-in* dSSKJrJrJrJr SSKJr SSKJr SSK J r S/r S SS SSSS S S .S jjr g))innerzerosinffinfo)norm)sqrt) make_systemminresNgh㈵>gF)rtolshiftmaxiterMcallbackshowcheckc0 [XX!5uppURn URn Sn SnURSnUcSU-n/SQnU(aD[U S-5 [U SUSS US 3-5 [U S USS US 3-5 [5 SnSnSnSnSnSnU Rn[ U5R nUcUR5nOXU -- nU "U5n[UU5nUS:a [S5eUS:XaU S4$[U5nUS:XaUn U S4$[U5nU (aU "U5nU "U5n[UU5n[UU5n [UU - 5n!UU-US--n"U!U":a [S5eU "U5n[UU5n[UU5n [UU - 5n!UU-US--n"U!U":a [S5eSn#Un$Sn%Sn&Un'Un(Un)Sn*Sn+Sn,[ U5Rn-Sn.Sn/[UUS9n[UUS9n0UnU(a[5 [5 [S5 UU:GaUS- nSU$- nUU-n1U "U15nUUU1-- nUS:a UU$U#- U-- n[U1U5n2UU2U$- U-- nUnUnU "U5nU$n#[UU5n$U$S:a [S5e[U$5n$U+U2S-U#S--U$S--- n+US:XaU$U- SU-::aSnU&n3U.U%-U/U2--n4U/U%-U.U2-- n5U/U$-n&U.*U$-n%[U5U%/5n6U(U6-n7[U5U$/5n8[U8U5n8U5U8- n.U$U8- n/U.U(-n9U/U(-n(SU8- n:U0n;Un0U1U3U;-- U4U0-- U:-nU U9U--n [U,U85n,[U-U85n-U)U8- n!U*U4U!-- n)U&*U!-n*[U+5n[U 5nUU-n"UU-U-nU>S:XaU"n>U(n'U'nUS:XdUS:Xa[ n?OUUU-- n?US:Xa[ n@OU6U- n@U,U-- nUS:XaESU?-nASW@-nBUBS::aSnWAS::aSnUU:aSnUSU- :aSnU0 : 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)HAbx0r r rrrrrxr!psolvefirstlastnmsgistopitnAnormAcondrnormynormxtyper$r1ybeta1bnormwr2stzepsaoldbbetadbarepslnqrnormphibarrhs1rhs2tnorm2gmaxgmincssnw2valfaoldepsdeltagbarrootArnormgammaphidenomw1epsxepsrdiagtest1test2t1t2prntstr1str2str3infosH U/var/www/html/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_isolve/minres.pyr r sZdQ2)JA! XXF XXF E D  Aa% CC  e445 e 1R&f~FFG e 72,od5\JJK  E C E E E E GGE ,  C  z VVX 1Wr A "aLE qy455 !1v GE z 1v KE  1I AY !AJ !BK AJC3>) t834 4AY !AJ "RL AJC3>) t8;< < D D D E F F D D F D <  D B B auA q B B    TU - q H aC 1I  M !8T$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$223 dSq\!" z d8O d8O)N) numpyrrrr numpy.linalgrmathrutilsr __all__r rjrirqs6** *j$c4 DuEjrj