_-Ph2dZddlmZddlZddlmZddlmZ ddl m Z ddl m Z mZmZmZmZmZmZmZddlmZdd lmZd Z ddZddZ ddZ ddZdZ ddZ ddZ dS)z3Methods to smooth tentative prolongation operators.)warnN)sparse)amg_core) scale_rows get_diagonalget_block_diagunamalfilter_operatorcompute_BtBinvfilter_matrix_rows truncate_rows)approximate_spectral_radius)upcastc|jd}|jd}t|jd|z }tj||z}t j||||jdtjtj||tj||j|j tj|j  |S)aU is the prolongator update. Project out components of U such that U*B = 0. Parameters ---------- U : bsr_matrix m x n sparse bsr matrix Update to the prolongator B : array n x k array of the coarse grid near nullspace vectors BtBinv : array Local inv(B_i.H*B_i) matrices for each supernode, i B_i is B restricted to the sparsity pattern of supernode i in U Returns ------- Updated U, so that U*B = 0. Update is computed by orthogonally (in 2-norm) projecting out the components of span(B) in U in a row-wise fashion. See Also -------- The principal calling routine, pyamg.aggregation.smooth.energy_prolongation_smoother r) blocksizeintshapenpravelrsatisfy_constraints_helper conjugateindptrindicesdata)UBBtBinvrows_per_blockcols_per_blocknum_block_rowsUBs X/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/aggregation/smooth.pysatisfy_constraintsr%s4[^N[^NN233N !A#B '(6 (* RXa[[(A(A(*BHV,<,<()!)(*(8(8 ::: HUUUUUU?rFdiagonalc:|dkr>tj|rd}n'tj|r|jddkrd}|r^tj|r|jd}nd}t |||}||}||dkr:t|d} t|| d} |t| z | z} n?|dkrt||jdd} tj | tj | jdtj | jddzf|j } | |z} |t| z | z} n|d krtj|tj|jddf|j z} tj| } d tj| | dkz | | dk<t|| d} || z} nt'd |rY|} t)|D]F} | | z| j}t-||}t/|||| |z } Gn|} t)|D] } | | | zz } | S)af Jacobi prolongation smoother. Parameters ---------- S : csr_matrix, bsr_matrix Sparse NxN matrix used for smoothing. Typically, A. T : csr_matrix, bsr_matrix Tentative prolongator C : csr_matrix, bsr_matrix Strength-of-connection matrix B : array Near nullspace modes for the coarse grid such that T*B exactly reproduces the fine grid near nullspace modes omega : scalar Damping parameter filter_entries : boolean If true, filter S before smoothing T. This option can greatly control complexity. weighting : string 'block', 'diagonal' or 'local' weighting for constructing the Jacobi D 'local' Uses a local row-wise weight based on the Gershgorin estimate. Avoids any potential under-damping due to inaccurate spectral radius estimates. 'block' uses a block diagonal inverse of A if A is BSR 'diagonal' uses classic Jacobi with D = diagonal(A) Returns ------- P : csr_matrix, bsr_matrix Smoothed (final) prolongator defined by P = (I - omega/rho(K) K) * T where K = diag(S)^-1 * S and rho(K) is an approximation to the spectral radius of K. Notes ----- If weighting is not 'local', then results using Jacobi prolongation smoother are not precisely reproducible due to a random initial guess used for the spectral radius approximation. For precise reproducibility, set numpy.random.seed(..) to the same value before each test. Examples -------- >>> from pyamg.aggregation import jacobi_prolongation_smoother >>> from pyamg.gallery import poisson >>> from scipy.sparse import coo_matrix >>> import numpy as np >>> data = np.ones((6,)) >>> row = np.arange(0,6) >>> col = np.kron([0,1],np.ones((3,))) >>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr() >>> T.toarray() array([[1., 0.], [1., 0.], [1., 0.], [0., 1.], [0., 1.], [0., 1.]]) >>> A = poisson((6,),format='csr') >>> P = jacobi_prolongation_smoother(A,T,A,np.ones((2,1))) >>> P.toarray() array([[0.64930164, 0. ], [1. , 0. ], [0.64930164, 0.35069836], [0.35069836, 0.64930164], [0. , 1. ], [0. , 0.64930164]]) blockr(rrT)inv)copyrinv_flagrlocaldtype?zIncorrect weighting option)r)risspmatrix_csrisspmatrix_bsrrr multiplyeliminate_zerosrrrr bsr_matrixrarangerabsonesr2 zeros_like ValueErrorrangetobsrr r%)STCromegadegreefilter_entries weightingnumPDEsD_invD_inv_SDP_rrs r$jacobi_prolongation_smootherrM=sNG   # # '"II  "1 % % '{1~""&     # # k!nGGG 1gw ' ' JJqMM JQD)))QD1114W===wF g  qAKNTJJJ!5")EKN*C*C#%9U[^A-=#>#>#@()111'4W===wF g   F1IIbgqwqz1oQW=== = a  bfQqAvY///a1f QD111-5666  v  A!!AK!88A $Aq))F 1f - - -AAA  v  AWQYAA Hr&ch|t|z }|}t|D] }||||zzz }|S)aRichardson prolongation smoother. Parameters ---------- S : csr_matrix, bsr_matrix Sparse NxN matrix used for smoothing. Typically, A or the "filtered matrix" obtained from A by lumping weak connections onto the diagonal of A. T : csr_matrix, bsr_matrix Tentative prolongator omega : scalar Damping parameter Returns ------- P : csr_matrix, bsr_matrix Smoothed (final) prolongator defined by P = (I - omega/rho(S) S) * T where rho(S) is an approximation to the spectral radius of S. Notes ----- Results using Richardson prolongation smoother are not precisely reproducible due to a random initial guess used for the spectral radius approximation. For precise reproducibility, set numpy.random.seed(..) to the same value before each test. Examples -------- >>> from pyamg.aggregation import richardson_prolongation_smoother >>> from pyamg.gallery import poisson >>> from scipy.sparse import coo_matrix >>> import numpy as np >>> data = np.ones((6,)) >>> row = np.arange(0,6) >>> col = np.kron([0,1],np.ones((3,))) >>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr() >>> T.toarray() array([[1., 0.], [1., 0.], [1., 0.], [0., 1.], [0., 1.], [0., 1.]]) >>> A = poisson((6,),format='csr') >>> P = richardson_prolongation_smoother(A,T) >>> P.toarray() array([[0.64930164, 0. ], [1. , 0. ], [0.64930164, 0.35069836], [0.35069836, 0.64930164], [0. , 1. ], [0. , 0.64930164]]) )rr>)r@rArCrDweightrKrLs r$ richardson_prolongation_smootherrPsKp.q11 1F A 6]] !  Hr&r0c 0 tjtj|jj|j|j|jf|j} |dkrt|dd} n|dkryt||j dd } tj| tj | jdtj | jdd zf|j} n|d krztj |tj|jdd f|jz} tj| } d tj | | dkz | | dk<nt!d tj|jj|j} tj| |j|jf|j} t#j|j|jtj|j|j|jtj|j| j| jtj| jt)|jd|j dz t)|jd |j d z |j d|j d |j d | xjdzc_t+| ||| jdkrt/d|S| j}d}||kr||kr|dvrt1| | }n| | z}| |}||krn|dkr|}|}n ||z }|||zz}|}d| jdd<t#j|j|jtj|j|j|jtj|j| j| jtj| jt)|jd|j dz t)|jd |j d z |j d|j d |j d t+| ||||| z }|||zz}|dr |d d|z|d dz}| || zz } |d z }| j}||kr||k|S)aTUse CG to smooth T by solving A T = 0, subject to nullspace and sparsity constraints. Parameters ---------- A : csr_matrix, bsr_matrix SPD sparse NxN matrix T : bsr_matrix Tentative prolongator, a NxM sparse matrix (M < N). This is initial guess for the equation A T = 0. Assumed that T B_c = B_f B : array Near-nullspace modes for coarse grid, i.e., B_c. Has shape (M,k) where k is the number of coarse candidate vectors. BtBinv : array 3 dimensional array such that, BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted to the neighborhood (in the matrix graph) of dof of i. pattern : csr_matrix, bsr_matrix Sparse NxM matrix This is the sparsity pattern constraint to enforce on the eventual prolongator maxiter : int maximum number of iterations tol : float residual tolerance for A T = 0 weighting : string 'block', 'diagonal' or 'local' construction of the diagonal preconditioning Cpt_params : tuple Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the standard SA prolongation smoothing is carried out. If True, then dict must be a dictionary of parameters containing, (1) P_I: P_I.T is the injection matrix for the Cpts, (2) I_F: an identity matrix for only the F-points (i.e. I, but with zero rows and columns for C-points) and I_C: the C-point analogue to I_F. Returns ------- T : bsr_matrix Smoothed prolongator using conjugate gradients to solve A T = 0, subject to the constraints, T B_c = B_f, and T has no nonzero outside of the sparsity pattern in pattern. See Also -------- The principal calling routine, pyamg.aggregation.smooth.energy_prolongation_smoother r1r/r(FTnorm_eqr+r*rr-rr0r3weighting value is invalid_Error in sa_energy_min(..). Initial R no nonzeros on a level. Returning tentative prolongator r0r(NI_FP_I)rr8rzerosrrr2rrrr rr9r:r;r<r=rincomplete_mat_mult_bsrrrr%nnzprintrrr6sum)ArArrpatternmaxitertolrF Cpt_paramsAPDinvrJuonesRresidiZnewsumrKoldsumbetaalphas r$cg_prolongation_smoothingrpsh  BHW\%7qwGGG#OW^=!( 0 0 0B JAu$777 g  a1;q>DIII $ $*Q-(@(@"$)DJqM!O"<"<">'(w000 g   F1IIbgqwqz1oQW=== =}QRVAa1fI...Q!V 5666 HW\'qw 7 7 7E5'/">+ '  / / /A $QXqy%'Xaf%5%5%&Xqy%'Xaf%5%5%&Xqy%'Xaf%5%5%(AKN)B%C%C%(AKN)B%C%C%&[^Q[^%&[^ 5 5 5FFdNFF1f%%%uzz 2 3 3 3 EE A g++%#++ - - -1d##AAQA++--((++0022 C<<  66AFFF?DDF A  (19)+!&)9)9)*19)+!&)9)9)+BJ)+"'):):),QWQZ A-F)G)G),QWQZ A-F)G)G)*QQ)*Q 9 9 9 B6*** ..r2277999 aK a= >1 e$Q&Au)==A bL Qs g++%#++z Hr&c |dkrtd|dd|j} | t j|jj|j} tj | |j |j f|j} t|dd } t j|jj|j} tj | |j |j f|j} d |z|z}d | jd d <tj| j | j t j| j|j |j t j|j| j | j t j| jt#|jd|jdz t#|jd|jdz | jd| jd|jdt'| ||| jdkrt+d|S| j}d}||kr||kr t-| | }| |}||krn|dkr|}|}n ||z }|||zz}|}||z}d | jd d <tj| j | j t j| j|j |j t j|j| j | j t j| jt#|jd|jdz t#|jd|jdz | jd| jd|jd~t'| ||||| z }|||zz}|dr |dd|z|ddz}| || zz } |dz }| j}||kr||k |S)aSmooth T with CGNR by solving A T = 0, subject to nullspace and sparsity constraints. Parameters ---------- A : csr_matrix, bsr_matrix SPD sparse NxN matrix Should be at least nonsymmetric or indefinite T : bsr_matrix Tentative prolongator, a NxM sparse matrix (M < N). This is initial guess for the equation A T = 0. Assumed that T B_c = B_f B : array Near-nullspace modes for coarse grid, i.e., B_c. Has shape (M,k) where k is the number of coarse candidate vectors. BtBinv : array 3 dimensional array such that, BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted to the neighborhood (in the matrix graph) of dof of i. pattern : csr_matrix, bsr_matrix Sparse NxM matrix This is the sparsity pattern constraint to enforce on the eventual prolongator maxiter : int maximum number of iterations tol : float residual tolerance for A T = 0 weighting : string 'block', 'diagonal' or 'local' construction of the diagonal preconditioning IGNORED here, only 'diagonal' preconditioning is used. Cpt_params : tuple Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the standard SA prolongation smoothing is carried out. If True, then dict must be a dictionary of parameters containing, (1) P_I: P_I.T is the injection matrix for the Cpts, (2) I_F: an identity matrix for only the F-points (i.e. I, but with zero rows and columns for C-points) and I_C: the C-point analogue to I_F. Returns ------- T : bsr_matrix Smoothed prolongator using CGNR to solve A T = 0, subject to the constraints, T B_c = B_f, and T has no nonzero outside of the sparsity pattern in pattern. See Also -------- The principal calling routine, pyamg.aggregation.smooth.energy_prolongation_smoother r(z Weighting of z unused.r) stacklevelr1r/rTrRrUrXNrz^Error in sa_energy_min(..). Initial R no nonzeros on a level. Returning tentative prolongatorrYrZ)rrAr sort_indicesrr[rrr2rr8rrrrr\rrrr%r]r^rr6r_)r`rArrrarbrcrFrdAhrgrerfrhATrirjrkrlrKrmrnAP_tempros r$cgnr_prolongation_smoothingrwsjJ 0Y 0 0 0Q???? BOO HW\'qw 7 7 7E  E7?GNC!( 0 0 0B 1$ / / /D HW\'qw 7 7 7E5'/7>B '  / / /A aBAF111I $RY %'Xbg%6%6%'Y %'Xbg%6%6%&Xqy%'Xaf%5%5%(AKN)B%C%C%(AKN)B%C%C%'\!_bl1o%&[^ 5 5 51f%%%uzz 0 1 1 1 EE A g++%#++ q$  ++--((++0022 C<<  66AFF&=DDF A A# (BJ)+"'):):)0)+',)?)?)+BJ)+"'):):),QWQZ A-F)G)G),QWQZ A-F)G)G)+a)+a!+a. J J J  B6*** ..r2277999 aK a= >1 e$Q&Au)==A bL Q{ g++%#++J Hr&ct|D]2}||}tj||||dz|||dz<3dS)aApply the first k Givens rotations in Q to v. Parameters ---------- Q : list list of consecutive 2x2 Givens rotations v : array vector to apply the rotations to k : int number of rotations to apply. Returns ------- v is changed in place Notes ----- This routine is specialized for GMRES. It assumes that the first Givens rotation is for dofs 0 and 1, the second Givens rotation is for dofs 1 and 2, and so on. rN)r>rdot)QvkjQlocs r$ apply_givensrgsW.1XX**t6$!AaC%))!AaC%**r&c `tj|jj|j} t j| |j|jf|j} t|j|j|j} g} g} tj|dz|dzf| }|dkrt|dd}n|dkryt||j d d }t j|tj |jd tj |jd dzf|j}n|d krztj|tj|jd df|jz}tj|}d tj||d kz ||d k<nt#d tj|jj|j} t j| |j|jf|j}t%j|j|jtj|j|j|jtj|j|j|jtj|jt+|jd |j d z t+|jd|j dz |j d |j d|j d|xjdzc_|dvrt-||}n||z}t/||||jd krt3d|Stj|j|jz}tj|dzf| }||d <|dkr| d |z |zd}||dz krG||kr@|dz}d| jdd<t%j|j|jtj|j| |j| |jtj| |j| j| jtj| jt+|jd |j d z t+|jd|j dz |j d |j d|j d|dvrt-| |} n|| z} t/| ||| | t?|dzD]t}| | | |dz|||f<| |dz|||f| |zz | |dz<utj| |dzj| |dzjz||dz|f<||dz|fdkr"d ||dz|fz | |dzz| |dz<|d krtC| |dd|f|||dz|fd kra|||f}||dz|f}tj|}tj|}||kr/||z }tj|tj|z }n||z }|tj|z }tj|dz|dzz}||z }||z|z }tj"|tj|g| |gg| }| |tj#||||dz|||dz<tj#|d ddf|||dz|f|||f<d||dz|f<tj||dz}||dz kr||k@|dkr[tIj%|d |dzd |dzf|d |dz} t?|dzD]}|| || |zz}|d r |dd|z|ddz}|S)a|Smooth T with GMRES by solving A T = 0 subject to nullspace and sparsity constraints. Parameters ---------- A : csr_matrix, bsr_matrix SPD sparse NxN matrix Should be at least nonsymmetric or indefinite T : bsr_matrix Tentative prolongator, a NxM sparse matrix (M < N). This is initial guess for the equation A T = 0. Assumed that T B_c = B_f B : array Near-nullspace modes for coarse grid, i.e., B_c. Has shape (M,k) where k is the number of coarse candidate vectors. BtBinv : array 3 dimensional array such that, BtBinv[i] = pinv(B_i.H Bi), and B_i is B restricted to the neighborhood (in the matrix graph) of dof of i. pattern : csr_matrix, bsr_matrix Sparse NxM matrix This is the sparsity pattern constraint to enforce on the eventual prolongator maxiter : int maximum number of iterations tol : float residual tolerance for A T = 0 weighting : string 'block', 'diagonal' or 'local' construction of the diagonal preconditioning Cpt_params : tuple Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the standard SA prolongation smoothing is carried out. If True, then dict must be a dictionary of parameters containing, (1) P_I: P_I.T is the injection matrix for the Cpts, (2) I_F: an identity matrix for only the F-points (i.e. I, but with zero rows and columns for C-points) and I_C: the C-point analogue to I_F. Returns ------- T : bsr_matrix Smoothed prolongator using GMRES to solve A T = 0, subject to the constraints, T B_c = B_f, and T has no nonzero outside of the sparsity pattern in pattern. See Also -------- The principal calling routine, pyamg.aggregation.smooth.energy_prolongation_smoother r1r/rr(FTrRr*rr-r0r3rTrUrWrVrXNrrYrZ)&rr[rrr2rr8rrrrr rr9r:r;r<r=rr\rrrr%r]r^sqrtrr_appendr,r>r6rarrayrylasolve)!r`rArrrarbrcrFrdrgAVxtyperzVHrfrJrhnormrgrjr}h1h2h1_magh2_magmutaudenomcsQblockys! r$gmres_prolongation_smoothingrsZn HW\'qw 7 7 7E  E7?GNC!( 0 0 0B 17AGQW - -E A A '!)WQY'u555AJAu$777 g  a1;q>DIII $ $*Q-(@(@"$)DJqM!O"<"<">'(w000 g   F1IIbgqwqz1oQW=== =}QRVAa1fI...Q!V 5666 HW\'qw 7 7 7E5'/7>B '  / / /A $QXqy%'Xaf%5%5%&Xqy%'Xaf%5%5%&Xqy%'Xaf%5%5%(AKN)B%C%C%(AKN)B%C%C%&[^Q[^%&[^ 5 5 5FFdNFF))) q$   F1f%%%uzz 2 3 3 3 GQV%%''.3355 6 6E '!)U+++A AaD s{{ #e)Q A gai--ECKK aC  (19)+!&)9)9)*1adl)+!A$))<)<)+BJ)+"'):):),QWQZ A-F)G)G),QWQZ A-F)G)G)*QQ)*Q 9 9 9 - - -B%%BBbB B6*** qs + +At~~''001Q388==??AadGqsVa1gadl*AacFFGQqsV[2244QqsV[@EEGGHH!A#q&  QqS!V9  Aac1fIo1Q3/AacF q55 AaaadGQ ' ' ' QqS!V9>>1a4B1Q36BVBZZFVBZZFUl2&&rvbzz1UmGFAI 122Eu As 5 AX2<??3qb!W=UKKKF HHV   vfa!A#h//Aa!eHfVAqqqD\1QqsUAX;77AadGAac1fIq1vQ gai--ECKKZ Bww HQq1ua!e|_a!A#h / /qs  AAaD1I AA!}: qM%  "Z]5%9 9 Hr&cg:0yE>c  |dkrtd|dkrtdtj|r|dd}n$tj|rnt dtj|r|dd}n$tj|rnt d |jd|jdkrtd |jd|jdkrtd t|j |j dkr|Stj|st d | i} | i} d| vr"| ddkr| dd d| vr"| ddkr| dd |tj tj t|j|j|jf|jd|jdz |jd|jdz f}t|j |j dkr|S| dkr|t'|jd|jdz t'|jd|jdz f} tj tj |jj|j|jf| }|}t)| D]}||z}d| vr6d| vr2t+|| d}t-|| d}||z }nXd| vrt-|| d}n=d| vrt+|| d}n"t| dkrtdt/||jd|jd}|n|}d| vr6d| vr2t+|| d}t-|| d}||z }nXd| vrt-|| d}n=d| vrt+|| d}n"t| dkrtdd|jd d <||dr"|dd|z}|dd|z}t3||}|dr|jd|jdksd| vr;t5|||||}|dr |dd|z|ddz}|dkrt7|||||||| | }n;|dkrt9|||||||| | }n|dkrt;|||||||| | }|t| dksd| vs |ddur|Sd| vrod| vrkt+|| d}t-|| d}d|jd d <d|jd d <||z}d|jd d <||}nEd| vrt-|| d}n*d| vrt+|| d}ntdtA|||||||ddd| iddi }|S)a-Minimize the energy of the coarse basis functions (columns of T). Both root-node and non-root-node style prolongation smoothing is available, see Cpt_params description below. Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix T : bsr_matrix Tentative prolongator, a NxM sparse matrix (M < N) Atilde : csr_matrix Strength of connection matrix B : array Near-nullspace modes for coarse grid. Has shape (M,k) where k is the number of coarse candidate vectors. Bf : array Near-nullspace modes for fine grid. Has shape (N,k) where k is the number of coarse candidate vectors. Cpt_params : tuple Tuple of the form (bool, dict). If the Cpt_params[0] = False, then the standard SA prolongation smoothing is carried out. If True, then root-node style prolongation smoothing is carried out. The dict must be a dictionary of parameters containing, (1) for P_I, P_I.T is the injection matrix for the Cpts, (2) I_F is an identity matrix for only the F-points (i.e. I, but with zero rows and columns for C-points) and I_C is the C-point analogue to I_F. See Notes below for more information. krylov : string 'cg' for SPD systems. Solve A T = 0 in a constraint space with CG 'cgnr' for nonsymmetric and/or indefinite systems. Solve A T = 0 in a constraint space with CGNR 'gmres' for nonsymmetric and/or indefinite systems. Solve A T = 0 in a constraint space with GMRES maxiter : integer Number of energy minimization steps to apply to the prolongator tol : scalar Minimization tolerance degree : int Generate sparsity pattern for P based on (Atilde^degree T) weighting : string 'block', 'diagonal' or 'local' construction of the diagonal preconditioning 'local' Uses a local row-wise weight based on the Gershgorin estimate. Avoids any potential under-damping due to inaccurate spectral radius estimates. 'block' Uses a block diagonal inverse of A if A is BSR. 'diagonal' Uses the inverse of the diagonal of A prefilter : dictionary Filter elements by row in sparsity pattern for P to reduce operator and setup complexity. If None or an empty dictionary, then no dropping in P is done. If postfilter has key 'k', then the largest 'k' entries are kept in each row. If postfilter has key 'theta', all entries such that :math:`P[i,j] < kwargs['theta']*max(abs(P[i,:]))` are dropped. If postfilter['k'] and postfiler['theta'] are present, then they are used with the union of their patterns. postfilter : dictionary Filters elements by row in smoothed P to reduce operator complexity. Only supported if using the rootnode_solver. If None or an empty dictionary, no dropping in P is done. If postfilter has key 'k', then the largest 'k' entries are kept in each row. If postfilter has key 'theta', all entries such that :math::`P[i,j] < kwargs['theta']*max(abs(P[i,:]))` are dropped. If postfilter['k'] and postfiler['theta'] are present, then they are used with the union of their patterns. Returns ------- T : bsr_matrix Smoothed prolongator Notes ----- Only 'diagonal' weighting is supported for the CGNR method, because we are working with A^* A and not A. When Cpt_params[0] == True, root-node style prolongation smoothing is used to minimize the energy of columns of T. Essentially, an identity block is maintained in T, corresponding to injection from the coarse-grid to the fine-grid root-nodes. See [2011OlScTu]_ for more details, and see util.utils.get_Cpt_params for the helper function to generate Cpt_params. If Cpt_params[0] == False, the energy of columns of T are still minimized, but without maintaining the identity block. See [1999cMaBrVa]_ for more details on smoothed aggregation. Examples -------- >>> from pyamg.aggregation import energy_prolongation_smoother >>> from pyamg.gallery import poisson >>> from scipy.sparse import coo_matrix >>> import numpy as np >>> data = np.ones((6,)) >>> row = np.arange(0,6) >>> col = np.kron([0,1],np.ones((3,))) >>> T = coo_matrix((data,(row,col)),shape=(6,2)).tocsr() >>> print(T.toarray()) [[1. 0.] [1. 0.] [1. 0.] [0. 1.] [0. 1.] [0. 1.]] >>> A = poisson((6,),format='csr') >>> B = np.ones((2,1),dtype=float) >>> P = energy_prolongation_smoother(A,T,A,B, None, (False,{})) >>> print(P.toarray()) [[1. 0. ] [1. 0. ] [0.66666667 0.33333333] [0.33333333 0.66666667] [0. 1. ] [0. 1. ]] References ---------- .. [1999cMaBrVa] Jan Mandel, Marian Brezina, and Petr Vanek "Energy Optimization of Algebraic Multigrid Bases" Computing 62, 205-228, 1999 http://dx.doi.org/10.1007/s006070050022 .. 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