S_fT?ddlZddlmZddlmZmZmZmZm Z m Z ddl m Z ddl mZddlmZmZdgZ dd Zed ded dddddd dddd dZy)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_system)_NoValue_deprecate_positional_argsgcrotmkFc |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 Dlib/python3.12/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolvez_fgmres..lpsolveAHc|Srrrs rrpsolvez_fgmres..rpsolveDrraxpydotscalnrm2)dtype)r r )r rFrr ignore)overdivideTFr!ordercol)which overwrite_qru check_finite)rr")rnpnanlenzerosr!onesfinfoepsrangecopy enumerateshapeerrstateisfiniteappendrabsrr conj)"matvecv0matolrrcsouter_vprepend_outer_vrrrr vszsyresBQRr4 breakdownjzww_normicalphahcurvQ2R2_s" r_fgmresrYsb  ++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 !  ! XXqsAaCjs ;4AaC419 1Q3qs7 XXqsAhaggS 94AaC46 Rq'+%A1!D'l : WKZ ;;q1v m$1Q3$t!t) a$1Q3$inn&67KAq!Q !DQqSD& A aBAs ""k  s N==O z1.14.0)versionioldestgh㈵>) tolmaxiterMcallbackr@kCU discard_CtruncaterArtolc  t||||\}}}}}tj|js t d| dvrt d| |j }|j }| g} ||}d\}}}||j }n |||z }tgd||f\}}}}||}td||| | \} } |dk(r |}||dfS| r| Dcgc] \}}d|f c}}| dd| rw| jd tj|jdt| f|jd }g}d}| r@| jd\}}|||}||dd|f<|d z }|j|| r@t!|ddd\}}}~t#|j$} g}!t't| D]}|||}t'|D]&}"||||"||jd||"|f }(t)|||fdt)|dzkrn$|d|||fz |}|!j|t#t+| |!ddd| dd| rVtddg|f\}}| D]?\}}|||}#||||jd|#}||||jd|# }At'|D]}$|||||}%t-| | |z}&|%|&kr|$dkDs| r|||z }||}%|%|&krd}$nc|t-|t| z dz}'| Dcgc]\}}| } }} t/|||%z |'|t-| | |z|%z | \}}}(})}*}+},|+|%z}+|*d|+dz}-t+|*d d|+d dD]\}.}#||.|-|-jd|#}-|(j3|+}/t+| |/D]#\}0}1|0\}}|||-|-jd|1 }-%|j3|j3|+}2|)d|2dz}3t+|)d d|2d dD]\}4}5||4|3|3jd|5}3 d ||3z }6tj|6s t5 ||6|3}3||6|-}-||3|}7||3||jd|7 }||-||jd|7}| dk(r)t| |k\r| r| d=t| |k\rn| rni| dk(rct| |k\rT| rQt9|ddddfj$|(j$j$}8t;|8\}9}:};g}}?|>\}@}A||@||jd|?}||A||jd|?}<|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, atol=1e-5) >>> 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. 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