VpfZdZddlmZddlmZddlZddlZddlmZ ddl m Z ddl m Z ddl mZddlmZdd lmZdd lmZddlZdd lmZmZ d5d6dZejejddg d7d8dZejejgd d7d9dZdZejj j!fdZ"d Z#d!Z$d"Z%d#Z&d$Z'd%Z(d&Z)d'Z*d(Z+d)Z,d*Z-d+Z.d,Z/d-Z0ej1d.Z2e23ejej4e2e25e*ej6e2e-dde.e0ej7e2<e j8e2e+d/0e j8e2e,d10d:d4Z9dS);zSparse linear algebra routines.) annotations)CallableN)sparse)mlir)xla)core)ad) gpu_solver) csr_matrixlinalgdA,jax.Array | Callable[[jax.Array], jax.Array]X jax.Arrayminttoljax.Array | float | Nonect|tjtjfrt ||||dSt ||||dS)a Compute the top-k standard eigenvalues using the LOBPCG routine. LOBPCG [1] stands for Locally Optimal Block Preconditioned Conjugate Gradient. The method enables finding top-k eigenvectors in an accelerator-friendly manner. This initial experimental version has several caveats. - Only the standard eigenvalue problem `A U = lambda U` is supported, general eigenvalues are not. - Gradient code is not available. - f64 will only work where jnp.linalg.eigh is supported for that type. - Finding the smallest eigenvectors is not yet supported. As a result, we don't yet support preconditioning, which is mostly needed for this case. The implementation is based on [2] and [3]; however, we deviate from these sources in several ways to improve robustness or facilitate implementation: - Despite increased iteration cost, we always maintain an orthonormal basis for the block search directions. - We change the convergence criterion; see the `tol` argument. - Soft locking [4] is intentionally not implemented; it relies on choosing an appropriate problem-specific tolerance to prevent blow-up near convergence from catastrophic cancellation of near-0 residuals. Instead, the approach implemented favors truncating the iteration basis. [1]: http://ccm.ucdenver.edu/reports/rep149.pdf [2]: https://arxiv.org/abs/1704.07458 [3]: https://arxiv.org/abs/0705.2626 [4]: DOI 10.13140/RG.2.2.11794.48327 Args: A : An `(n, n)` array representing a square Hermitian matrix or a callable with its action. X : An `(n, k)` array representing the initial search directions for the `k` desired top eigenvectors. This need not be orthogonal, but must be numerically linearly independent (`X` will be orthonormalized). Note that we must have `0 < k * 5 < n`. m : Maximum integer iteration count; LOBPCG will only ever explore (a subspace of) the Krylov basis `{X, A X, A^2 X, ..., A^m X}`. tol : A float convergence tolerance; an eigenpair `(lambda, v)` is converged when its residual L2 norm `r = |A v - lambda v|` is below `tol * 10 * n * (lambda + |A v|)`, which roughly estimates the worst-case floating point error for an ideal eigenvector. If all `k` eigenvectors satisfy the tolerance comparison, then LOBPCG exits early. If left as None, then this is set to the float epsilon of `A.dtype`. Returns: `theta, U, i`, where `theta` is a `(k,)` array of eigenvalues, `U` is a `(n, k)` array of eigenvectors, `i` is the number of iterations performed. Raises: ValueError : if `A,X` dtypes or `n` dimensions do not match, or `k` is too large (only `k * 5 < n` supported), or `k == 0`. F)debug) isinstancejaxArraynpndarray_lobpcg_standard_matrix_lobpcg_standard_callable)rrrrs ^/var/www/html/nettyfy-visnx/env/lib/python3.11/site-packages/jax/experimental/sparse/linalg.pylobpcg_standardr %sRBCIrz*++> "1aCu = = == "1aCu = = ==r)static_argnamesFboolcXttjt|||||S)z.conds/"'Ar2r9a ?1q5)a- 0 00r!c |\}}}}}}ttj||fd|}tj|||fd}t|\}}|dddf} tj| ddd} | | z} t || }tj|ddd} || z}tj|ddfj\} }t |dddf| } t || }tj|ddd}|tj |dkd|z}|}||tj df|zz }tj|dd}tj|dd|dz}|z}|d z}||zk}tj |}|dz|||||tj dff}rt||||||||z }||f}|S) Nr*r+rTordr+r,?)r>r+ ) _project_outr/ concatenate_rayleigh_ritz_orthr normr'qrTwherenewaxissum_generate_diagnostics)r1r2rPRr7thetaXPRQBnormBnormXqdiff_rayleigh_orthonormPAX resid_normsreltol res_convergedr6 new_state diagnosticsrrr8nrs rbodyz'_lobpcg_standard_callable..bodyse Aq!Q5 S_aV!444a88A /1a)! , , ,C#1c**HE1 !!!RaR%A JOOA11tO < @@kk*i r!c|Sr.)r1r7r]s rz+_lobpcg_standard_callable..ser!)xslength) shapedtype _check_inputsr/finfoeps_orthonormalize _extend_basisrIrlaxscan while_loop)rrrrrdtrKrVrMrLr9r6r1r[r2r4r5 _convergedr]r8r\s` ``` @@@rrrus $!Qw"1[ )B-- Ca!Aqwqz""!  qtt" '!b&q4 0 0 0%519n!111111MMMMMMMMM^) aAy% (% 2$$$$eQ&@@E;; G  tT5 1 1E$)!!QB E * AAA;1k )) q!!!ta r!c|j\}}|j}|dkrtd||dz|krtd|dzd|d|tj|df|j}|j|krtd |jd|d|j|dfkr|j}td |d|d |dS) Nrzmust have search dim > 0, got z*expected search dim * 5 < matrix dim (got z, )r*rdz!A, X must have same dtypes (were z A must be (z) matrix A, got output )rcrd ValueErrorr/zeros)rrr\r8rm test_outputss rreres $!Qw"!VV 9a99 : ::UaZZ O!a%OO1OOO P PP#)QF!'22233+" FK,=FFFFF H HH1a&  A E1EEEE!EE F FF! r!cHtj||||fSr.)rrjdot)ab precisions rr'r' s QIy1 2 22r!c &|jd|j|jksJd}fd}t|j|} || } || | z } t|j|} || } || | z }||j|z}|jdtjtj|dkdz }|tjtj|dkdtjtj|dkd|dtj|dd|tj|tj |tj || ||d S) Nr*cNtjtj|Sr.)r/diag)xs rr`z'_generate_diagnostics..ssx ,,r!c\tj|dzz SNr<)r/absrI)rr8s rr`z'_generate_diagnostics..s"SWQZZ^^%%a0r!rr;r<)r+r>) z basis rankzX zeroszP zeroszlambda historyzresidual historyr6zadjusted residual maxzadjusted residual p50zadjusted residual minzX orthzP orthzP.X) rcr'rFr/rIallr rDmaxmedianmin)prev_XPRrrKrLrMr6 adj_residdiagdiagabserrXTXDXorthXPTPDPorthPPX prev_basisr8s @rrJrJ sgaj! AG     , ,( 0 0 0 0& AC #x}}" &r  % AC #x}}" &r  % vacAg"~a 37378s?+K+K+K#L#LL*c22233c22233bqb *//!!/;;"wy11"z)44"wy11    r!cztj|\}}|ddd|dddddffS)N)r/r eigh)rwVs r_eigh_ascendingr.s@   $!Q 44R4!AAAtttG* r!ctj|ddd}|tj|dkd|z}t |j|}t |\}}tj|jj |dz}tj ||}tj|dk|ddz}||tj ddfz}t ||} ||ktj |dkztj ddf} | | | jz} tj| ddd}| |dkz} | tj| |dz} | S) aADerives a truncated orthonormal basis for `X`. SVQB [1] is an accelerator-friendly orthonormalization procedure, which squares the matrix `C = X.T @ X` and computes an eigenbasis for a smaller `(k, k)` system; this offloads most of the work in orthonormalization to the first multiply when `n` is large. Importantly, if diagonalizing the squared matrix `C` reveals rank deficiency of X (which would be evidenced by near-0 then), eigenvalues corresponding columns are zeroed out. [1]: https://sdm.lbl.gov/~kewu/ps/45577.html Args: X : An `(n, k)` array which describes a linear subspace of R^n, possibly numerically degenerate with some rank less than `k`. Returns: An orthonormal space `V` described by a `(n, k)` array, with trailing columns possibly zeroed out if `X` is of low rank. r<rTr=r?Nr)r/r rDrGr'rFrrfrdrgmaximumrHr~astype) rnormsinnerrrtaupaddedsqrtedscaledVorthoXkeeps r_svqbr3sY4 *//!T/ : :%sy!S%(((! ac1++%   $!Q !'1%# ;q#  & 9S1Wfc * *t 4& s{AAA~& &' q'??& s7sx, -s{AAA~ >$DKK % %%& *//&aa$/ ? ?%%#+$CIdE3 ' ''& -r!c tdD]7}|t|t|j|z}t|}8tdD](}|t|t|j|z})tj|ddd}||dk|jz}|S)aDerives component of U in the orthogonal complement of basis. This method iteratively subtracts out the basis component and orthonormalizes the remainder. To an extent, these two operations can oppose each other when the remainder norm is near-zero (since normalization enlarges a vector which may possibly lie in the subspace `basis` to be subtracted). We make sure to prioritize orthogonality between `basis` and `U`, favoring to return a lower-rank space thank `rank(U)`, in this tradeoff. Args: basis : An `(n, m)` array which describes a linear subspace of R^n, this is assumed to be orthonormal but zero columns are allowed. U : An `(n, k)` array representing another subspace of R^n, whose `basis` component is to be projected out. Returns: An `(n, k)` array, with some columns possibly zeroed out, representing the component of `U` in the complement of `basis`. The nonzero columns are mutually orthonormal. r<rTr=gGz?) ranger'rFrhr/r rDrrd)basisUr7normUs rrArAlsL 88aUCOO $ $$AAA 88%%aUCOO $ $$AA *//!T/ : :% ag&&&! (r!cHtdD]}t|}|Sr)rr)rr7s rrhrhs) 88a %LLEE ,r!c\t|j||}t|S)aSolve the Rayleigh-Ritz problem for `A` projected to `S`. Solves the local eigenproblem for `A` within the subspace `S`, which is assumed to be orthonormal (with zero columns allowed), identifying `w, V` satisfying (1) `S.T A S V ~= diag(w) V` (2) `V` is standard orthonormal Note that (2) is simplified to be standard orthonormal because `S` is. Args: A: An operator representing the action of an `n`-sized square matrix. S: An orthonormal subspace of R^n represented by an `(n, k)` array, with zero columns allowed. Returns: Eigenvectors `V` and eigenvalues `w` satisfying the size-`k` system described in this method doc. Note `V` will be full rank, even if `S` isn't. )r'rFr)rSSASs rrCrCs+, AC1#   r!c|j\}}tj||gd\}}tj|\}}}tj|t ||z|gd} tjtj||jtj ||z |z |f|jgd} t | |j dd|zzdztj ddfz} dtj | | |dddfj | gtjjj z} | j|d| S) aExtend the basis of `X` with `m` addition dimensions. Given an orthonormal `X` of dimension `k`, a typical strategy for deriving an extended basis is to generate a random one and project it out. We instead generate a basis using block householder reflectors [1] [2] to leverage the favorable properties of determinism and avoiding the chance that the generated random basis has overlap with the starting basis, which may happen with non-negligible probability in low-dimensional cases. [1]: https://epubs.siam.org/doi/abs/10.1137/0725014 [2]: https://www.jstage.jst.go.jp/article/ipsjdc/2/0/2_0_298/_article Args: X : An `(n, k)` array representing a `k`-rank orthonormal basis for a linear subspace of R^n. m : A nonnegative integer such that `k + m <= n` telling us how much to extend the basis by. Returns: An `(n, m)` array representing an extension to the basis of `X` such that their union is orthonormal. rr;rrr<r*rN)r{)rcr/splitr svdrBr'eyerdrtrFrH multi_dotrrj PrecisionHIGHESTatadd) rrr\r8XupperXlowerurvvtyotherrhs rriris\0 $!Q9Q!,,,.&& Z^^F # #(!Q  ovAr *F3!<<>!3:  !ABBE(*e(9(A  C CC! abbe  r!c |j|jkrtd|jd|jtj|jtjr$tj|jtjstd|jd|j|j|jcxkr|jcxkr|jcxkrdks1ntd|jd|jd|jd |j|j|jdzks|j|jkr'td |jd|jd|j|d vrtd |d t|}|S)Nz$data types do not match: data.dtype=z b.dtype=z6index arrays must be integer typed; got indices.dtype=z indptr.dtype=r*z/Arrays must be one-dimensional. Got data.shape=z indices.shape=z indptr.shape=z b.shape=z%Invalid CSR buffer sizes: data.shape=)rr*r<zreorder=z' not valid, must be one of [1, 2, 3, 4]) rdrsr/ issubdtypeintegerndimrcsizefloatdataindicesindptrrzrreorders r_spsolve_abstract_evalrs Z17 IIIqwII J JJ . 4 4b VYVa9b9bb ` ``QWQ]`` a aa gl @ @ @ @fk @ @ @ @QV @ @ @ @q @ @ @ @ V JVV*1-VV;A<VVKL7VV W WW [AFQJ4:#>#> `$*`` ``QWQ]`` a aa L  IIII J JJ c # (r!c X|j\}}}}tj|j||||||Sr.)avals_inr cuda_csrlsvqrrd) ctxrrrrzrr data_avalr7s r_spsolve_gpu_loweringrs8 )Q1  !)/4"(!S' ; ;;r!c n~~||||g}d}tj||d||j|jd\} } } | S)Nct|||f|j|jf}tj|||jfS)N)rc)r rr spsolverrd)rrrrzkwargsrs r _callbackz(_spsolve_cpu_lowering.._callback!sID'6*16162BCCCA N1a ' ' 0 0 22r!F)has_side_effect)remit_python_callbackr avals_out) rrrrrzrrargsrresultr7s r_spsolve_cpu_loweringrs^ 7  #$333* 9dD#, ,&!Q -r!c t||||fi|}tj|||||jdz t |fd}t||||fi| S)Nr*F)rc transpose)rr csr_matvec_pbindrlen)data_dotrrrrzkwdsprSs r_spsolve_jvp_lhsr+szgvq11D11A  7FA(. aQ'@+0 ! 2 2A D'61 5 5 5 5 55r!c "t||||fi|Sr.)r)b_dotrrrrzrs r_spsolve_jvp_rhsr4s 4&% 8 84 8 88r!c|jdz }tjtj|j|ddz }t j|||fd\}}}tj|jdd tjtj || |j }|||fS)Nr*r<)num_keys)rb) rr/cumsum zeros_likerrrrjsortsetbincountrrd) rrrrrowrow_T indices_Tdata_Tindptr_Ts r_csr_transposer9s kAo! 3>'**-f599!<<==A# W\\7C*>\KK%F ^F # # &qrr * . . jeA...//66v|DDFF( H $$r!c tj|rJtj|rJtj|r+t|||\}}}t||||fi|} |||| fSt d)Nz&spsolve transpose with respect to data)r is_undefined_primalrrNotImplementedError) ctrrrrzrrrrct_outs r_spsolve_transposerCs  #G , ,,,,  #F + ++++AH"0w"G"GFIx VY" = = = =F && (( F G GGr!rcuda)platformcpuư>r*cBt||||||S)a A sparse direct solver using QR factorization. Accepts a sparse matrix in CSR format `data, indices, indptr` arrays. Currently only the CUDA GPU backend is implemented. Args: data : An array containing the non-zero entries of the CSR matrix. indices : The column indices of the CSR matrix. indptr : The row pointer array of the CSR matrix. b : The right hand side of the linear system. tol : Tolerance to decide if singular or not. Defaults to 1e-6. reorder : The reordering scheme to use to reduce fill-in. No reordering if ``reorder=0``. Otherwise, symrcm, symamd, or csrmetisnd (``reorder=1,2,3``), respectively. Defaults to symrcm. Returns: An array with the same dtype and size as b representing the solution to the sparse linear system. )rr) spsolve_prrs rrrYs"( gvqc7 K KKr!)r N)rrrrrrrr)F) rrrrrrrrrr#) rr(rrrrrrrr#)rr*):__doc__ __future__rcollections.abcrr%r jax.numpynumpyr/jax.experimentalrjax.interpretersrrjax._srcrjax._src.interpretersr jax._src.libr r scipy.sparser r r r&jitrrrerjrrr'rJrrrArhrCrirrrrrrr Primitiverdef_implapply_primitivedef_abstract_evaldefjvpprimitive_transposesregister_loweringrr_r!rrs}&%""""""$$$$$$ ######!!!!!! $$$$$$######++++++++ $( C>C>C>C>C>J37S'N;;;  3333<;337,?,?,?@@@  ||||A@|~GGG*)13333B 666r; ; ; z:666t    ;;;   666999 %%% H H H DN9 % % $9$S%8)DDEEE 2333  )%tT3CDDD%7 "y"7&IIIIy"7%HHHHLLLLLLr!