VpfVBdZddlmZddlmZddlmZddlZddlm cm cm Z ddl m cm cmZddlmZddlmZddlmZdd lmZdd lmZdd lmZd Zdd Z ddZdZddZd dZGddeZeej ddZ!dddddddZ"dS)!aSymmetric (Hermitian) eigendecomposition using QDWH References: Nakatsukasa, Yuji, and Nicholas J. Higham. "Stable and efficient spectral divide and conquer algorithms for the symmetric eigenvalue decomposition and the SVD." SIAM Journal on Scientific Computing 35, no. 3 (2013): A1325-A1349. https://epubs.siam.org/doi/abs/10.1137/120876605 This implementation is primarily used on TPU, but it can in principle work on CPU and GPU also. ) annotations)partial) NamedTupleN) reductions)ufuncs)lax)qdwh)linalg)Stackc||zdz |z|zSN)ins Q/var/www/html/nettyfy-visnx/env/lib/python3.11/site-packages/jax/_src/lax/eigh.py _round_upr4s Q3q5Q,! ctj|t|ksJd}t|D]9\}}|2t jtj|j||k}||n||z}:||ntj|||S)zMasks `x` up to the dynamic shape `dims`. Replaces values outside those dimensions with `alternative`. `alternative` is broadcast with `x`. N) jnpndimlen enumeraterbroadcasted_iotaint32shapewhere)xdims alternativemaskrd mask_dim_is r_maskr$7s !D ! ! ! ! $ooAAda}' 17A>>Bjz_slice..[s:::!Q1I:::rc3>K|]}tj|VdSNrrr)rs r z_slice..\s*'L'L ! 'L'L'L'L'L'Lr)rpadrarraydtype dynamic_slicetupler$)operand start_indicesdynamic_slice_sizesstatic_slice_sizes fill_valuepaddedouts r_slicer<Es( 779Q ..::'9::: < <& &%'L'Lm'L'L'L"L"L, . .# s' 4 44rc~|j}tj|tjd|jd|jD}t d|D}tj|||j}t|||}tj |||}tj |dg|j z|S)a Similar to lax.dynamic_update_slice, but handles padded updates where padding values should not overwrite existing values in the array. Args: operand: the array to update update: the padded array to write start_indices: the offset at which to write `update`. update_dims: the true dimensions of the padded update `update`. Only values inside the rectangle given by `update_dims` will be overwritten.rcg|]}d|dfSr'rr(s rr*z!_update_slice..ns5551aAY555rc3>K|]}tj|VdSr,r-r.s rr/z _update_slice..os*<< ! <<<<< >>rFcRj\}tjd }t|ftj}t j|}dd|f} t| f} tj} dtt j| j j z| zfdfd} fd} | \} }}t j dtj }tj| | | |||f\} }}}|r|| fS| |fS) a}Decomposes the `n x n` rank `rank` Hermitian projector `P` into an `n x rank` isometry `V_minus` such that `P = V_minus @ V_minus.conj().T` and an `n x (n - rank)` isometry `V_minus` such that -(I - P) = V_plus @ V_plus.conj().T`. The subspaces are computed using the naiive QR eigendecomposition algorithm, which converges very quickly due to the sharp separation between the relevant eigenvalues of the projector. Args: P: A rank-`rank` Hermitian projector into the space of `H`'s first `rank` eigenpairs. `P` is padded to NxN. H: The aforementioned Hermitian matrix, which is used to track convergence. n: the true (dynamic) shape of `P`. rank: Rank of `P`. maxiter: Maximum number of iterations. swap: If true, the two outputs spaces are swapped. Returns: V_minus, V_plus: Isometries into the eigenspaces described in the docstring. r)axisNg$@cDtj|d\}}t| f}t|d f z ff}t j|j}t j||}tj|}|||fS)Ncomplete)moder) jnp_linalgqrr$r<rdotconjTnorm) XQ_V1V2 error_matrixerrorHNrranks rbody_f_after_matmulz0_projector_subspace..body_f_after_matmuls = , , ,DAq q1d)  B At9q!d(maV 4 4B727799;**L7<,,L OL ) )E r5=rc`|\}}}}|k}|k}tj||dS)Nr)r logical_and)argsrUjrYstill_counting unconvergedmaxiterthreshs rcond_fz#_projector_subspace..cond_fs:NAq!U[N&.K  nk : :1 ==rcl|\}}}}tj|}|\}}}|||dz|fSr )rrO) r`rVrUrarSrWrYPr]s rbody_fz#_projector_subspace..body_fsHKB1a 2A''**MBE r1q5% rr2)rrMrRr$rnanargsortfloatfinfor2epsonesrr while_loop)rhrZrr\rdswaprUnegative_column_norms sort_idxsrSH_normrfrirVrWrYoner[r]res````` @@@r_projector_subspacerwvs0 $!Q%?115555 5tSWEEk/00)9 o! A4y! ?1  & % !'**.// /& 8&        >>>>>>       &%a((-"b% #)$$$#^FFRS%4HII"b!U  r6M r6Mrc2j\}}|tj||jzjz }t j|df\}}}}t tj|jf}d||z ztjtjtj tj d||zzz k} tj | fdfd\} } | jz| z} | jz| z} |*tj|| } tj|| } | | | | fS)a The Hermitian matrix `H` is split into two matrices `H_minus` `H_plus`, respectively sharing its eigenspaces beneath and above its `split_point`th eigenvalue. Returns, in addition, `V_minus` and `V_plus`, isometries such that `Hi = Vi.conj().T @ H @ Vi`. If `V0` is not None, `V0 @ Vi` are returned instead; this allows the overall isometries mapping from an initial input matrix to progressively smaller blocks to be formed. Args: H: The Hermitian matrix to split. split_point: The eigenvalue to split along. V0: Matrix of isometries to be updated. Returns: H_minus: A Hermitian matrix sharing the eigenvalues of `H` beneath `split_point`. V_minus: An isometry from the input space of `V0` to `H_minus`. H_plus: A Hermitian matrix sharing the eigenvalues of `H` above `split_point`. V_plus: An isometry from the input space of `V0` to `H_plus`. rank: The dynamic size of the m subblock. rjT) is_hermitian dynamic_shapegg?c,tdS)NTrrrw)rZP_plusr rank_plussrz split_spectrum..s!&!Q EEErc,tdS)NFr|r})rZP_minusr rank_minussrrz split_spectrum..s!'1a%HHHr)rreyer2astyper r$roundtracerrealrrcondrPrQrO)rZr split_pointV0r[rUH_shiftUIrrV_minusV_plusH_minusH_plusrr~rrs`` @@@@rsplit_spectrumrs. $!Q swq 0ABBBBJJ17SS S'ytAq6JJJ*!Q1 CGAQW % % %1v..! AEN'y6;w#7#78899@@KK* !a%=&*n) Z $H EEEEEEEHHHHHHH/'6 \\^^  !W ,' KKMMOa 6 )&^gb'""G WR F '66: 55rc(eZdZUdZded<ded<dS) _SubproblemzDescribes a subproblem of _eigh_work. Each subproblem is a `size` x `size` Hermitian matrix, starting at `offset` in the workspace. z jax.ArrayoffsetsizeN)__name__ __module__ __qualname____doc____annotations__rrrrrs3 /////rrtermination_sizesubset_by_index)static_argnamesc `|j\}tjtjt jdzt tjdtjtjdtj}|t tjd}tj |j }tj t|f|}fd}fd} d} t|} | gt|| g|krt| d} d } t#| z }|| krSt%|| }|t| ||d z}|| kStjd fd }t'j| ||||f\}}}|d d df|fS)a The main work loop performing the symmetric eigendecomposition of H. Each step recursively computes a projector into the space of eigenvalues above jnp.mean(jnp.diag(H)). The result of the projections into and out of that space, along with the isometries accomplishing these, are then computed. This is performed recursively until the projections have size 1, and thus store an eigenvalue of the original input; the corresponding isometry is the related eigenvector. The results are then composed. This function cannot be Jitted because the internal split_spectrum cannot be. Args: H: The Hermitian input. n: The true (dynamic) shape of H. Returns: H, V: The result of the projection. rr)rrrjct||df||f||f}t|d|f |f |f}t|||f}tj|d\}} t|||f}t| |f} t j||}| |j} t|| dddf|df|df}t||d|f |f}|||fS)NrFsort_eigenvaluesr) r<r$rr eighrrOrr2rF) Brbagendablocks eigenvectorsrZVeig_vecseig_valsr[rs r base_casez_eigh_work..base_case^s  v{QFQF33A|a[1a&1a&99A a!QAUCCHhX1v&&HXt$$Hwq(##Hx~..H 68AAAtG#4vqkAq6 J JF x!Vq!fMML 6< ''rc , t|dfff fd}d fd}tj }tjtj jj|j} tj tjtjtj  jz } | d| z|zk} || zk} tj | | z|||||S)Nrcrt|tjdddfdfdf}|||fS)Nrr)rFrdiag)rrrrZrrs rnearly_diagonal_casez@_eigh_work..recursive_case..nearly_diagonal_cases@VSXa[[D%9FA;AOOf V\ ))rc<|dn||dk||dkzS)NTrrr)startendrs rshould_update_rangez?_eigh_work..recursive_case..should_update_ranges2 $ $++oa6H0HIrc N tdfff}tjtt jt jftj}t||\}}}}} t|df| | ft|df| f t| f | z} tj| fdfd\t|| zdf| z | z ft|d| zf| z f t| z| z f | zz} tj| fdfd\fS)Nr)rcSr,r)updated_minus_statesrrzJ_eigh_work..recursive_case..default_case..s%rcfSr,rrrrsrrzJ_eigh_work..recursive_case..default_case..6<0rcSr,r)updated_plus_statesrrzJ_eigh_work..recursive_case..default_case..s$rcfSr,rrsrrzJ_eigh_work..recursive_case..default_case..rr)r<r nanmedianr$rrrrrkrrFpushrrr)rrrrrrrrrr\should_update_minusshould_update_plusrrrrZr[rrrrrs``` @@r default_casez8_eigh_work..recursive_case..default_cases! 6{QFQF ; ;a(sx A/G/G!sw)W)WXXk/= Q 0#0#0#,gw &!tTl C C  g6{QI F F ++k&$// 0 0 0/ &4-&)X  % % % % 0 0 0 0 0 0&&"flF fv}a01t8QX2F   FQ $6AH    ++k&4-!d(<< = =/. 4-!_&)X  $ $ $ $ 0 0 0 0 0 0&&"flF V\ ))rrj)r<rMrRrasarrayrnr2rorrrrrr)rrrrrrrrrRro off_diag_normnearly_diagonaltinyrZrH0_normr[rrs``` @@rrecursive_casez"_eigh_work..recursive_casezs` v{QFQF33A*******************************` ?1  D +ci((,DJ ? ? ?CO CHSXfk!nn--44QW== > >>@@M#q3w~5O #- D 8$   rc8|\}}}|Sr,)empty)staterrUs r loop_condz_eigh_work..loop_condsLFAq LLNN?rgGz? rGrc |\}}}|\\}}}tj |ktjtjj }tj|}tj||||||Sr,) poprriinformaxargminrswitch) rrrrrrwhichchoicebranchesbucketss r loop_bodyz_eigh_work..loop_bodysv#( FFL **,,KVQ Igk39SY#7#7#;W E EE Z  F :fh66< P PPrN)rrrrr createrr1rrr2rMrRr$minrappendintrrrq)rZrrrrUrrrrrrcutoff multiplier granularityr bucket_sizerrr[rrs ` ` @@@@r _eigh_workr s, $!Q k!SY! <E;syCI.. !SY0G0G H H J J& ;;{#)A,,Q??? @ @& !'***, OE!aV,, - -'0 &((((((8MMMMMMMM^ q" # #& H'i(( )(  NN1 OOGNA..///JK A NA f**a--k nn[!!!oognk::;;; q&a f** IgW - - -'QQQQQQ NVV\:<<!V\ 1| ##rfloat32T) precisionrrrrc|j\}}||krtd|jd|||krtdd}||d|fk}t|dkrtd|d|d krtd |ddkrtd ||n|} |d | krtd ||kre|t |||f}t j||p| \} } |r0| |d|d } | dd|d|d f} | | fS||n|}tj|5t||||\} } dddn #1swxYwYt tj | |ftj } |s|r@tj| } |r| |d|d } | | } | dd| f} | | fS)aComputes the eigendecomposition of the symmetric/Hermitian matrix H. Args: H: The `n x n` Hermitian input, padded to `N x N`. precision: :class:`~jax.lax.Precision` object specifying the matmul precision. termination_size: Recursion ends once the blocks reach this linear size. n: the true (dynamic) size of the matrix. sort_eigenvalues: If `True`, the eigenvalues will be sorted from lowest to highest. subset_by_index: Optional 2-tuple [start, end] indicating the range of indices of eigenvalues to compute. For example, is ``range_select`` = [n-2,n], then ``eigh`` computes the two largest eigenvalues and their eigenvectors. Returns: vals: The `n` eigenvalues of `H`. vecs: A unitary matrix such that `vecs[:, i]` is a normalized eigenvector of `H` corresponding to `vals[i]`. We have `H @ vecs = vals * vecs` up to numerical error. zInput H of shape z must be square.Nz5Static size must be greater or equal to dynamic size.FrrGz*subset_by_index must be a tuple of size 2.rz)Got empty index range in subset_by_index.z0Indices in subset_by_index must be non-negative.z0Index in subset_by_index[1] exceeds matrix size.rr)r TypeError ValueErrorrr$ lax_linalg eigh_jacobijaxdefault_matmul_precisionrrrrrkrl) rZrrrrrMr[ compute_slice range_maxrrrts rrrs< $!Q!VV AAAA B BB]q1uu L M MM-  #1v-M ?q  C D DDq_Q/// B C CCqA I J JJYAIqI%% I J JJ } Aq6  a#/ ->HhF/!,q/AABh!!!_Q//!2DDDEh X  9aa!! #I..# 1/Hh6;x((1$ 8 8(&& H%%IEOA.1CCDi "H9 %H 8 sE33E7:E7r')rGFr,)#r __future__r functoolsrtypingrrjax._src.numpy.lax_numpy_srcnumpy lax_numpyrjax._src.numpy.linalgr rMjax._src.numpyrrr jax._src.laxr rjax._src.lax.stackr rr$r<rFrwrrjitrrrrrrs  #""""" &&&&&&&&&&&&************%%%%%%!!!!!!------$$$$$$ @ @ @ @55556???,DDDDN06060606l     *    "IJJJO$O$KJO$j IIIIIIIr