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Sjjrg)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_systemgcrotmkFc UcSnUcSn[/SQU45uppU/n /nSn[RnU[U5-n[R"[U5U4UR S9n[R "SUR S9n[R"SUR S9n[R"UR 5RnSn[U5GHxnU(aU[U5:a UUunnO_U(aU[U5:Xa U"U5nSnO>U(d*UU[U5- :aUUU[U5- - unnO U"U S 5nSnUcU"U"U55nOUR5nU "U5n[U5H/unnU "UU5nUUUU4'U "UUURS U*5nM1 [R"US -UR S9n[U 5H-unnU "UU5nUUU'U "UUURS U*5nM/ U "U5UWS -'[R"S S S9 S US - nSSS5 [R"W5(a U "UU5nUS UU-:dSnU RU5 URU5 [R"US -US -4UR SS9nUUSUS -2SUS -24'S UUS -US -4'[R"US -U4UR SS9n UU SUS -2SS24'[!UU UUSSSS9unn[#US5nUU:d U(dGMy O [R"UWU45(d [%5e['USUS -2SUS -24US SUS -24R)55un n!USS2SUS -24nUUUXUU4$!,(df  GN=f)a FGMRES Arnoldi process, with optional projection or augmentation Parameters ---------- matvec : callable Operation A*x v0 : ndarray Initial vector, normalized to nrm2(v0) == 1 m : int Number of GMRES rounds atol : float Absolute tolerance for early exit lpsolve : callable Left preconditioner L rpsolve : callable Right preconditioner R cs : list of (ndarray, ndarray) Columns of matrices C and U in GCROT outer_v : list of ndarrays Augmentation vectors in LGMRES prepend_outer_v : bool, optional Whether augmentation vectors come before or after Krylov iterates Raises ------ LinAlgError If nans encountered Returns ------- Q, R : ndarray QR decomposition of the upper Hessenberg H=QR B : ndarray Projections corresponding to matrix C vs : list of ndarray Columns of matrix V zs : list of ndarray Columns of matrix Z y : ndarray Solution to ||H y - e_1||_2 = min! res : float The final (preconditioned) residual norm NcU$Nxs a/home/mande/repo/quber/.venv/lib/python3.13/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolve_fgmres..lpsolve@HcU$rrrs rrpsolve_fgmres..rpsolveCrraxpydotscalnrm2)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  ++JRERDt B B A &&C CLA #b'1RXX.A bhh'A rxx(A ((288  CI1X q3w</1:DAq c'l!2 AA Q!c'l*:%:1CL 012DAq2AA 9q "AAabMDAq1IEAacFQ1771:v.A" xx!177+bMDAq1IEDGQ1771:v.A"GQqS [[hx 8d2hJE9 ;;u  UAAR3<'I !  ! 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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)r[smallestzInvalid value for 'truncate': N)NNNrr rcUSSL$)Nrr)cus rgcrotmk..=s r!uD0r)keyr'r(r Teconomic) overwrite_amodepivotingg-q=)rrg?r"rr)rrArBr$)invalidr[re) r r.r:all ValueErrorr>r6rr sortemptyr8r0r!popr;rlistTr5r<zipmaxrYrrr9FloatingPointErrorZeroDivisionErrorrrr7)BAbx0r\rAr]r^r_r@r`rarbrcrr>psolverrrrr b_normrRuCusrMrJrKPrBnew_usrQycj_outerbetabeta_tolmlrIrErFrGpresuxrNbyrgbychycxrUhycrSgammaDWsigmaVnew_CUrOcupwpcpupczuzsB rr r sx!b#GA ;;q>    ?@@--9(FGG XXF XXF z y &OD#t z FFH q M*+JQPQFSD#tT !WF 64>JD { 1v ')*rtq!$r*1  01 HHaggaj#b'*!'' E  66!9DAqy1IAacF FA IIaLbQDzDI1a !##Ys2wA1Q4A1XAaD1aggaj1QqS6':1QqS6{US3[00S1Q3Z#A MM!  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