U Gv f,@sddlmZmZmZmZddlmZddlmZddl m Z dgZ dd dZ e d krdd lmZddlmZdZgZddZeedededgdgeeddZededededgdgeeddZeje_eejdZe eeddedZdS))innerzerosinffinfo)norm)sqrt) make_systemminresNh㈵>Fc I Cst||||\}}} }} |j} |j} d}d}|jd}|dkrFd|}dddd d d d d dddg }|rt|dt|d||ft|d||ftd}d}d}d}d}d}| j}t|j}|dkr|}n ||| }| |}t||}|dkr t dn|dkr | | dfSt |}|dkrB|} | | dfSt |}| r| |}| |}t||} t||}!t | |!}"| ||d}#|"|#krt d| |}t||} t||}!t | |!}"| ||d}#|"|#krt dd}$|}%d}&d}'|}(|})|}*d}+d},d}-t|j }.d}/d}0t||d}t||d}1|}|r\tttd||kr:|d7}d|%} | |}2| |2}|||2}|dkr||%|$|}t|2|}3||3|%|}|}|}| |}|%}$t||}%|%dkrt dt |%}%|,|3d|$d|%d7},|dkr:|%|d|kr:d}|'}4|/|&|0|3}5|0|&|/|3}6|0|%}'|/ |%}&t |6|&g}7|)|7}8t |6|%g}9t |9|}9|6|9}/|%|9}0|/|)}:|0|)})d|9};|1}<|}1|2|4|<|5|1|;}| |:|} t |-|9}-t|.|9}.|*|9}"|+|5|"}*|' |"}+t |,}t | }||}#|||}=|||}>|6}?|?dkrb|#}?|)}(|(}|dks~|dkrt}@n |||}@|dkrt}An|7|}A|-|.}|dkr0d|@}Bd|A}C|Cdkrd}|Bdkrd}||krd}|d |krd!}|=|krd"}|A|kr"d}|@|kr0d}d#}D|d$krBd%}D|dkrPd%}D||dkrbd%}D|ddkrtd%}D|(d|=krd%}D|(d|>krd%}D|d&|krd%}D|dkrd%}D|r|Drd'|| d|@f}Ed(|Af}Fd)|||6|f}Gt|E|F|G|ddkrt|dk r(|| |dkr\q:q\|rtt|d*||ft|d+||ft|d,||ft|d-|8ft|||d|dkr|}Hnd}H| | |HfS).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 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. tol : float Tolerance to achieve. The algorithm terminates when the relative residual is below `tol`. 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``. 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 zEnter minres. zExit minres. rNz3 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 = %3g shift = %23.14ez"itnlim = %3g rtol = %11.2ezindefinite preconditionergUUUUUU?znon-symmetric matrixznon-symmetric preconditionerdtypezD Itn x(1) Compatible LS norm(A) cond(A) gbar/|A|r? g?F(Tg{Gz?z%6g %12.5e %10.3ez %10.3ez %8.1e %8.1e %8.1ez( istop = %3g itn =%5gz' Anorm = %12.4e Acond = %12.4ez' rnorm = %12.4e ynorm = %12.4ez Arnorm = %12.4e)r matvecshapeprintrrepscopyr ValueErrorrrabsmaxrminr)IAbZx0shifttolmaxiterMcallbackshowcheckx postprocessrpsolvefirstlastnmsgistopitnZAnormZAcondZrnormZynormZxtyperZr1yZbeta1ZbnormwZr2stzZepsaZoldbbetaZdbarZepslnZqrnormZphibarZrhs1Zrhs2Ztnorm2ZgmaxZgmincsZsnZw2vZalfaZoldepsdeltaZgbarrootZArnormgammaphiZdenomZw1ZepsxZepsrZdiagZtest1Ztest2t1t2ZprntZstr1Zstr2Zstr3inforBF/tmp/pip-unpacked-wheel-96ln3f52/scipy/sparse/linalg/_isolve/minres.pyr sR                                                            __main__)arange)spdiagsrcCstttt|dS)N) residualsappendrr"r!)r*rBrBrCcbsrIrZcsr)formatrg-q=)r$r%r')Nr r NNNFF)ZnumpyrrrrZ numpy.linalgrmathrutilsr __all__r __name__rEZ scipy.sparserFr/rGrIfloatr!r&rr,rr"r*rBrBrBrCs.    o   $(