_-PhLdZddlZddlmZddlZddlmZddlmZ d dZ dS) z#Minimum Residual projection method.N)warn)norm) make_systemh㈵>ct||||\}}}}} tjdd|#tdt |zdz}n|dkrt d|||zz } || z} t | } || g|dd<t |} | d krd }nt ||z}| ||zkr | |d fSd }d } ||| zz}tj| | }|d krtd| |dfS|tj| |z }||| zz}|dz }tj ||r|d kr|||zz } || z} n| ||zz } t | } || | | ||| ||zkr | |d fS||kr | ||fS)a Minimal residual (MR) algorithm. 1D projection method. Solves the linear system Ax = b. Left preconditioning is supported. 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. maxiter : int maximum number of iterations allowed 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 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. .. minimal residual algorithm: Preconditioned version: r = b - A x r = b - A x, z = M r while not converged: while not converged: p = A r p = M A z alpha = (p,r) / (p,p) alpha = (p, z) / (p, p) x = x + alpha r x = x + alpha z r = r - alpha p z = z - alpha p See Also -------- _steepest_descent Examples -------- >>> from pyamg.krylov import minimal_residual >>> 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) = minimal_residual(A,b, maxiter=2, tol=1e-8) >>> print(f'{norm(b - A*x):.6}') 7.26369 References ---------- .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems, Second Edition", SIAM, pp. 137--142, 2003 http://www-users.cs.umn.edu/~saad/books.html alwayszpyamg.krylov._minimal_residual)moduleNg?rz%Number of iterations must be positivegg?r2Tz; Indefinite matrix detected in minimal residual, stopping. ) rwarningsfilterwarningsintlen ValueErrorrnpinner conjugatermodappend)Abx0tolmaxiterMcallback residualsx postprocessrznormrnormbnormMb recompute_ritppzalphas ^/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_minimal_residual.pyminimal_residualr- sXb*!QA66Aq!Q  H-MNNNNc#a&&j//A% 1@AAA AE A AA GGEw !!!  GGE ||a!e sV| A""K B ( QKXakkmmQ ' ' 88 P Q Q QKNNB' 'RXakkmmQ///  M a 6"k " " rAvvAE AAAAEAI AQ    U # # #   HQKKK 3<  KNNA& & ==KNNB' 'A ()NrNNNN) __doc__rrnumpyr util.linalgrutilrr-r.r,r4s)))-%).2U(U(U(U(U(U(r.