B `< @sfddlZddlZddlmZddlmZmZmZm Z m Z m Z ddl m Z dgZddd ZdddZdS)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq) make_systemgcrotmkFc " Cs|dkrdd}|dkr dd}tddddg|f\} } } } |g} g}d}tj}|t|}tjt||f|jd }tjd |jd }tjd |jd }t|jj}d }x|t |D]n}|r|t|kr||\}}n`|r|t|kr||}d}nB|s&||t|kr&|||t|\}}n|| d }d}|dkrN|||}n| }| |}xBt |D]6\}}| ||}||||f<| |||j d| }qhWtj|d|jd }x>t | D]2\}}| ||}|||<| |||j d| }qW| |||d<tj dddd|d }WdQRXt|rB| ||}|d ||ksXd}| |||tj|d|df|jdd}||d|dd|df<d||d|df<tj|d|f|jdd} || d|dddf<t|| ||ddd d\}}t|d}||ks |rPqWt|||fs@tt|d|dd|df|dd|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 CU : 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|S)Nr )xr r R/home/dcms/DCMS/lib/python3.7/site-packages/scipy/sparse/linalg/isolve/_gcrotmk.py@z_fgmres..cSs|S)Nr )r r r r rBraxpydotscalnrm2)dtype)r)rrFrrignore)ZoverdivideTF)rordercol)whichZ overwrite_qruZ check_finite)rr)rnpnanlenzerosrZonesZfinfoepsrangecopy enumerateshapeZerrstateisfiniteappendrabsrrZconj)"matvecZv0matolZlpsolverpsolvecsZouter_vZprepend_outer_vrrrrvszsyresBQRr"Z breakdownjzwZw_normicalphaZhcurvZQ2ZR2_r r r _fgmress|1            >r>h㈵>oldestc A Cs*t||||\}}} }}t|s.td| dkrDtd| f| dkr`tjdtdd|} |j}|j}| dkrxg} |dkr|}d\}}}||| }t d d d d g| |f\}}}}||}| rd d| D| dd<| rP| j dddtj |j dt | f|jdd}g}d}xN| rf| d\}}|dkr@||}||dd|f<|d7}||qWt|dddd\}}}~t|j}g} xtt |D]}|||}x8t|D],}!||||!||j d||!|f }qWt|||fdt|dkr P|d|||f|}| |qWtt|| ddd| dd<| rt d d g|f\}}xF| D]>\}}|||}"||| | j d|"} ||||j d|" }qpWx6t|D](}#|dk r|| ||}$t| ||}%|$|%kr|#dks| r||| }||}$|$|%kr*d}#P|t|t | d}&dd| D}y@t|||$|&|t| |||$|d\}}}'}(})}*}+|*|$9}*Wntk rPYnX|)d|*d},x|=\}?}@||?||j d|>}||@||j d|>}q WxH|;D]@\}?}@||?|}5||?||j d|5 }||@||j d|5 }qPW||}5|d|5|}|d|5|}|;||fqW|;| dd<| |2|,fqW| d| f| rd!d| D| dd<|| |#dfS)"a Solve a matrix equation using flexible GCROT(m,k) algorithm. Parameters ---------- A : {sparse matrix, dense matrix, 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 : {array, matrix} Right hand side of the linear system. Has shape (N,) or (N,1). x0 : {array, matrix} 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, dense matrix, 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 : array or matrix 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). z$RHS must contain only finite numbers)rBsmallestz Invalid value for 'truncate': %rNzscipy.sparse.linalg.gcrotmk called without specifying `atol`. The default value will change in the future. 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