueErrorwarningswarnDeprecationWarningr;r3rsortemptyr5r-rpopr8rlistTr2r9zipmaxrVrrFloatingPointErrorZeroDivisionErrorrrr4)CAbx0tolmaxiterMcallbackr=kCU discard_Ctruncater>r postprocessr;psolverrrrrb_normrOuCusrJrGrHPr?new_usrNycj_outerbetabeta_tolmlrFrBrCrDpresuxrKbyr]bychycxrRhycrPgammaDWsigmaVnew_CUrLcupwpcpupczuzsC rr r s@&a"Q/Aa!K ;;q>   ?@@--9(FGG | < 2a A XXF XXF z y &OD#t z FFH q M*+JQPQFSD#tT !WF { A""')*tq!$*1  01 HHaggaj#b'*!'' E  66!9DAqy1IAacF FA IIaLQDzDI1a !##Ys2w A1Q4A1X ;AaD1aggaj1QqS6': ;1QqS6{US3[00S1Q3Z#A MM!  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