_-PhLdZddlZddlZddlmZddlmZddlm Z  d d Z dS) zbicgstab Krylov solver.N)sparse)norm) make_systemh㈵>rrc `t||||\}}} }} tjdd|t| dz}n|dkrt d||| zz } t | } || g|dd<t |} t |} | dkrd } |d kr|| z}n|d krt j|jrt |jj }nUt|jtj r't tj |j}nt d ||tj| z| zz}nt d | |kr | | dfS|jddkrAtj |tjd g| jz}| ||z dfS| }| }tj|| }d} ||z}||z}|tj||z }| ||zz }||z}||z}tj||tj||z }| ||zz||zz} |||zz } tj|| }||z ||z z}|}| ||||zz zz}|dz }t | } ||| | || |d kr|| z}n.|d kr(||tj| z| zz}| |kr | | dfS||kr | | |fS)aBiconjugate Gradient Algorithm with Stabilization. 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 criteria : string Stopping criteria, let r=r_k, x=x_k 'rr': ||r|| < tol ||b|| 'rr+': ||r|| < tol (||b|| + ||A||_F ||x||) if ||b||=0, then set ||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 residual history in the 2-norm, including the initial 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. Examples -------- >>> from pyamg.krylov import bicgstab >>> 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) = bicgstab(A,b, maxiter=2, tol=1e-8) >>> print(f'{norm(b - A*x):.6}') 4.68163 References ---------- .. [1] Yousef Saad, "Iterative Methods for Sparse Linear Systems, Second Edition", SIAM, pp. 231-234, 2003 http://www-users.cs.umn.edu/~saad/books.html alwayszpyamg.krylov._bicgstab)moduleNz%Number of iterations must be positivegg?rzrr+z5Unable to use ||A||_F with the current matrix format.zInvalid stopping criteria.r)dtype)rwarningsfilterwarningslen ValueErrorrrissparseAdata isinstancenpndarrayravellinalgshapearrayrcopyinner conjugateappend)rbx0tolcriteriamaxiterMcallback residualsx postprocessrnormrnormbrtolnormAentryrstarp rrstarOlditMpAMpalphasMsAMsomega rrstarNewbetas V/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/krylov/_bicgstab.pybicgstabr? sN*!QA66Aq!Q  H-EFFFFa&&1* 1@AAA AE A GGEw !!!  GGE GGE ||4U{ U   ?13   VNNEE RZ ( ( V!#''EETUU UebinnQ///%785666 t|| A"" wqzQRXse17;;;;<< AeG$$a(( FFHHE A**A..I B1( U"f"(5??#4#4c:::  O U"f!,,RXcmmoos-K-KK  NURZ '  OHU__..22 I%%%-8  ECK( ( aQ    U # # #   HQKKK t  ;DD   %").."3"33e;rGs04!&*o(o(o(o(o(o(r@