3 Od< @spddlZddlZddlmZddlmZmZmZm Z m Z m Z ddl m Z dgZddffdfddZdd dZdS)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 }tj|jj}d }x~t |D]p}|r|t|kr||\}}nb|r|t|kr||}d}nD| r(||t|kr(|||t|\}}n|| d}d}|dkrP|||}n|j }| |}xBt |D]6\}}| ||}||||f<| |||j d | }qjWtj|d |jd }x>t | D]2\}}| ||}|||<| |||j d | }qW| |||d <tj dddd |d}WdQRXtj|rD| ||}|d||ksZd}| j||j|tj|d |d f|jdd}||d|d d|d f<d ||d |d f<tj|d |f|jdd} || d|d ddf<t|| ||ddd d\}}t|d}||ks"|rPqWtj|||fsBtt|d|d d|d f|d d|d fj\}}!}!}!|ddd|d f}|||| |||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)N)xr r W/var/www/html/virt/lib64/python3.6/site-packages/scipy/sparse/linalg/isolve/_gcrotmk.py@sz_fgmres..cSs|S)Nr )r r r r rBsaxpydotscalnrm2)dtyperFignore)ZoverdivideTF)rordercol)whichZ overwrite_qruZ check_finite)rr)rrrrr)rr)rnpnanlenZzerosrZonesZfinfoepsrangecopy enumerateshapeZerrstateisfiniteappendrabsrrZconj)"matvecZv0matolZlpsolverpsolvecsZouter_vZprepend_outer_vrrrrvszsyresBQRr Z breakdownjzwZw_normicalphaZhcurvZQ2ZR2_r r r _fgmress|1            >r<h㈵>oldestc ACs*t||||\}}} }}tj|js.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| jd\}}|dkr@||}||dd|f<|d7}|j|qWt|dddd\}}}~t|j}g} xtt |D]}|||}x8t|D],}!||||!||j d||!|f }qWt|||fdt|d kr P|d|||f|}| j|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\}}}'}(})}*}+|*|$9}*Wntk rPYnX|)d|*d},x|=\}?}@||?||j d|>}||@||j d|>}q WxH|;D]@\}?}@||?|}5||?||j d|5 }||@||j d|5 }qPW||}5|d|5|}|d|5|}|;j||fqW|;| dd<| j|2|,fqW| jd| j f| rdd| D| dd<|| |#dfS)$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$RHS must contain only finite numbersr@smallestz Invalid value for 'truncate': %rNzscipy.sparse.linalg.gcrotmk called without specifying `atol`. The default value will change in the future. To preserve current behavior, set ``atol=tol``.r)category stacklevelrrrrcSsg|]\}}d|fqS)Nr ).0r8ur r r )szgcrotmk..cSs |ddk S)Nrr )cur r r r.szgcrotmk..)keyrr)rrrTZeconomic)Z overwrite_amodeZpivotingg-q=g?cSsg|] \}}|qSr r )rDr8rEr r r rFys)r+r*r,cSsg|]\}}d|fqS)Nr )rDczuzr r r rFs)r@rA)NNN)rrrrr)!r rr%all ValueErrorwarningswarnDeprecationWarningr(rsortemptyr$rrpopr&rlistTr!r'zipmaxr<rrFloatingPointErrorZeroDivisionErrorrrr#r")AAbZx0ZtolmaxiterMcallbackr)kZCUZ discard_Ctruncater*r postprocessr(ZpsolverrrrrZb_normCusr4r8rEr2r3Pr,Znew_usr7ZycZj_outerbetaZbeta_tolmlr1r-r.r/ZpresZuxr5ZbyrGZbycZhyZcxr:Zhycr9gammaDWsigmaVZnew_CUr6cupZwpcpZupr r r r sR       ,"       $ $          ,   $  ) Nr=r>NNr?NNFr@N)rNZnumpyrZ numpy.linalgrZ scipy.linalgrrrrrrZ scipy.sparse.linalg.isolve.utilsr __all__r<r r r r r s    (