_-Ph,jdZddlZddlmZddlZddlZddlmZm Z ddl m Z ddl m Z dZ d d ZdS) z(GMRES Gram-Schmidt-based implementation.N)warn)get_blas_funcsget_lapack_funcs)norm) make_systemct|D]2}||}tj||||dz|||dz<3dS)aApply the first k Givens rotations in Q to v. Parameters ---------- Q : list list of consecutive 2x2 Givens rotations v : array vector to apply the rotations to k : int number of rotations to apply. Returns ------- v is changed in place Notes ----- This routine is specialized for GMRES. It assumes that the first Givens rotation is for dofs 0 and 1, the second Givens rotation is for dofs 1 and 2, and so on. rN)rangenpdot)QvkjQlocs W/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_gmres_mgs.py apply_givensr sW.1XX**t6$!AaC%))!AaC%**h㈵>Fc p| 1|td| }d} tj| tdt ||||\}}} }} |jd}tjdd td g| g\}tj tj d | j rtgd | g\}}}}ntgd| g\}}}}|r!|r|}nd}||krtd|}|}n.d}|t|d}n||krtd|}|}|dkrAtj|tjdg| j z}| ||z dfS||| zz }||z}t!|}||g|dd<t!|}|dkrd}nt!||z}|||zkr | | dfSd}t#|D]9}g}tj |dz|dzf| j }tj |dz|f| j }g} |d|z ||dddf<| |dddftj |f| j }!||!d<t#|D]6}"||"dzddf}#tj||| dzz|#dd<| |#t!|#}$t#|"dzD]1}%| |%}&||&|#}'|'||"|%f<||&|#||' |#dd<2t!|#}(|(||"|"dzf<| dur[|$|$d|(zzkrOt#|"dzD]<}%| |%}&||&|#}'||"|%f|'z||"|%f<||&|#||' |#dd<=||"|"dzfdkr|d||"|"dzfz |#|#dd<|"dkrt'|||"ddf|"|"|dz kr||"|"dzfdkr|||"|"f||"|"dzf\})}*}tj|)|*gtj|* |)gg| j }+||+tj|+|!|"|"dz|!|"|"dz<||+dddf||"|"|"dzf||"|"f<d||"|"dzf<|dz }|"|dz krtj|!|"dz}|||zkrn||||t.j|d|"dzd|"dzfj|!d|"dz},tj|d|"dzddfj|,dd}-|| |-z8t.j|d|"dzd|"dzfj|!d|"dz},tj|d|"dzddfj|,dd}-| |-z} ||| zz }||z}t!|}| || |||| dk}.|.rJtjtj|-|.| |.z }/|/dkr| | dfcS|||zkr| | dfcS;| | |fS)a Generalized Minimum Residual Method (GMRES) based on MGS. GMRES iteratively refines the initial solution guess to the system Ax = b. Modified Gram-Schmidt version. 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='mgs') >>> 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 .. [2] C. T. Kelley, http://www4.ncsu.edu/~ctk/matlab_roots.html Nz*Only use restart, not restrt (deprecated).zaThe keyword argument "restrt" is deprecated and will be removed in 2024. 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