Vpf$dZddlmZddlZddlZddlmZddlmZddl m Z ddl m Z ddZddZdd Zd Zd Zd Zejejd dddddddZdS)u0A JIT-compatible library for QDWH-based polar decomposition. QDWH is short for QR-based dynamically weighted Halley iteration. The Halley iteration implemented through QR decmopositions does not require matrix inversion. This is desirable for multicore and heterogeneous computing systems. Reference: Nakatsukasa, Yuji, Zhaojun Bai, and François Gygi. "Optimizing Halley's iteration for computing the matrix polar decomposition." SIAM Journal on Matrix Analysis and Applications 31, no. 5 (2010): 2700-2720. https://epubs.siam.org/doi/abs/10.1137/090774999 ) annotationsN)lax)core)linalgctj|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 alternativemaskid mask_dim_is Q/var/www/html/nettyfy-visnx/env/lib/python3.11/site-packages/jax/_src/lax/qdwh.py_maskr's !D ! ! ! ! $ooAAda}' 17A>>Bjrr?zs r _use_choleskyrPWs /)Q $!Q1388::>SWQcill;;;;! A1vswq00011!!e444!!Dd<<<<@DFF!!Q$d.2FFFFG QQ rc |+ttjjj}tjd}tjtj}tj |tj |z}tj |dkd|}| jz}|} d|zdz } tj |  d} dd } g} g}d}| | zdkr|kr|dz }| | z}d d|z dz z|z d z}d |zd z}|d|z dd|z z||zz zd zz}|dz dzd z }||zdz }| |||zzzd||zzz } || kr!| | |||n | |||| | zdkr|k fdfd}||| td\}}|||td\}}fd}fd}t!| t!|z}tj|||||f\}}}d|zd |z|j|zzz }|jz}||jzdz }tj|}||||fS)zGQR-based dynamically weighted Halley iteration for polar decomposition.N)ordrg$@g@c0||z }||z }|dz}||z ||fS)N?)r)r*rNr<rMr;s r get_qr_paramsz_qdwh..get_qr_paramss/ AAAI 5\F   **rc ||z }||z }|||fSNrV)r)r*rNr<rMs rget_chol_paramsz_qdwh..get_chol_paramss! AAAI q! rdgUUUUUU?g?rUc|\}}| ddd}ntj||d}|}|| |}d} |r&tj||z  k} || fS)NrRF)keepdimsT)rdynamic_index_in_dimrrnorm)kstate update_fncoefstest_convergencer7rAr9u_previs_not_convergedrZr+r8tol_norms r iterationz_qdwh..iterations DAq }q!Q''ff'q5AAAf F !Q6""A@V44x?  rc|s|dfStj|j}t jfd|i|}t jdt|||dfS)NTrfr) rrastyper functoolspartialr fori_loopr )r7rfkwargsbodyrkrs riteratez_qdwh..iteratesp  Wn Ie   # #AG , ,E  Y > >e >v > >D =CJJq$i 8 88rFrfrergTcB|\}}}tj||kSrY)r logical_and)rdrcrArimax_iterationss rcond_funz_qdwh..cond_funs("Aq  ?+Q-? @ @@rcT|\}}}|||fdtd\}}|dz||fS)NTrtrR)rP)rdrcr7rirks rbody_funz_qdwh..body_funsU"Aq #)  A q5!% %%rg?)floatrfinforepsrrbinfrrsqrtrrmcbrtappendrDrPr while_loopr5r6 logical_not)!rr+r8rwr}one_norminf_norm alpha_inverser7ltol_lrWCHOLESKY_CUTOFFqr_coefs chol_coefsrcl2ddsqdr)r*rNrsrArirxrz num_itersh is_convergedrZrkrjs!```` @@@r_qdwhrss  [  !'""& ' 'C Z__QA_ & &( Z__QCG_ , ,()H%% ((;(;;-)HM1m<<--  qw ' ''! ! *s % Xe__(+++  / (*! E A !n,,FA QB q2vz R U +B 8 C q2vQV S11u==A Q1 qA A A QRZABJ'A?oommAq!,,----1a00111 E A !n,,"999999 x7U   $!Q z]T! AAAAA & & & & & (mmc*oo%!#&>1a!12$$ )Q  Aga1388::>**!chhjj1n!1388::~!!122, Ay, &&r) is_hermitianrwr})static_argnamesF)rrwr} dynamic_shaperboolrw int | Noner} float | Nonertuple[int, int] | Nonectjt|d}|d}ntjt|d}|j\}}||krt d||\}}t |||f}n||}}tjd5t|||||\} } } } dddn #1swxYwY| | | | fS)aQR-based dynamically weighted Halley iteration for polar decomposition. Args: x: A full-rank matrix, with shape `M x N`. The matrix may be padded up to that size from a smaller true shape (``dynamic_shape``). is_hermitian: True if `x` is Hermitian. Default to `False`. This parameter is currently unused, but exists for backward compatibility. eps: The final result will satisfy ``|x_k - x_k-1| < |x_k| * (4*eps)**(1/3)`` where `x_k` is the iterate. max_iterations: Iterations will terminate after this many steps even if the above is unsatisfied. dynamic_shape: the unpadded shape as an ``(m, n)`` tuple; optional. Returns: A four-tuple of (u, h, num_iters, is_converged) containing the polar decomposition of `x = u * h`, the number of iterations to compute `u`, and `is_converged`, whose value is `True` when the convergence is achieved within the maximum number of iterations. zbThe `is_hermitian` argument must be statically specified to use `qdwh` within JAX transformations.N zdThe `max_iterations` argument must be statically specified to use `qdwh` within JAX transformations.z1The input matrix of shape M x N must have M >= N.float32) rconcrete_or_errorrintr ValueErrorrjaxdefault_matmul_precisionr) rrrwr}rr=r>r+r8r7rrrs rqdwhrsC>' L<==,NN+ ^>??N $!QUU H I II DAq a!QAA aqA #I..HH$)!Q>3$G$G!Aq)\HHHHHHHHHHHHHHH Ay, &&s B22B69B6)r)rrrrr)rrrwrr}rrr)__doc__ __future__rrnr jax.numpynumpyrrjax._srcr jax._src.laxrr1rr$r,rDrPrrojitrrVrrrsV  #""""" ------ @ @ @ @:::: AAAA111*8p'p'p'hGF !%,0 3'3'3'3'3'3'3'3'r