_-Ph2rdZddlZddlmZddlZddlmZddlZddl m Z ddl m Z ddl mZd Z d d ZdS) zDFlexible Generalized Minimum Residual Method (fGMRES) Krylov solver.N)warn)get_lapack_funcs)norm) make_system)amg_corec@|dkrdS|tj|z S)zReturn complex sign of x.?)npabs)xs T/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_fgmres.py_mysignr s"Cxxs RVAYY;h㈵>c  | 1|td| }d} tj| tdt ||||\}}} }} |jd} tjdd td g| g\}|r!|r|}nd}|| krtd | }|}n.d}|t| d }n|| krtd | }|}| dkrAtj |tj dg| j z}| ||z dfS||| zz }t|}||g|dd<t|}|dkrd}|||zkr | | dfSd}t|D]>}|}t|d|z}|dxx|z cc<|t|z}tjd|zf| j }tj||f| j }tj|| f| j }tj| |f| j }||dddf<tj| f| j }| |d<t|D]0}dtj||z|z}||xxdz cc<t%j|tj || |dz dd||z}||dd|f<||z}t%j|tj || d|dzd|| dz kr||dz kr||dzddf}||dzd} t| }!|!dkrct| d|!z}!||dz kr/| ||dzd<||dzxx|!z cc<|t|z}|! ||dz<d||dzd<|dkrt%j||| ||| dz kr||dzdkr|||||dz\}"}#}tj |"|#gtj|# |"gg| j }$tj |$||dz|dzdz<tj|$|||dz|||dz<tj|$dddf|||dz||<d||dz<|d||dd|f<||dz krtj||dz}|||zkrn||||ot2j|d|dzd|dzf|d|dz}%tj|ddd|dzf|%}&|| |&z|dz }2t2j|d|dzd|dzf|d|dz}%tj|ddd|dzf|%}&| |&z} ||| zz }| || t|}|||| dk}'|'rJtjtj|&|'| |'z }(|(dkr| | dfcS|||zkr| | dfcS@| | |fS)a" Flexible Generalized Minimum Residual Method (fGMRES). fGMRES iteratively refines the initial solution guess to the system Ax = b. fGMRES is flexible in the sense that the right preconditioner (M) can vary from iteration to iteration. Parameters ---------- A : array, matrix, sparse matrix, LinearOperator n x n, linear system to solve b : array, matrix right hand side, shape is (n,) or (n,1) x0 : array, matrix initial guess, default is a vector of zeros tol : float Tolerance for stopping criteria, let r=r_k ||r|| < tol ||b|| if ||b||=0, then set ||b||=1 for these tests. restart : None, int - if int, restart is max number of inner iterations and maxiter is the max number of outer iterations - if None, do not restart GMRES, and max number of inner iterations is maxiter maxiter : None, int - if restart is None, maxiter is the max number of inner iterations and GMRES does not restart - if restart is int, maxiter is the max number of outer iterations, and restart is the max number of inner iterations - defaults to min(n,40) if restart=None M : array, matrix, sparse matrix, LinearOperator n x n, inverted preconditioner, i.e. solve M A x = M b. M need not be stationary for fgmres callback : function User-supplied function is called after each iteration as callback(xk), where xk is the current solution vector residuals : list residual history in the 2-norm, including the initial residual reorth : boolean If True, then a check is made whether to re-orthogonalize the Krylov space each GMRES iteration restrt : None, int Deprecated. See restart. Returns ------- xk, info xk : an updated guess after k iterations to the solution of Ax = b info : halting status == ======================================= 0 successful exit >0 convergence to tolerance not achieved, return iteration count instead. <0 numerical breakdown, or illegal input == ======================================= Notes ----- The LinearOperator class is in scipy.sparse.linalg. Use this class if you prefer to define A or M as a mat-vec routine as opposed to explicitly constructing the matrix. fGMRES allows for non-stationary preconditioners, as opposed to GMRES For robustness, Householder reflections are used to orthonormalize the Krylov Space Givens Rotations are used to provide the residual norm each iteration Flexibility implies that the right preconditioner, M, can vary from iteration to iteration Examples -------- >>> from pyamg.krylov import fgmres >>> from pyamg.util.linalg import norm >>> import numpy as np >>> from pyamg.gallery import poisson >>> A = poisson((10,10)) >>> b = np.ones((A.shape[0],)) >>> (x,flag) = fgmres(A,b, maxiter=2, tol=1e-8) >>> print(f'{norm(b - A*x):.6}') 6.54282 References ---------- .. 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