o h|= @sddlZddlmZddlmZmZmZmZm Z m Z ddl m Z ddl mZdgZ  dd d Zdd d ddddddd dd ddZdS)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_systemgcrotmkFc " Cs|durdd}|durdd}tgd|f\} } } } |g} g}d}tj}|t|}tjt||f|jd}tjd|jd}tjd |jd}t|jj}d }t |D];}|rg|t|krg||\}}n.|rv|t|krv||}d}n|s||t|kr|||t|\}}n|| d }d}|dur|||}n| }| |}t |D]\}}| ||}||||f<| |||j d | }qtj|d |jd}t | D]\}}| ||}|||<| |||j d | }q| |||d<tj dddd|d }Wdn 1swYt|r| ||}|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|rnqUt|||fstt|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 NcS|SNr xr r X/var/www/vscode/kcb/lib/python3.10/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolve@z_fgmres..lpsolvecSrrr rr r rrpsolveCrz_fgmres..rpsolveaxpydotscalnrm2)dtype)r r )r rFrr ignore)overdivideTFrordercol)which overwrite_qru check_finite)rr)rnpnanlenzerosronesfinfoepsrangecopy enumerateshapeerrstateisfiniteappendrabsrrconj)"matvecv0matolrrcsouter_vprepend_outer_vrrrrvszsyresBQRr. breakdownjzww_normicalphahcurvQ2R2_r r r_fgmress1           >rSgh㈵>gioldest) rtolr;maxiterMcallbackr:kCU discard_Ctruncatec A Cst||||\}}} }}t|std| dvr"td| |j}|j}| dur.g} | dur4|} d\}}}|durB|}n||| }tgd| |f\}}}}||}td|||\}}|dkrn|} || dfS| r{d d | D| dd<| r-| j d d d tj |j dt | f|j dd}g}d}| r| d\}}|dur||}||dd|f<|d7}||| st|dddd\}}}~t|j}g} tt |D]E}|||}t|D]}!||||!||j d||!|f }qt|||fdt|dkr n|d|||f|}| |qtt|| ddd| dd<| r\tddg|f\}}| D]\}}|||}"||| | j d|"} ||||j d|" }q|=\}?}@||?||j d|>}||@||j d|>}q|;D] \}?}@||?|}5||?||j d|5 }||@||j d|5 }q"||}5|d|5|}|d|5|}|;||fq|;| dd<| |2|,fq`| d| f| rdd | D| dd<|| |#dfS)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)rUsmallestzInvalid value for 'truncate': N)NNNrr rcSg|]\}}d|fqSrr .0rLur r r 8zgcrotmk..cSs |dduS)Nrr )cur r r=s zgcrotmk..)keyr!r"r Teconomic) overwrite_amodepivotingg-q=)rrg?rrrcSsg|]\}}|qSr r r`r r rrcs)rr;r<rUr^cSr_rr )raczuzr r rrcrd)r r(r4all ValueErrorr8r0rr sortemptyr2r*rpopr5rlistTr/r6zipmaxrSrrFloatingPointErrorZeroDivisionErrorrrr1)AAbx0rVr;rWrXrYr:rZr[r\r]r postprocessr8psolverrrrrb_normCusrGrLrbrDrEPr<new_usrKycj_outerbetabeta_tolmlrCr?r@rApresuxrHbyrebychycxrOhycrMgammaDWsigmaVnew_CUrIcupwpcpupr r rr s\          ("       " "        &   "   )NNr r Fr)numpyr( numpy.linalgr scipy.linalgrrrrrr iterativer !scipy.sparse.linalg._isolve.utilsr __all__rSr r r r rs      +