_-Phv4rdZddlZddlmZddlZddlmZddlZddl m Z ddl m Z ddl mZd Z d d ZdS) z'GMRES Housholder-based implementations.N)warn)get_lapack_funcs)norm) make_system)amg_corec@|dkrdS|tj|z S)zReturn complex sign of x.?)npabs)xs _/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_gmres_householder.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 }||z}t|}||g|dd<t|}|dkrd}nt||z}|||zkr | | dfSd}t|D]}|}t|d|z}|d|z|d<|t|z |dd<tjd|zf| j }tj||f| j }tj|dz| f| j }||dddf<tj| f| j }| |d<t|D]d}dtj||z|z}||dz||<t%j|tj || |dz dd||z}||z}t%j|tj || d|dzd|| dz kr||dz kr||dzddf}||dzd} t| }!|!dkrht| d|!z}!||dz kr4| ||dzd<||dzxx|!z cc<|t|z |dd<|! ||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 }||dz krtj||dz}|||zkrn||||t2j|d|dzd|dzf|d|dz}%tj| j| j }&t%j|&tj |tj |%| |dd|| |&zft2j|d|dzd|dzf|d|dz}%tj| j| j }&t%j|&tj |tj |%| |dd| |&z| dd<||| zz }||z}t|}| || |||| dk}'|'rJtjtj|&|'| |'z }(|(dkr| | dfcS|||zkr| | dfcS| | |fS)a Generalized Minimum Residual Method (GMRES) based on Housholder. GMRES iteratively refines the initial solution guess to the system Ax = b. Householder reflections are used for orthogonalization. Left preconditioning, leading to preconditioned residuals. 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 ||M r|| < tol ||M b|| if ||b||=0, then set ||M 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. callback : function User-supplied function is called after each iteration as callback(xk), where xk is the current solution vector residuals : list preconditioned residual history in the 2-norm, including the initial preconditioned 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. For robustness, modified Gram-Schmidt is used to orthogonalize the Krylov Space Givens Rotations are used to provide the residual norm each iteration The residual is the *preconditioned* residual. Examples -------- >>> from pyamg.krylov import gmres >>> 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) = gmres(A, b, maxiter=2, tol=1e-8, orthog='householder') >>> print(f'{norm(b - A*x):.6}') 6.54282 References ---------- .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems, Second Edition", SIAM, pp. 151-172, pp. 272-275, 2003 http://www-users.cs.umn.edu/~saad/books.html Nz*Only use restart, not restrt (deprecated).zaThe keyword argument "restrt" is deprecated and will be removed in 2024. 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