_-Ph6dZddlZddlmZddlmZ d dZdS) z:Generalized Minimum Residual Method (GMRES) Krylov solver.N) gmres_mgs)gmres_householderh㈵> householderc | 1|td| }d} tj| td| dkrt ||f|||||||d| \} }n/| dkrt ||f|||||||d| \} }ntd | |fS) a Generalized Minimum Residual Method (GMRES). GMRES iteratively refines the initial solution guess to the system Ax = b. Left preconditioned. Residuals are 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 residual orthog : string 'householder' calls _gmres_householder which uses Householder reflections to find the orthogonal basis for the Krylov space. 'mgs' calls _gmres_mgs which uses modified Gram-Schmidt to find the orthogonal basis for the Krylov space 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. The orthogonalization method, orthog='householder', is more robust than orthog='mgs', however for the majority of problems your problem will converge before 'mgs' loses orthogonality in your basis. orthog='householder' has been more rigorously tested, and is therefore currently the default 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) >>> 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. Use "restart" instead.r) stacklevelr)x0tolrestartmaxiterMcallback residualsmgsz,orthog must be either "mgs" or "householder") ValueErrorwarningswarnDeprecationWarningrr)Abr r r r rrrorthogrestrtkwargsmsgxflags S/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_gmres.pygmresrs|  IJJ J> c-!<<<<%a0rsG.5/7900)/00 DD 5aPrsG&-'/9PPHNPP DDGHHH t9) NrNNNNNrN)__doc__r _gmres_mgsr_gmres_householderrrr rr%sd@@!!!!!!111111:>NRssssssr