g|= 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 W/var/www/html/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 !  ! 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 gh㈵>gioldest) rtolr?maxiterMcallbackr>kCU discard_Ctruncatec 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 |\}}}'}(})}*}+|*|$z}*|)d|*dz},t+|)d d|*d dD]\}-}"||-|,|,jd|"},|'j3|*}.t+| |.D]#\}/}0|/\}}|||,|,jd|0 },%|j3|j3|*}1|(d|1dz}2t+|(d d|1d dD]\}3}4||3|2|2jd|4}2 d ||2z }5tj|5s t5 ||5|2}2||5|,},||2|}6||2||jd|6 }||,| | jd|6} | dk(r)t| | k\r| r| d=t| | k\rn| rni| dk(rct| | k\rT| rQt9|ddddfj$|'j$j$}7t;|7\}8}9}:g};t=|8ddd| d z fj$D]\}}<| d\}}|||=\}?}@||?||jd|>}||@||jd|>}<|;D]@\}?}@||?|}5||?||jd|5 }||@||jd|5 }B||}5|d|5z |}|d|5z |}|;j||f|;| dd| j|2|,f| jd| j f| r| DABcgc] \}A}Bd|Bf c}B}A| dd|| #d zfScc}}wcc}}w#t0$rYbwxYw#t4t6f$rY)wxYwcc}B}Aw)a Solve a matrix equation using flexible GCROT(m,k) algorithm. 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@rYrc)r r,r8all ValueErrorr<r4rr sortemptyr6r.rpopr9rlistTr3r:zipmaxrWrrFloatingPointErrorZeroDivisionErrorrrr5)CAbx0rZr?r[r\r]r>r^r_r`rar postprocessr<psolverrrrrb_normrPuCusrKrHrIPr@new_usrOycj_outerbetabeta_tolmlrGrCrDrEpresuxrLbyrebychycxrShycrQgammaDWsigmaVnew_CUrMcupwpcpupczuzsC rr r sx&a"Q/Aa!K ;;q>   ?@@--9(FGG XXF XXF z y &OD#t z FFH q M*+JQPQFSD#tT !WF 64>JD$ { A""')*tq!$*1  01 HHaggaj#b'*!'' E  66!9DAqy1IAacF FA IIaLQDzDI1a !##Ys2w A1Q4A1X ;AaD1aggaj1QqS6': ;1QqS6{US3[00S1Q3Z#A MM!  SV_%dd+1 "FE?QD9 c ,DAqQBQ1771:r*AQ1771:s+A , >|   QKAwtTF]+ 8 1F1I A7D 8 G  QR[!$ $ DAqa   '.v/0v/17=47d6k4J44O24 (6 $Aq!RQ IA4U1Q4ZAB12' .EAraRXXa["-B . UU1X2r{ 0GBDAqaRXXa[3$/B 0 UU1558_ URU]"QR&"QR&) /FAsaRXXa[#.B /  d2hJE;;u%(**& %_ %_B  Q UF + Q E * x b'Q,2qEb'Q,2  #2w!|!CRCE(**acc*,,!!f 5!%a$1Q3$ikk2*DAqa5DAqAaDAAaDA#&r!"vqu#58R!$B Q B7 Q B78#)<B #B  Q UF; Q UF;<!GESY*ASY*AMM1a&))**1 2r(y|~IItQVVX*,-B$-1 q>7Q; &&}+`    Z#$56   j.s6[ ? [1[*[(*[> [%$[%([;:[;)NNrrFr)numpyr, numpy.linalgr scipy.linalgrrrrrr iterativer !scipy.sparse.linalg._isolve.utilsr __all__rWr rrrrsU$KK%9 +MO!g#T~'4b$$QUDTUX~'r