_-PhdZddlmZddlZddlmZddlmZddl m Z ddl m Z dd l mZmZmZmZmZdd lmZd!d Zd"dZd#dZd$dZ d%dZ d%dZdZd&dZd'dZd ZdS)(aoStrength of Connection functions. Requirements for the strength matrix C are: 1) Nonzero diagonal whenever A has a nonzero diagonal 2) Non-negative entries (float or bool) in [0,1] 3) Large entries denoting stronger connections 4) C denotes nodal connections, i.e., if A is an nxn BSR matrix with row block size of m, then C is (n/m) x (n/m) )warnN)sparse)amg_core)jacobi)approximate_spectral_radius)scale_rows_by_largest_entry amalgamate scale_rowsget_block_diag scale_columns)set_tol@Tcptj|rrt|jd|jdz }t j|jjdf}tj||j |j f||f}tj |s.tdtj tj|}|jd}|j }t jt j|jd|j dd|j ddz }||df||dfz dz} t!d|D]} | ||| f||| fz dzz } t j| } d| | dk<tj| |j |j f|j} |d urC|t jkr2t)j| jd|| j | j | jn2t)j| jd|| j | j | j| | tj| jd| jdd z} d | jz | _t3| } | S) aDistance based strength-of-connection. Parameters ---------- A : csr_matrix or bsr_matrix Square, sparse matrix in CSR or BSR format V : array Coordinates of the vertices of the graph of A relative_drop : bool If false, then a connection must be within a distance of theta from a point to be strongly connected. If true, then the closest connection is always strong, and other points must be within theta times the smallest distance to be strong Returns ------- C : csr_matrix C(i,j) = distance(point_i, point_j) Strength of connection matrix where strength values are distances, i.e. the smaller the value, the stronger the connection. Sparsity pattern of C is copied from A. Notes ----- - theta is a drop tolerance that is applied row-wise - If a BSR matrix given, then the return matrix is still CSR. The strength is given between super nodes based on the BSR block size. Examples -------- >>> from pyamg.gallery import load_example >>> from pyamg.strength import distance_strength_of_connection >>> data = load_example('airfoil') >>> A = data['A'].tocsr() >>> S = distance_strength_of_connection(data['A'], data['vertices']) rshapeImplicit conversion of A to csrrNgư>Tcsrformat?)risspmatrix_bsrintr blocksizenponesdata csr_matrixindicesindptrisspmatrix_csrrSparseEfficiencyWarningrepeatarangerangesqrtcopyinfrapply_distance_filterapply_absolute_distance_filtereliminate_zeroseyer ) AVtheta relative_dropsnudimcolsrowsCds N/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/strength.pydistance_strength_of_connectionr;sNQH ak!n, - - GQV\!_& ' '  q!)QX6r2h G G G   # #! .0NOOO  a  '!*C 9D 9RYqwqz**AHQRRL18AbD>,I J JD 47aaj 1$A 1c]]** aaj1T1W:% ))  AAa$hK1ainn.. @ ! ) ) )A BF??  *171:uah+,9af > > > / E1801 16 C C C FJqwqz171:e < < <= theta * max(|A[i,k]|), where k != i (norm='abs') -A[i,j] >= theta * max(-A[i,k]), where k != i (norm='min') Parameters ---------- A : csr_matrix or bsr_matrix Square, sparse matrix in CSR or BSR format theta : float Threshold parameter in [0,1] block : bool, default True Compute strength of connection block-wise norm : 'string', default 'abs' Measure used in computing the strength:: 'abs' : |C[i,j]| >= theta * max(|C[i,k]|), where k != i 'min' : -C[i,j] >= theta * max(-C[i,k]), where k != i where C = A for non-block-wise computations. For block-wise:: 'abs' : C[i, j] is the maximum absolute value in block A[i, j] 'min' : C[i, j] is the minimum (negative) value in block A[i, j] 'fro' : C[i, j] is the Frobenius norm of block A[i, j] Returns ------- S : csr_matrix Matrix graph defining strong connections. S[i,j] ~ 1.0 if vertex i is strongly influenced by vertex j, or block i is strongly influenced by block j if block=True. See Also -------- symmetric_strength_of_connection : symmetric measure used in SA evolution_strength_of_connection : relaxation based strength measure Notes ----- - A symmetric A does not necessarily yield a symmetric strength matrix S - Calls C++ function classical_strength_of_connection - The version as implemented is designed for M-matrices. Trottenberg et al. use max A[i,k] over all negative entries, which is the same. A positive edge weight never indicates a strong connection. - See [2000BrHeMc]_ and [2001bTrOoSc]_ References ---------- .. [2000BrHeMc] Briggs, W. L., Henson, V. E., McCormick, S. F., "A multigrid tutorial", Second edition. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. xii+193 pp. .. [2001bTrOoSc] Trottenberg, U., Oosterlee, C. W., Schuller, A., "Multigrid", Academic Press, Inc., San Diego, CA, 2001. xvi+631 pp. Examples -------- >>> import numpy as np >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import classical_strength_of_connection >>> n=3 >>> stencil = np.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = classical_strength_of_connection(A, 0.0) rrzMatrix must have square blockszexpected theta in [0,1]r>axisminfrozInvalid choice of norm.gؗҜrCz*Unrecognized option for norm for strength.r)rrr ValueErrorrrrmaxr>rrB conjugatesumr#rr$r empty_liker"r!r$classical_strength_of_connection_abs$classical_strength_of_connection_minr r-r ) r/r1blocknormrNrSpSjSxSs r: classical_strength_of_connectionrSosPQ KNak!n , ,!+a.12D2D=>> >KN    UQYY2333 &q))  Y& ' ' 5==6"&a888qAAADD U]]6"&a000q999DD U]]<''!&0D6"&A...Q777DD677 7&)RVD\\E !""$Q'' % 2F4R S S S!!$$Av GAJ qx B qy ! !B t  B ~5 uah 4R = = = = 5 uah 4R = = = =EFFF2r2,q!f555AVAF^^AF#A&&A1}}U} q) $ $ Hr<c 6|dkrtdtj|rtj|j}tj|j}tj|j}tj }||j d||j|j|j|||tj |||f|j }ntj |r{|j \}}|j \} } | | krtd|dkrtjt|j|j} tj | |j|jft%|| z t%|| z f}ntj|j|jzd| | z} tj| d} tj | |j|jft%|| z t%|| z f}t||St/d tj|j|_t3|}|S) a;Symmetric Strength Measure. Compute strength of connection matrix using the standard symmetric measure An off-diagonal connection A[i,j] is strong iff:: abs(A[i,j]) >= theta * sqrt( abs(A[i,i]) * abs(A[j,j]) ) Parameters ---------- A : csr_matrix Matrix graph defined in sparse format. Entry A[i,j] describes the strength of edge [i,j] theta : float Threshold parameter (positive). Returns ------- S : csr_matrix Matrix graph defining strong connections. S[i,j]=1 if vertex i is strongly influenced by vertex j. See Also -------- symmetric_strength_of_connection : symmetric measure used in SA evolution_strength_of_connection : relaxation based strength measure Notes ----- - For vector problems, standard strength measures may produce undesirable aggregates. A "block approach" from Vanek et al. is used to replace vertex comparisons with block-type comparisons. A connection between nodes i and j in the block case is strong if:: ||AB[i,j]|| >= theta * sqrt( ||AB[i,i]||*||AB[j,j]|| ) where AB[k,l] is the matrix block (degrees of freedom) associated with nodes k and l and ||.|| is a matrix norm, such a Frobenius. - See [1996bVaMaBr]_ for more details. References ---------- .. [1996bVaMaBr] Vanek, P. and Mandel, J. and Brezina, M., "Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems", Computing, vol. 56, no. 3, pp. 179--196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html Examples -------- >>> import numpy as np >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import symmetric_strength_of_connection >>> n=3 >>> stencil = np.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = symmetric_strength_of_connection(A, 0.0) rexpected a positive thetarzmatrix must have square blocksdtyperrr@!expected csr_matrix or bsr_matrix)rErr#rrIr"r!rr symmetric_strength_of_connectionrr rrrlenrWr)rrGreshaper(rH TypeErrorr>r ) r/r1rOrPrQfnrRMrNRr8rs r:rYrYsI~ qyy4555 Q!=]18 $ $ ]19 % % ]16 " "  6 171:uah 162r2FFF  r2rl!' : : :  q ! !=w1{1 66=>> > A::73qy>>999D!4)9)918==??"K),QUSQZZ(@BBBAA L((161::2q1uEED74888++,,D!4AH"=),QUSQZZ(@BBBA3Au== =;<<<VAF^^AF $A&&A Hr<rDrc . |dkrtdtj|std|dkrtdt|tstdtj|r;d}|jd}|jd|jdkrtdnd }d}tj|r)|}| }n<| }|}| }| }d |z }d ||dk<tj |tj|jdtj|jdff|j }||z}d t!|z } ~tj |j|j } tj|jd|jdd} t'|dzD]} | | || || zz zzz} t'|jdD]:} | dd| f}|}||z}tjtj||}t'|j| |j| dzD]}|j|}||}d ||<tjtj|||z|z d z }|dkrt7||j|<n d |j|<|||<_A / _A, where v_j = v, such that entry j in v has been zeroed out. As is common, larger values imply a stronger connection. Current implementation is a very slow pure-python implementation for experimental purposes, only. See [2006BrBrMaMaMc]_ for more details. References ---------- .. [2006BrBrMaMaMc] Brannick, Brezina, MacLachlan, Manteuffel, McCormick. "An Energy-Based AMG Coarsening Strategy", Numerical Linear Algebra with Applications, vol. 13, pp. 133-148, 2006. Examples -------- >>> import numpy as np >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import energy_based_strength_of_connection >>> n=3 >>> stencil = np.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = energy_based_strength_of_connection(A, 0.0) rrUzexpected sparse matrixz!expected positive number of stepszexpected integerTrz&expected square blocks in BSR matrix AFrrDrrVcscrNg{Gz)r1r)$rEr isspmatrix isinstancerrrr#r)tocsctocsrdiagonal csc_matrixrr&rrrWr.r'toarrayravelr(innerconjr"r!r>rrSr- sort_indicestobsrrr r )r/r1kbsr_flagnumPDEsAtildeDDinvDinvAomegarRId_iivAvdenomjcolvjvalnblocksuones r:#energy_based_strength_of_connectionrcs\t qyy4555  Q  312221uu<=== a  -+,,, Q+a. ;q>Q[^ + +EFF F ,Q  GGII GGII A 7DDaL  dRYqwqz%:%:%'Yqwqz%:%:%<=DEG M M MD 1HE -e44 4E  !'111A AGAJ 5 9 9 9BAEll.. a!e,- -6<? # # aaadGOO   GGII U2..//v}Q'q1u)=>>  A.#C3BAcF'"(16688QU3344u>> import numpy as np >>> from pyamg.gallery import stencil_grid >>> from pyamg.strength import evolution_strength_of_connection >>> n=3 >>> stencil = np.array([[-1.0,-1.0,-1.0], ... [-1.0, 8.0,-1.0], ... [-1.0,-1.0,-1.0]]) >>> A = stencil_grid(stencil, (n,n), format='csr') >>> S = evolution_strength_of_connection(A, np.ones((A.shape[0],1))) rzexpected epsilon > 1.0rz number of time steps must be > 0)rD_Azproj_type must be "l2" or "D_A"rXNrrVFT)rinv_flagrrD)r)rrr)rrWrzmThe most efficient time stepping for the Evolution Strength Method is done in powers of two. You have chosen z time steps.g-C6?r?rb)>rErr#rr\rrrrWasarrayrrgr bsr_matrixr&rf zeros_liker r-rmrspdiagsr.rlog2Tr)diffr"modr%rr!rr'multiplyrerincomplete_mat_mult_csrrjr realimagarrayboolr>r(finfofloatepsrHzerosrGrevolution_strength_helperrlr*r+rn min_blocksr r )*r/rrrorrrBmatcsrflagrqrsrtDinv_AmaskdimenNullDim rho_DinvArnsquarenincrwrr row_lengthmy_pderx JacobiStep AtildeCSCBmat_forscalingDAtilde DAtildeDivBrangle weak_ratioBDBColsBDBcounterryr}toln_blocksrCSRdatas* r:rrs h}}1222Avv;<<< %%:;;;  !! $ $=v/DQ/G/G=;<<<  yw Aag666z!}}   # #0+a. JJLL  4!!wFFFD$dBIdjm,D,D&(i 1 0A&B&B&D+,7444DQh%%''FF=##D8DqwDJDaL4d333F GGII JJLL}QCx1T7]T Q!V At$///NN GAJEjmG,F33IEnaS1#ueEBBBje17CCC"'!**ooG q'z>D E5ag > > >B 3?f, ,F X^^  F 6688D{{WT[))  %(('226:..=@ "&w//69:    axx RBCRRR S S S.. % %Bf_FFC)Ov558>>@@ ++ ) )Bj(FF  !!! &&    A Q;;DIaaaL__T**F  " " $ $ $    ! ! ! ! $$ % %Bf_FFLLNN     ()/i6F)2):IN)-dlDI).  0 0 0 v    fd*!||(4..031,-//##hw''/9 {!! AAAFK00vrx'@'@AA  BGFK00"'$--@@ K ,,bgdmm < <= d+++kD( VFK((4/ # +,, #& J  E    BF FK"'"(5//*=">">>? *eebfRYw{334455hw'qw777w & &A1g&& & &"%\"(41:"6"677"(3aaaQRdCS:T:TT#VAAAwJ!A+ & fl## *6;+1=+1>+1<?+-8D>>+-8S16688^4F+G+G+-8C==+2GS B B B    (276;//u===FK"&&ugv}'-~v{ D D D   +)* E5 / / /BGGv   GG b[F  K'(:;;>'*$Q'&*:1*== (H;''HiHRZ %<%<==w H H H"GV^V]#K*-fl1o.G*H*H*-fl1o.G*H*H*JKKK  #FK) 0 0F Mr<c*|jd}tj||zdz }tj|||fd}tj|df}t d|D] }t||dd|f|||!|S)aGenerate test vectors by relaxing on Ax=0 for some random vectors x. Parameters ---------- A : csr_matrix Sparse NxN matrix alpha : scalar Weight for Jacobi R : integer Number of random vectors k : integer Number of relaxation passes Returns ------- x : array Dense array N x k array of relaxation vectors rrF)orderrN) iterationsrv)rrrandomrandr[rr'r)r/r_roalphanxbrs r:relaxation_vectorsrSs*  A q1u#A 1q!fC(((A !QA 1a[[99q!AAAqD'1%88888 Hr<rctjstj|dkrtd|dkst |t std|dkst |t std|dkrtdfd}t |||||S)aAffinity Distance Strength Measure. Parameters ---------- A : csr_matrix Sparse NxN matrix alpha : scalar Weight for Jacobi R : integer Number of random vectors k : integer Number of relaxation passes epsilon : scalar Drop tolerance Returns ------- C : csr_matrix Sparse matrix of strength values References ---------- .. [LiBr] Oren E. Livne and Achi Brandt, "Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver" Notes ----- No unit testing yet. Does not handle BSR matrices yet. See [LiBr]_ for more details. rexpected alpha>0expected integer R>0expected integer k>0rexpected epsilon>1.0c\}}dtj||||zddztj||dzdtj||dzdzz z S)Nrr@r)nonzerorrH)rr7r6r/s r:distancez#affinity_distance..distancesyy{{ t26!D'AdG+!444a7 VAdGQJQ ' ' '"&4!!*D*D*D DFF Fr<rr#r rErdrdistance_measure_common)r/rr_rorrs` r:affinity_distancervsF   # #!  a  qyy+,,,AvvZ3''v/000AvvZ3''v/000{{/000FFFFF #1hq!W E EEr<ctjstj|dkrtddkst t std|dkst |t std|dkrtddkrtdfd}t ||||S) aAlgebraic Distance Strength Measure. Parameters ---------- A : csr_matrix Sparse NxN matrix alpha : scalar Weight for Jacobi R : integer Number of random vectors k : integer Number of relaxation passes epsilon : scalar Drop tolerance p : scalar or inf p-norm of the measure Returns ------- C : csr_matrix Sparse matrix of strength values References ---------- .. [SaSaSc] Ilya Safro, Peter Sanders, and Christian Schulz, "Advanced Coarsening Schemes for Graph Partitioning" Notes ----- No unit testing yet. Does not handle BSR matrices yet. See [SaSaSc]_ for more details. rrrrrrz"expected p>1 or equal to numpy.infcJ\}}tjkrEtjtj||||z zdz }|dz zStj||||z dS)Nrr@r)rrr*rHr>rF)rr7r6avgr/r_ps r:rz$algebraic_distance..distancesyy{{ t ;;&$!D' 122A5A>>>BC37# #vag$'((,,!,444r<r)r/rr_rorrrs` ` ` r:algebraic_distancersJ   # #!  a  qyy+,,,AvvZ3''v/000AvvZ3''v/000{{/0001uu=>>>5555555 #1hq!W E EEr<cBt||||}||}|\}} tj|| kd} d|| <t j||| ff|j} | tj | jd|| j | j | j | d| j z | _ | t j | jd| jddz} t| } | S)zRCreate strength of connection matrixfrom a function applied to relaxation vectors.rrrrrr)rrrwhererr rr-rr+r"r!rr.r ) r/funcrr_rorrr9r7r6weakr8s r:rrs 1aE**A QA99;;LT4 8DDL ! !! $DAdG1tTl+17;;;A "171:w#$9af66616\AF FJqwqz171:e < < <)r)rDr)NrrrFT)rrrr)rrrrr)__doc__warningsrnumpyrscipyrrrelaxation.relaxationr util.linalgr util.utilsr r r r r util.paramsrr;rSrYrrrrrrrr<r:rs  ))))))44444488888888888888 T T T T nC C C C Lk k k k \PPPPfGKDHLLLL@A@E8<QQQQh    F7F7F7F7Ft?F?F?F?FD     r<