Q(h* `ddlmZmZmZmZddlmZddlmZddl m Z dgZ d dd dddd d d d Z y))innerzerosinffinfo)norm)sqrt) make_systemminresNgh㈵>gF)rtolshiftmaxiterMcallbackshowcheckc t||||\}}} }} |j} |j} d}d}|jd}|d|z}gd}|rDt|dzt|d|dd |d zt|d |dd |d ztd}d}d}d}d}d}| j}t |j }||j}n||| zz }| |}t||}|dkr td|dk(r | | dfSt|}|dk(r |} | | dfSt|}| r| |}| |}t||} t||}!t| |!z }"| |z|dzz}#|"|#kDr td| |}t||} t||}!t| |!z }"| |z|dzz}#|"|#kDr tdd}$|}%d}&d}'|}(|})|}*d}+d},d}-t |j}.d}/d}0t||}t||}1|}|rtttd||kr|dz }d|%z } | |z}2| |2}|||2zz }|dk\r ||%|$z |zz }t|2|}3||3|%z |zz }|}|}| |}|%}$t||}%|%dkr tdt|%}%|,|3dz|$dzz|%dzzz },|dk(r |%|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}?|?dk(r|#}?|)}(|(}|dk(s|dk(rt }@n|||zz }@|dk(rt }An|7|z }A|-|.z }|dk(r>d@z}BdAz}C|Cdkrd}Bdkrd}||k\rd}|d|z k\rd}|=|k\rd}A|krd}@|krd}d}D|dkrd}D|dkrd}D||dz k\rd}D|dzdk(rd}D|(d|=zkrd}D|(d|>zkrd}D|d |z krd}D|dk7rd}D|rLDrJ|d!d"| dd#d"@d$}Ed"Ad$}Fd"|d%d"|d%d"|6|z d%}Gt|E|Fz|Gz|dzdk(r t||| |dk7rn||kr|rrtt|d&|dd'|d(zt|d)|d*d+|d*zt|d,|d*d-|d*zt|d.8d*zt|||dzz|dk(r|}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 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. 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