(ph* dSSKJrJrJrJr SSKJr SSKJr SSK J r S/r S SS SSSS S S .S jjr g))innerzerosinffinfo)norm)sqrt) make_systemminresNgh㈵>gF)rtolshiftmaxiterMcallbackshowcheckcP [XX!5uppn URn URn SnSnURSnUcSU-n/SQnU(aD[US-5 [USUSS US 3-5 [US USS US 3-5 [5 SnSnSnSnSnSnU Rn[ U5R nUcUR5nOXU -- nU "U5n[UU5nUS:a [S5eUS:Xa U "U 5S4$[U5nUS:Xa Un U "U 5S4$[U5nU (aU "U5nU "U5n[UU5n [UU5n![U U!- 5n"U U-US--n#U"U#:a [S5eU "U5n[UU5n [UU5n![U U!- 5n"U U-US--n#U"U#:a [S5eSn$Un%Sn&Sn'Un(Un)Un*Sn+Sn,Sn-[ U5Rn.Sn/Sn0[UUS9n[UUS9n1UnU(a[5 [5 [S5 UU:GaUS- nSU%- n U U-n2U "U25nUUU2-- nUS:a UU%U$- U-- n[U2U5n3UU3U%- U-- nUnUnU "U5nU%n$[UU5n%U%S:a [S5e[U%5n%U,U3S-U$S--U%S--- n,US:XaU%U- SU-::aSnU'n4U/U&-U0U3--n5U0U&-U/U3-- n6U0U%-n'U/*U%-n&[U6U&/5n7U)U7-n8[U6U%/5n9[U9U5n9U6U9- n/U%U9- n0U/U)-n:U0U)-n)SU9- n;U1nU6n?U?S:XaU#n?U)n(U(nUS:XdUS:Xa[ n@OUUU-- n@US:Xa[ nAOU7U- nAU-U.- nUS:XaESW@-nBSWA-nCUCS::aSnWBS::aSnUU:aSnUSU- :aSnU=U:aSnWAU::aSnW@U::aSnSnDUS::aSnDUS::aSnDUUS- :aSnDUS-S:XaSnDU(SU=-::aSnDU(SU>-::aSnDUS U- ::aSnDUS:waSnDU(aRWD(aKUS!S"U SS#S"W@S$3nES"WAS$3nFS"US%S"US%S"U6U- S%3nG[UEUF-UG-5 US-S:Xa [5 UbU"U 5 US:waO UU:aGMU(ar[5 [US&USS'US(3-5 [US)US*S+US*3-5 [US,US*S-US*3-5 [US.W8S*3-5 [UUUS--5 US:XaUnHOSnHU "U 5WH4$)/a Use MINimum RESidual iteration to solve Ax=b MINRES minimizes norm(Ax - b) for a real symmetric matrix A. 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 array, 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 array, 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``. Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import minres >>> A = csc_array([[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 zEnter 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 matvecshapeprintrrepscopyr ValueErrorrrabsmaxrminr)IAbx0r r rrrrrx postprocessr!psolvefirstlastnmsgistopitnAnormAcondrnormynormxtyper$r1ybeta1bnormwr2stzepsaoldbbetadbarepslnqrnormphibarrhs1rhs2tnorm2gmaxgmincssnw2valfaoldepsdeltagbarrootArnormgammaphidenomw1epsxepsrdiagtest1test2t1t2prntstr1str2str3infosI U/var/www/html/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_isolve/minres.pyr r sbd*!6A! XXF XXF E D  Aa% CC  e445 e 1R&f~FFG e 72,od5\JJK  E C E E E E GGE ,  C  z VVX 1Wr A "aLE qy455 !A"" GE z A"" KE  1I AY !AJ !BK AJC3>) t834 4AY !AJ "RL AJC3>) t8;< < D D D E F F D D F D <  D B B auA q B B    TU - q H aC 1I  M !8T$YN"AQqz dB    2JR{ !834 4Dz$'D!G#dAg-- !8EzRV# T BI%Dy29$T td{T4L!$dD\"E3 E\ E\6kfE    ]U2X % . AI44 5LeAg~wqyV Qs{u}s"u}t# 19D A:!EU5[)E A:E5LET  A:UBUBQwQwg~Cu}}} 7D "9D '"* D 8q=D RW D RW D DH D A:D D"XQqtEl!E%=9DuUm$DuTl!E$223 dSq\!" z N4  )N) numpyrrrr numpy.linalgrmathrutilsr __all__r rkrjrrs6** *j!$c4 DuEj!rk