a Rep> @sfddlZddlZddlmZddlmZmZmZm Z m Z m Z ddl m Z dgZddd ZdddZdS)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq) make_systemgcrotmkFc " Cs|durdd}|dur dd}tgd|f\} } } } |g} g}d}tj}|t|}tjt||f|jd}tjd|jd}tjd |jd}t|jj}d }t |D]}|r|t|kr||\}}n`|r|t|kr||}d}nB|s||t|kr|||t|\}}n|| d }d}|durF|||}n| }| |}t |D]6\}}| ||}||||f<| |||j d | }q^tj|d |jd}t | D]2\}}| ||}|||<| |||j d | }q| |||d<tj dddd|d }Wdn1s(0Yt|rH| ||}|d ||ks^d}| |||tj|d |d f|jdd}||d|dd|df<d||d|df<tj|d |f|jdd} || d|dddf<t|| ||ddd d\}}t|d}||ks&|rq.qt|||fsHtt|d|dd|df|d d|df\}}!}!}!|ddd|df}|||| |||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 NcSs|SNr xr r X/var/www/sunrise/env/lib/python3.9/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolve@sz_fgmres..lpsolvecSs|Sr r r r r rrpsolveCsz_fgmres..rpsolveaxpydotscalnrm2)dtype)r)rrFrrignore)ZoverdivideTFrordercol)whichZ overwrite_qruZ check_finite)rr)rnpnanlenzerosrZonesZfinfoepsrangecopy enumerateshapeZerrstateisfiniteappendrabsrrZconj)"matvecZv0matolrrcsZouter_vZprepend_outer_vrrrrvszsyresBQRr&Z breakdownjzwZw_normicalphaZhcurvZQ2ZR2_r r r_fgmress~1      ,       >rAh㈵>oldestc A Cs(t||||\}}} }}t|s.td| dvrDtd| | dur`tjdtdd|} |j}|j}| durxg} |dur|}d\}}}|dur| }n ||| }t gd | |f\}}}}||}|d kr|} || d fS| rd d | D| dd<| rp| j d ddtj |j d t| f|jdd}g}d }| r| d \}}|durf||}||dd|f<|d7}||q@t|dddd\}}}~t|j}g} tt|D]}|||}t|D],}!||||!||j d ||!|f }qt|||fdt|dkr,qP|d|||f|}| |qtt|| ddd| dd<| rt ddg|f\}}| D]>\}}|||}"||| | j d |"} ||||j d |" }qt|D]}#|dur|| ||}$t| ||}%|$|%kr2|#d ks| r2||| }||}$|$|%krFd}#q|t|t| d }&dd | D}z@t|||$|&|t| |||$|d\}}}'}(})}*}+|*|$9}*WntyYqYn0|)d |*d },t|)dd|*ddD]\}-}"||-|,|,j d |"},q|'|*}.t| |.D](\}/}0|/\}}|||,|,j d |0 },q&|||*}1|(d |1d }2t|(dd|1ddD]\}3}4||3|2|2j d |4}2qz"d||2}5t|5stWnttfyYqYn0||5|2}2||5|,},||2|}6||2||j d |6 }||,| | j d |6} | dkr`t| |kr| r| d =q>nx| dkrt| |kr| rt|ddddfj|'jj}7t|7\}8}9}:g};t |8ddd|dfjD]\}}<| d \}}|||=\}?}@||?||j d |>}||@||j d |>}q|;D]@\}?}@||?|}5||?||j d |5 }||@||j d |5 }qT||}5|d|5|}|d|5|}|;||fq|;| dd<| |2|,fq| d| f| rd d | D| dd<|| |#dfS)!a, Solve a matrix equation using flexible GCROT(m,k) algorithm. 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)rEsmallestzInvalid value for 'truncate': Nzscipy.sparse.linalg.gcrotmk called without specifying `atol`. The default value will change in the future. 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