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Unlike the Conjugate Gradient method, A can be indefinite or singular. If shift != 0 then the method solves (A - shift*I)x = b Parameters ---------- A : {sparse matrix, ndarray, LinearOperator} The real symmetric 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 : ndarray Right hand side of the linear system. Has shape (N,) or (N,1). Returns ------- x : ndarray The converged solution. info : integer Provides convergence information: 0 : successful exit >0 : convergence to tolerance not achieved, number of iterations <0 : illegal input or breakdown Other Parameters ---------------- x0 : ndarray Starting guess for the solution. shift : float Value to apply to the system ``(A - shift * I)x = b``. Default is 0. rtol : float Tolerance to achieve. The algorithm terminates when the relative residual is below ``rtol``. maxiter : integer Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. M : {sparse matrix, ndarray, LinearOperator} Preconditioner for A. The preconditioner should approximate the inverse of A. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function User-supplied function to call after each iteration. It is called as callback(xk), where xk is the current solution vector. show : bool If ``True``, print out a summary and metrics related to the solution during iterations. Default is ``False``. check : bool If ``True``, run additional input validation to check that `A` and `M` (if specified) are symmetric. Default is ``False``. tol : float, optional, deprecated .. deprecated:: 1.12.0 `minres` keyword argument ``tol`` is deprecated in favor of ``rtol`` and will be removed in SciPy 1.14.0. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import minres >>> A = csc_matrix([[3, 2, 0], [1, -1, 0], [0, 5, 1]], dtype=float) >>> A = A + A.T >>> b = np.array([2, 4, -1], dtype=float) >>> x, exitCode = minres(A, b) >>> print(exitCode) # 0 indicates successful convergence 0 >>> np.allclose(A.dot(x), b) True References ---------- Solution of sparse indefinite systems of linear equations, C. C. Paige and M. A. Saunders (1975), SIAM J. Numer. Anal. 12(4), pp. 617-629. https://web.stanford.edu/group/SOL/software/minres/ This file is a translation of the following MATLAB implementation: https://web.stanford.edu/group/SOL/software/minres/minres-matlab.zip z'scipy.sparse.linalg.minres' keyword argument `tol` is deprecated in favor of `rtol` and will be removed in SciPy v1.14. Until then, if set, it will override `rtol`.)category stacklevelzEnter minres. zExit minres. r) z3 beta2 = 0. If M = I, b and x are eigenvectors z/ beta1 = 0. The exact solution is x0 z3 A solution to Ax = b was found, given rtol z3 A least-squares solution was found, given rtol z3 Reasonable accuracy achieved, given eps z3 x has converged to an eigenvector z3 acond has exceeded 0.1/eps z3 The iteration limit was reached z3 A does not define a symmetric matrix z3 M does not define a symmetric matrix z3 M does not define a pos-def preconditioner zSolution of symmetric Ax = bz n = 3gz shift = z23.14ez itnlim = z rtol = z11.2ezindefinite preconditionergUUUUUU?znon-symmetric matrixznon-symmetric preconditioner)dtypezD Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|r g? g?F(Tg{Gz?6g z12.5ez10.3ez8.1ez istop = z itn =5gz Anorm = z12.4ez Acond = z rnorm = z ynorm = z Arnorm = )r r warningswarnDeprecationWarningfloatmatvecshapeprintrrepscopyr ValueErrorrrabsmaxrminr)JAbx0rrrrrrrrx postprocessmsgr+psolvefirstlastnistopitnAnormAcondrnormynormxtyper.r1ybeta1bnormwr2stzepsaoldbbetadbarepslnqrnormphibarrhs1rhs2tnorm2gmaxgmincssnw2valfaoldepsdeltagbarrootArnormgammaphidenomw1epsxepsrdiagtest1test2t1t2prntstr1str2str3infosJ Blib/python3.12/site-packages/scipy/sparse/linalg/_isolve/minres.pyr r sp*!QA6Aq!Q  (E  c$61E _uSz$ XXF XXF E D  Aa% CC  e445 e 1R&f~FFG e 72,od5\JJK  E C E E E E GGE ,  C  z VVX 1Wr A "aLE qy455 !A"" GE z A"" KE  1I AY !AJ !BK AJC3>) t834 4AY !AJ "RL AJC3>) t8;< < D D D E F F D D F D <  D B B auA q B B    TU - q H aC 1I  M !8T$YN"AQqz dB    2JR{ !834 4Dz$'D!G#dAg-- !8EzRV# T BI%Dy29$T td{T4L!$dD\"E3 E\ E\6kfE    ]U2X % . 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