Џkh= ddlZddlmZddlmZmZmZmZm Z m Z ddl m Z ddl mZdgZ dd Zdd d d ddd ddddd dZy)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_systemgcrotmkFc |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 `/var/www/teggl/fontify/venv/lib/python3.12/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_fgmresrWsb  ++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 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). 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 z$RHS must contain only finite numbers)rYsmallestzInvalid value for 'truncate': N)NNNrr rc|dduS)Nrr)cus rzgcrotmk..=sr!uD0r)keyr%r&r Teconomic) overwrite_amodepivotingg-q=)rrg?r rr)rr?r@r")invalidrYrc) r r,r8all ValueErrorr<r4rr sortemptyr6r.rpopr9rlistTr3r:zipmaxrWrrr7FloatingPointErrorZeroDivisionErrorrrr5)BAbx0rZr?r[r\r]r>r^r_r`rarr<psolverrrrrb_normrPuCusrKrHrIPr@new_usrOycj_outerbetabeta_tolmlrGrCrDrEpresuxrLbyrebychycxrShycrQgammaDWsigmaVnew_CUrMcupwpcpupczuzsB rr r sx!Ab#GAa! ;;q>   ?@@--9(FGG XXF XXF z y &OD#t z FFH q M*+JQPQFSD#tT !WF 64>JD$ { 1v ')*tq!$*1  01 HHaggaj#b'*!'' E  66!9DAqy1IAacF FA IIaLQDzDI1a !##Ys2w A1Q4A1X ;AaD1aggaj1QqS6': ;1QqS6{US3[00S1Q3Z#A MM!  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