_-PhzHdZddlmZddlZddlmZddlmZddl m Z m Z m Z m Z ddlmZdd Zd Zdd ZddZ ddZddZdZd dZd!dZdS)"zLinear Algebra Helper Routines.)warnN)sparse)aslinearoperator)lapackget_blas_funcseigsvd)set_tol2c(tj|}|dkr>tjtj||jS|dkr&tjtj|Std)ai2-norm of a vector. Parameters ---------- x : array_like Vector of complex or real values pnorm : string '2' calculates the 2-norm 'inf' calculates the infinity-norm Returns ------- n : float 2-norm of a vector Notes ----- - currently 1+ order of magnitude faster than scipy.linalg.norm(x), which calls sqrt(numpy.sum(real((conjugate(x)*x)),axis=0)) resulting in an extra copy - only handles the 2-norm and infinity-norm for vectors See Also -------- scipy.linalg.norm : scipy general matrix or vector norm r infz/Only the 2-norm and infinity-norm are supported) npravelsqrtinnerconjrealmaxabs ValueError)xpnorms Q/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/util/linalg.pynormr sq:  A ||wrx!,,1222 ~~vbfQii   F G GGctj|stj|r{|t j|j|j|jf|j }|t j |j d|j z Stj |rHt |t j |j d|j z St jt j|t j |j df|j  S)a Infinity norm of a matrix (maximum absolute row sum). Parameters ---------- A : csr_matrix, csc_matrix, sparse, or numpy matrix Sparse or dense matrix Returns ------- n : float Infinity norm of the matrix Notes ----- - This serves as an upper bound on spectral radius. - csr and csc avoid a deep copy - dense calls scipy.linalg.norm See Also -------- scipy.linalg.norm : dense matrix norms Examples -------- >>> import numpy as np >>> from scipy.sparse import spdiags >>> from pyamg.util.linalg import infinity_norm >>> n=10 >>> e = np.ones((n,1)).ravel() >>> data = [ -1*e, 2*e, -1*e ] >>> A = spdiags(data,[-1,0,1],n,n) >>> print(infinity_norm(A)) 4.0 )shaper dtype)risspmatrix_csrisspmatrix_csc __class__rrdataindicesindptrronesr r isspmatrixdot)Aabs_As r infinity_normr,5sHQD6#8#;#;D RVAF^^QYA"#'++AG<<<<AACCC EA!'!*QW====BBDDD 6"&))RWagaj]!'BBB C C G G I IIr?cRtdg||gd}||||dS)aQuick level-1 call to BLAS y = a*x+y. Parameters ---------- x : array_like nx1 real or complex vector y : array_like nx1 real or complex vector a : float real or complex scalar Returns ------- y : array_like Input variable y is rewritten Notes ----- The call to get_blas_funcs automatically determines the prefix for the blas call. axpyrN)r)ryafns rr/r/es3. 1a& ) )! ,BBq!QKKKKKrct|}t|j}d}|jd|jdkrt dt |jd|}|mt j|jdd}|jtkr1|dt j|jddzz}n|}|t|z}t j |dz|ft j |j|j}|g}d} t|D]} ||d z} |r| dkr| || dz | f<| | |d zz} t jt j| |d } | || | f<| | |d zz} t| } | || dz| f<|| dz| f|krd }n#| | z} || |d d}t%|D]e\} }t jt j|| || | f<| || | f|zz } ft| || dz| f<|| dz| f|kr:d }|| dz| fdkr| || dz| fz } || n'| || dz| fz } || t'|d| dzd| dzfdd \}}|||||fS) aApprixmate eigenvalues. Used by approximate_spectral_radius and condest. Returns [W, E, H, V, breakdown_flag], where W and E are the eigenvectors and eigenvalues of the Hessenberg matrix H, respectively, and V is the Krylov space. breakdown_flag denotes whether Lanczos/Arnoldi suffered breakdown. E is therefore the approximate eigenvalues of A. To obtain approximate eigenvectors of A, compute V*W. Frr expected square matrixN?rT)leftright)rr r rrminrrandomrandcomplexrzeros result_typeranger) conjugaterappend enumerater)r*maxiter symmetric initial_guess breakdownbreakdown_flagv0HVbetajwalphaivEigsVectss r_approximate_eigenvaluesrUsW A  INwqzQWQZ1222!'!*g&&G Y^^AGAJ * * 7g  dRY^^AGAJ::::B $r((NB '!)W%~bh88 : : :A A D 7^^(( "I % Avv !A#q& TAbE\!F2< 22AbEKKMMBBEAadG 2 A77DAac1fI!A#q& I%%!% IA HHQKKK"##AA"!  " "1&aggii!8!8!''))DD!Q$!Q$ MQAac1fI!A#q& I%%!%QqS!V9##!AaCF) A !AaCF) A HHQKKKKa1dqsd m%t<<>> from pyamg.util.linalg import approximate_spectral_radius >>> import numpy as np >>> from scipy.linalg import eigvals, norm >>> A = np.array([[1.,0.],[0.,1.]]) >>> sr = approximate_spectral_radius(A,maxiter=3) >>> print(f'{sr:2.6}') 1.0 >>> print(max([norm(x) for x in eigvals(A)])) 1.0 rhoFr zexpected maxiter > 0rzexpected restart >= 0z(expected A to be float (complex or real)zexpected square ANr5z(initial_guess and A must have same shapezNinitial_guess must be an (n,1) or (n,) vectorr7r)rGzBreakdown occurred in step )hasattrrr intrrr<r=r>lenreshapearrayrArUrargmaxr)hstackrrr(rZ)r*tolrErestartrFrG return_vectorrJrNevectevrKrLrInvecs max_indexerrorrZs rapproximate_spectral_radiusrjs~ 1e  ? ?  Q;;344 4 Q;;455 5 7c>>GHH H 71: # #011 1   A..Bw'!!$ A!>!>>>"1%33 !KLLLM'((1,,=3Fq3IA3M3M "/000&&r1--B"AG,,,Bwqy!!  A(GYbQQQ .UB1nHQKEr ))++IeU1Wn%b)m(<+F+Fr1+M+MNNBve}}RVByM222S88 6166777  fR ]##  Q   AE  9  5Lrc t| d}|s fd}| _d}t ||\}}}}} ~~~~ tjd|Dt d|Dz |zS)aKEstimates the condition number of A. Parameters ---------- A : {dense or sparse matrix} e.g. array, matrix, csr_matrix, ... maxiter: {int} Max number of Arnoldi/Lanczos iterations symmetric : {bool} If symmetric use the far more efficient Lanczos algorithm, Else use Arnoldi. If hermitian, use symmetric=True. If complex symmetric, use symmetric=False. Returns ------- Estimate of cond(A) with \|lambda_max\| / \|lambda_min\| or simga_max / sigma_min through the use of Arnoldi or Lanczos iterations, depending on the symmetric flag Notes ----- The condition number measures how large of a change in the the problems solution is caused by a change in problem's input. Large condition numbers indicate that small perturbations and numerical errors are magnified greatly when solving the system. Examples -------- >>> import numpy as np >>> from pyamg.util.linalg import condest >>> c = condest(np.array([[1.,0.],[0.,2.]])) >>> print(f'{c:2.6}') 2.0 r c>j|zSN)rmatvecr*)rRCs rmatveczcondest..matvecs99QS1W%% %rg?c,g|]}t|Sr.0rs r zcondest..s(((DGG(((rc34K|]}t|VdSrnrtrus r zcondest..s(-B-B!d1gg-B-B-B-B-B-Br)rrqrUrrr;) r*rErFpowerrqrerfrKrLrIrps @rcondestr{sJ A E  & & & & & !GY77&UB1n q!^ F((R((( ) )#-B-Br-B-B-B*B*B BU JJrc|jd|jdkrtdtj|r|}t |\}}}~~t j|t|z S)aReturn condition number of A. Parameters ---------- A : {dense or sparse matrix} e.g. array, matrix, csr_matrix, ... Returns ------- 2-norm condition number through use of the SVD Use for small to moderate sized dense matrices. For large sparse matrices, use condest. Notes ----- The condition number measures how large of a change in the problems solution is caused by a change in problem's input. Large condition numbers indicate that small perturbations and numerical errors are magnified greatly when solving the system. Examples -------- >>> import numpy as np >>> from pyamg.util.linalg import condest >>> c = condest(np.array([[1.0,0.],[0.,2.0]])) >>> print(f'{c:2.6}') 2.0 rr r4) rrrr(toarrayr rrr;)r*USigmaVhs rcondrs{< wqzQWQZ1222  IIKKq66LAub 2 6%==U ##rTư>c ,tj|stj|}|rtj|jd}tj|jd}|jtkr`|dtj|jdzz}|dtj|jdzz}tj | | j |}tj | j | |}ttj ||z tjtj ||zz }ntj|r4tj||j z j}n.tj||j z }tj|jdkrd}n&tjtj |}||krd}dS|rt'|dS)a)Return True if A is Hermitian to within tol. Parameters ---------- A : {dense or sparse matrix} e.g. array, matrix, csr_matrix, ... fast_check : {bool} If True, use the heuristic < Ax, y> = < x, Ay> for random vectors x and y to check for conjugate symmetry. If False, compute A - A.conj().T. tol : {float} Symmetry tolerance verbose: {bool} prints max( \|A - A.conj().T\| ) if nonhermitian and fast_check=False.. \| - ) \| / sqrt( \| * \| ) if nonhermitian and fast_check=True Returns ------- True if hermitian False if nonhermitian Notes ----- This function applies a simple test of conjugate symmetry Examples -------- >>> import numpy as np >>> from pyamg.util.linalg import ishermitian >>> ishermitian(np.array([[1,2],[1,1]])) False >>> from pyamg.gallery import poisson >>> ishermitian(poisson((10,10))) True rr5TF)rr(rasarrayr<r=rr r>r)rBTfloatrrrrr$rprint) r* fast_checkrbverboserr0xAyxAtydiffs r ishermitianrsT  Q   JqMM( INN171: & & INN171: & & 7g  D 3333AD 3333AfaeeAhh))++-q11vakkmmoquuQxx00RVC$J''"'"&T2B2B*C*CCDD  Q   ,8Q^122DD8A N++D 6$*   " "DD6"&,,''D czzt d 5rc  |jd}|jd}|dkr5|dkd}d||<d|z |dd<d||<~dStjdt jd|j\}}t j||j}tj||||}|t|j}t|D]$} ||| |||d d } | d|| <%dS) a+Calculate the Moore-Penrose pseudo inverse of each block of the 3D array a. Parameters ---------- a : {dense array} Is of size (n, m, m) tol : {float} Used by gelss to filter numerically zeros singular values. If None, a suitable value is chosen for you. Returns ------- Nothing, a is modified in place so that a[k] holds the pseudoinverse of that block. Notes ----- By using lapack wrappers, this can be much faster for large n, than directly calling a pseudoinverse (SVD) Examples -------- >>> import numpy as np >>> from pyamg.util.linalg import pinv_array >>> a = np.array([[[1.,2.],[1.,1.]], [[1.,1.],[3.,3.]]]) >>> ac = a.copy() >>> # each block of a is inverted in-place >>> pinv_array(a) rr r6r-N)gelss gelss_lwork)r rTF)rlwork overwrite_a overwrite_b) rnonzerorget_lapack_funcsrr'r eye_compute_lworkr rA) r1rbnm zero_entriesrrRHSrkk gelssoutputs r pinv_arrayr0s5>  A  AAvvS))++A. ,1u!!!, LL $45M68gd!'6R6R6RUU{fQag&&&%k1a;; ;!'""C(( # #B%"sE,0eEEEKNAbEE # #r)r )r-)NN)rVrWrXNNF)rkF)TrFrn)__doc__warningsrnumpyrscipyrscipy.sparse.linalgr scipy.linalgrrrr paramsr rr,r/rUrjr{rrrrsrrrse%%000000999999999999%H%H%H%HP-J-J-J`nb/b/b/b/JBC>B.3@@@@F1K1K1K1Kh($($($VJJJJZ=#=#=#=#=#=#r