_Mh* bddlmZmZmZmZddlmZddlmZddl m Z dgZ d dd dddd d d d Z dS))innerzerosinffinfo)norm)sqrt) make_systemminresNgh㈵>gF)rtolshiftmaxiterMcallbackshowcheckc t||||\}}} }} |j} |j} d}d}|jd}|d|z}gd}|rTt|dzt|d|d d |d zt|d |d d |dztd}d}d}d}d}d}| j}t |j}||}n||| zz }| |}t||}|dkrtd|dkr | | dfSt|}|dkr|} | | dfSt|}| r| |}| |}t||} t||}!t| |!z }"| |z|dzz}#|"|#krtd| |}t||} t||}!t| |!z }"| |z|dzz}#|"|#krtdd}$|}%d}&d}'|}(|})|}*d}+d},d}-t |j }.d}/d}0t||}t||}1|}|r+tttd||kr|dz }d|%z } | |z}2| |2}|||2zz }|dkr ||%|$z |zz }t|2|}3||3|%z |zz }|}|}| |}|%}$t||}%|%dkrtdt|%}%|,|3dz|$dzz|%dzzz },|dkr|%|z d|zkrd}|'}4|/|&z|0|3zz}5|0|&z|/|3zz }6|0|%z}'|/ |%z}&t|6|&g}7|)|7z}8t|6|%g}9t|9|}9|6|9z }/|%|9z }0|/|)z}:|0|)z})d|9z };|1}<|}1|2|4||6}?|?dkr|#}?|)}(|(}|dks|dkrt }@n|||zz }@|dkrt }An|7|z }A|-|.z }|dkrEd|@z}Bd|Az}C|Cdkrd}|Bdkrd}||krd}|d|z krd}|=|krd}|A|krd}|@|krd}d}D|dkrd }D|dkrd }D||dz krd }D|dzdkrd }D|(d|=zkrd }D|(d|>zkrd }D|d!|z krd }D|dkrd }D|rX|DrV|d"d#| dd$d#|@d%}Ed#|Ad%}Fd#|d&d#|d&d#|6|z d&}Gt|E|Fz|Gz|dzdkrt| || |dkrn||k|rtt|d'|d d(|d)zt|d*|d+d,|d+zt|d-|d+d.|d+zt|d/|8d+zt|||dzz|dkr|}Hnd}H| | |HfS)0a 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. rN) z3 beta2 = 0. If M = I, b and x are eigenvectors z/ beta1 = 0. 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