ó ¦–Õ\c@`såddlmZmZmZddlZddlmZddlmZddl Z ddl Z ddl m Z ddlmZddlmZdd lmZd d lmZmZmZd d lmZd d lmZd„Zd„Zd„Zd„Zee d„Z!e d„Z"ddd„Z#de d„Z$de d„Z%d„Z&d„Z'd„Z(d„Z)ed„Z*ed„Z+d„Z,d „Z-ed!„Z.d"„Z/d#„Z0d$„Z1ee j2j3ƒe e ed%„ƒZ3dS(&i(tabsolute_importtdivisiontprint_functionN(twraps(tNumberi(ttokenize(t blockwise(tapply(tHighLevelGraphi(tdotmanytArrayt concatenate(teye(t RandomStatecc`s6d}x)|D]!}|}||7}||fVq WdS(Ni((titttotaltxttotal_previous((s0lib/python2.7/site-packages/dask/array/linalg.pyt_cumsum_blockss   cC`s|d|d|fS(Ni((tlasttnew((s0lib/python2.7/site-packages/dask/array/linalg.pyt _cumsum_partscC`sDt||gƒ}tj|ƒr'|n|}tj|ƒr@|S|S(N(tmintnptisnan(tmtntk_0tk_1((s0lib/python2.7/site-packages/dask/array/linalg.pyt_nanminscC`sP|jddkr<tjdƒtjd|jdfƒfStjj|ƒSdS(sê A wrapper for np.linalg.qr that handles arrays with 0 rows Notes: Created for tsqr so as to manage cases with uncertain array dimensions. In particular, the case where arrays have (uncertain) chunks with 0 rows. iiN(ii(tshapeRtzerostlinalgtqr(ta((s0lib/python2.7/site-packages/dask/array/linalg.pyt _wrapped_qr%s )cX `sªtˆjdƒtˆjdƒ}}tˆjdƒˆjdd}}ˆjdkod|dksˆtdjˆjˆjƒƒ‚ndtˆ|ƒ}ˆj\}‰|df} t j j t j dd5dˆj ƒƒ\} } ˆjƒjjƒ} ˆjƒjjƒ} d|‰ ttˆ d ˆjd d i| ˆj6ƒ}|| ˆ <ˆjƒ| ˆ nR|9rc |: rc ˆj};ˆjdˆff}<ˆˆf}=ˆˆf}>n |9 r¿ |:r¿ ˆj};ˆjdt j2ff}<t j2t j2f}=t j2ft j2ff}>n¯ˆjdˆjdkrâ ˆjnˆjdˆjdf};ˆjdˆjdkr ˆjnˆjdˆjdf}<ˆjdˆjdkr_ ˆˆfnˆj}=|=}>t"| | ƒ}!t#|!|'d|;d|<d| j ƒ}?t#|!|%d|=d|>d| j ƒ}@|?|@fSd,|d-}Att j j3|Ad |%d d id7|%6ƒ}Bd |d.}Cit*j+|Addfdf|Cddf6}Dd |d/}Eit*j+|Addfdf|Edf6}Fd |d0}Git*j+|Addfdf|Gddf6}Hd |d1}Itt4|Id |'d2|Cd3d i| |'6d8|C6ƒ}J|B| |A<|%h| |A<|D| |C<|Ah| |C<|J| |I<|'|Ch| |I<|F| |E<|Ah| |E<|H| |G<|Ah| |G= 2``). In particular, this is done to ensure that single core computations do not have to work on blocks larger than ``(m, n)``. Where blocks are irregular, the above logic is applied with the "height" of the "tallest" block used in place of ``m``. Consider use of the ``rechunk`` method to control this behavior. Blocks that are as tall as possible are recommended. See Also -------- dask.array.linalg.qr - Powered by this algorithm dask.array.linalg.svd - Powered by this algorithm dask.array.linalg.sfqr - Variant for short-and-fat arrays iiisInput must have the following properties: 1. Have two dimensions 2. Have only one column of blocks Note: This function (tsqr) supports QR decomposition in the case of tall-and-skinny matrices (single column chunk/block; see qr)Current shape: {}, Current chunksize: {}t-RtdtypeR!tijt numblockstgetitems-q1c3`s9|]/}ˆ|dftjˆ|dfdffVqdS(iN(toperatorR((t.0ti(t name_q_st1t name_qr_st1(s0lib/python2.7/site-packages/dask/array/linalg.pys ‰ss-r1c3`s9|]/}ˆ|dftjˆ|dfdffVqdS(iiN(R)R((R*R+(R-t name_r_st1(s0lib/python2.7/site-packages/dask/array/linalg.pys ‘scs`s+|]!}|D]}tj|ƒVq qdS(N(RR(R*tcstc((s0lib/python2.7/site-packages/dask/array/linalg.pys ststackc 3`s[|]Q\}}ˆ|dftjtg|D]\}}ˆ|df^q+fffVqdS(iN(Rtvstackttuple(R*R+tsub_block_infotidxt_(R.tname_r_stacked(s0lib/python2.7/site-packages/dask/array/linalg.pys ¹scS`s|dS(Ni((R((s0lib/python2.7/site-packages/dask/array/linalg.pytÂttchunkst_max_vchunk_sizesgetitem-s-q2c 3`s²|]¨\}}tg|D]}|d^qtg|D]}|d^q6ƒƒD]X\}}ˆ|dftjˆj|dft|d|dƒtdˆƒfffVqPqdS(iiN(tzipRR)R(tnametslice(R*R+R4Rtjte(Rt name_q_st2tq_inner(s0lib/python2.7/site-packages/dask/array/linalg.pys Ïs sr-inner-sdot-s-q3s-qr2s-q2-auxcs`s+|]!}|D]}tj|ƒVq qdS(N(RR(R*R/R0((s0lib/python2.7/site-packages/dask/array/linalg.pys ÿsc`s:i|]0}ttˆj|dfdffˆ|f“qS(iR(RtgetattrR=(R*R+(tdatat name_q2bs(s0lib/python2.7/site-packages/dask/array/linalg.pys s s-ntcumsumc`s8i|].}tˆ|dfˆ|ffˆ|f“qS(i(R(R*R+(REt name_q2cs(s0lib/python2.7/site-packages/dask/array/linalg.pys s R>s-qc`sCi|]9}tttˆ|fftdˆfgfˆ|f“qS(i(R3RR>(R*R+(tname_blockslicetname_nRG(s0lib/python2.7/site-packages/dask/array/linalg.pys s s q-blocksizesc3`s?|]5\}}ˆ|dftjˆddf|ffVqdS(iN(R)R((R*R+tb(RAtname_q_st2_aux(s0lib/python2.7/site-packages/dask/array/linalg.pys -stdots-r2cs`s|]}tj|ƒVqdS(N(RR(R*R0((s0lib/python2.7/site-packages/dask/array/linalg.pys Bscs`s|]}tj|ƒVqdS(N(RR(R*R0((s0lib/python2.7/site-packages/dask/array/linalg.pys Cstsvds-2s-u2s-s2s-v2s-u4tiktkjN(ii(ii(ii(ii(ii(6tlenR:tmaxtndimt ValueErrortformatRt chunksizeRRR R!tonesR%t__dask_graph__tlayerstcopyt dependenciesRR#R=t__dask_layers__tdicttrangetanytceiltNonetintt enumerateRtappendR3tsumtmapRR ttsqrttoolztmergetdaskRLR2R)R(RR>tsetRCtupdateRtnanRMR R(XRDt compute_svdR;tnrtnctcr_maxtccttokenRR'tqqtrrRXRZt dsk_qr_st1t dsk_q_st1t dsk_r_st1tsingle_core_compute_mtchunks_well_definedtprospective_blockstmeaningful_reduction_possibletcan_distributet all_blockst curr_blockt curr_block_szR5ta_mtm_qtn_qtm_rt dsk_r_stackedR4tvchunks_rstackedtgrapht r_stackedtr_innert dsk_q_st2t name_r_st2t dsk_r_st2t name_q_st3t dsk_q_st3R+tto_stacktname_r_st1_stackedtdsk_r_st1_stackedt name_qr_st2t dsk_qr_st2t dsk_q_st2_auxR@tq2_block_sizest block_slicestdsk_q_blockslicestdepst dsk_q2_shapestdsk_nt dsk_q2_cumsumtdsk_block_slicest is_unknown_mt is_unknown_ntq_shapetq_chunkstr_shapetr_chunkstqtrt name_svd_st2t dsk_svd_st2t name_u_st2t dsk_u_st2t name_s_st2t dsk_s_st2t name_v_st2t dsk_v_st2t name_u_st4t dsk_u_st4tuutsstvvhtktm_utn_utn_stm_vhtn_vhtd_vhtutstvh(( RDRRHRIRERGR,RARKR-R.R7RBs0lib/python2.7/site-packages/dask/array/linalg.pyRf4s’4'%  -       "     1 *  "   ,      "  )<  %  )   +  33       ==/    "  "          0!:-6c `s„tˆjdƒtˆjdƒ}}ˆjddˆjdd}}ˆjdkoz|dkoz||kpz|dksŒtdƒ‚n|pŸdtˆƒ}|d7}ˆj\}}tjjtj dddˆj ƒƒ\} } ˆj ƒj j ƒ} ˆj ƒjj ƒ} |d } |d ‰iˆjddf| ddf6| | <ˆjƒ| | <‡‡fd †t|dƒDƒ| ˆ<ˆjƒ| ˆ<|d }|d }|d}itjj| ddff|ddf6| |<| h| |Ïs tQ_R_1tQtR_1R:taxis(ii(RPR:RRRSRRRR R!RVR%RWRXRYRZR=R[R]R)R(RR RRctTRLR (RDR=RnRotcrRqtprefixRRRsRtRXRZtname_A_1t name_Q_R1tname_Qtname_R_1R†R¿RÀtRsR¼tR((RDR½s0lib/python2.7/site-packages/dask/array/linalg.pytsfqr˜sT'#     -  &   / / /  6 -   <i icC`stt|||ƒ|ƒS(s  Compression level to use in svd_compressed Given the size ``n`` of a space, compress that that to one of size ``q`` plus oversampling. The oversampling allows for greater flexibility in finding an appropriate subspace, a low value is often enough (10 is already a very conservative choice, it can be further reduced). ``q + oversampling`` should not be larger than ``n``. In this specific implementation, ``q + oversampling`` is at least ``min_subspace_size``. >>> compression_level(100, 10) 20 (RRQ(RR¢t oversamplingtmin_subspace_size((s0lib/python2.7/site-packages/dask/array/linalg.pytcompression_levelôsc C`s°|jd}t||ƒ}t|ƒ}|jd||fd|jd|ffƒ}|j|ƒ}x/t|ƒD]!} |j|jj|ƒƒ}qrWt|ƒ\}} |jS(sa Randomly sample matrix to find most active subspace This compression matrix returned by this algorithm can be used to compute both the QR decomposition and the Singular Value Decomposition. Parameters ---------- data: Array q: int Size of the desired subspace (the actual size will be bigger, because of oversampling, see ``da.linalg.compression_level``) n_power_iter: int number of power iterations, useful when the singular values of the input matrix decay very slowly. References ---------- N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., Survey and Review section, Vol. 53, num. 2, pp. 217-288, June 2011 https://arxiv.org/abs/0909.4061 itsizeR:( RRÎR tstandard_normalR:RLR]RÂRf( RDR¢t n_power_itertseedRt comp_leveltstatetomegatmat_hR?R6((s0lib/python2.7/site-packages/dask/array/linalg.pytcompression_matrixs  c C`s²t||d|d|ƒ}|j|ƒ}t|jdtƒ\}}}|jj|ƒ}|j}|dd…d|…f}|| }|d|…dd…f}|||fS(sé Randomly compressed rank-k thin Singular Value Decomposition. This computes the approximate singular value decomposition of a large array. This algorithm is generally faster than the normal algorithm but does not provide exact results. One can balance between performance and accuracy with input parameters (see below). Parameters ---------- a: Array Input array k: int Rank of the desired thin SVD decomposition. n_power_iter: int Number of power iterations, useful when the singular values decay slowly. Error decreases exponentially as n_power_iter increases. In practice, set n_power_iter <= 4. Examples -------- >>> u, s, vt = svd_compressed(x, 20) # doctest: +SKIP Returns ------- u: Array, unitary / orthogonal s: Array, singular values in decreasing order (largest first) v: Array, unitary / orthogonal References ---------- N. Halko, P. G. Martinsson, and J. A. Tropp. Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev., Survey and Review section, Vol. 53, num. 2, pp. 217-288, June 2011 https://arxiv.org/abs/0909.4061 RÑRÒRmN(R×RLRfRÂtTrue( R"R±RÑRÒtcompt a_compressedtvR¹R¸((s0lib/python2.7/site-packages/dask/array/linalg.pytsvd_compressed-s'  cC`sot|jdƒdkr<t|jdƒdkr<t|ƒSt|jdƒdkr_t|ƒStdƒ‚dS(s© Compute the qr factorization of a matrix. Examples -------- >>> q, r = da.linalg.qr(x) # doctest: +SKIP Returns ------- q: Array, orthonormal r: Array, upper-triangular See Also -------- np.linalg.qr : Equivalent NumPy Operation dask.array.linalg.tsqr: Implementation for tall-and-skinny arrays dask.array.linalg.sfqr: Implementation for short-and-fat arrays iis}qr currently supports only tall-and-skinny (single column chunk/block; see tsqr) and short-and-fat (single row chunk/block; see sfqr) matrices Consider use of the rechunk method. For example, x.rechunk({0: -1, 1: 'auto'}) or x.rechunk({0: 'auto', 1: -1}) which rechunk one shorter axis to a single chunk, while allowing the other axis to automatically grow/shrink appropriately.N(RPR:RfRËtNotImplementedError(R"((s0lib/python2.7/site-packages/dask/array/linalg.pyR!_s 2  cC`st|dtƒS(sÆ Compute the singular value decomposition of a matrix. Examples -------- >>> u, s, v = da.linalg.svd(x) # doctest: +SKIP Returns ------- u: Array, unitary / orthogonal s: Array, singular values in decreasing order (largest first) v: Array, unitary / orthogonal See Also -------- np.linalg.svd : Equivalent NumPy Operation dask.array.linalg.tsqr: Implementation for tall-and-skinny arrays Rm(RfRØ(R"((s0lib/python2.7/site-packages/dask/array/linalg.pyRM…scC`s%ddl}|jj||dtƒS(Nitlower(t scipy.linalgR tsolve_triangularRØ(R"RJtscipy((s0lib/python2.7/site-packages/dask/array/linalg.pyt_solve_triangular_loweržs c C`s—ddl}|jdkr*tdƒ‚n|j\}}||krTtdƒ‚ntt|jd|jdƒƒdks“d}t|ƒ‚nt|jdƒ}t|jdƒ}t|ƒ}d|}d |} d |} d |} d |} d |} d|}d|}d|}i}xÉtt ||ƒƒD]²}|j ||f}|dkrág}x^t|ƒD]P}|||||f}t j | ||f| ||ff||<|j |ƒqrWtj|t|ff}n|jj|f||||f>> p, l, u = da.linalg.lu(x) # doctest: +SKIP Returns ------- p: Array, permutation matrix l: Array, lower triangular matrix with unit diagonal. u: Array, upper triangular matrix iNis/Dimension must be 2 to perform lu decompositions9Input must be a square matrix to perform lu decompositionisqAll chunks must be a square matrix to perform lu decomposition. 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