ukiQ%dZddlmZddlZddlZddlmZddlmZddlm Z ddlm Z ddlm Z dd lmZ dd l mZdd lmZdd Zdd Zdd ZdZdZdZej0ej2dddddd ddZy)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)api)config)core)dtypes)lax)numpy)linalgc$tj|t|k(sJd}t|D]C\}}| t j tj |j||k}||n||z}E||Stj|||S)zMasks `x` up to the dynamic shape `dims`. Replaces values outside those dimensions with `alternative`. `alternative` is broadcast with `x`. N) npndimlen enumeraterbroadcasted_iotaint32shapejnpwhere)xdims alternativemaskid mask_dim_is S/home/cdr/jupyterlab/.venv/lib/python3.12/site-packages/jax/_src/tpu/linalg/qdwh.py_maskr,s s4y  $oAda}''!''1=Aj a_minus_ecrArFrCrrDzs r _use_choleskyrU\s/)Q $!Q13388:>SWWQfll1o>>! A1vswwq01!!e4!!!Dd<<@DF!!!Q$d.2FFGa QQ rcf |2ttjjj}t j d}t j tj}tj|tj|z}tj|dk(d|}|jjz}|} 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 d d |z z||zz zd zz}|dz d zd z }||zdz }| |||zzzd||zzz } || kDr| j| |||n|j|||| | zdkr|kr fdfd}||| t d\}}|||t"d\}}fd}fd}t%| t%|z}tj&|||||f\}}}d|zd |z|j(j+|zzz }|j(j+z}||j(j+zd z }tj,|}||||fS)zGQR-based dynamically weighted Halley iteration for polar decomposition.)ordrg$@g@c0||z }||z }|dz}||z ||fS)N?)r.r/rSrArRr@s r get_qr_paramsz_qdwh..get_qr_paramss/ AAAI 5\F   **rc ||z }||z }|||fSNr[)r.r/rSrArRs rget_chol_paramsz_qdwh..get_chol_paramss! AAAI q! rdgUUUUUU?g?rZc|\}}|  ddd}ntj||d}|}|| |}d} |rtj||z  kD} || fS)NrWF)keepdimsT)rdynamic_index_in_dim jnp_linalgnorm)kstate update_fncoefstest_convergencer<rFr>u_previs_not_convergedr_r0r=tol_norms r iterationz_qdwh..iterationsu DAq }q!Q'f''q5Af F!Q6"A#V4x?  rc|s|dfStj|jj}t j fd|i|}t jdt|||dfS)NTrlr) rr!astyper" functoolspartialr fori_loopr)r<rlkwargsbodyrqrs riteratez_qdwh..iteratesc  Wn IIe  # #AGG ,E   Y >e >v >D ==CJq$i 88rFrlrkrmTcB|\}}}tj||kSr^)r logical_and)rjrirFromax_iterationss rcond_funz_qdwh..cond_funs'"Aq  ??+Q-? @@rcL|\}}}|||fdtd\}}|dz||fS)NTrzrW)rU)rjrir<rorqs rbody_funz_qdwh..body_funsI"Aq #  A q5!% %%rg?)floatrfinfor"epsrgrhr infrrsqrtrrrscbrtappendrIrUr while_loopr:r; logical_not)!rr0r=r}rone_norminf_norm alpha_inverser<ltol_lr\CHOLESKY_CUTOFFqr_coefs chol_coefsril2ddsqdr.r/rSryrFror~r num_itersh is_convergedr_rqrps!```` @@@r_qdwhrxs  [  QWW%)) *C __QA &( __QBFF +())H% ((;;-))HM1m<--  qww ''! ! *s % XXe_(+  / (*! E A !n,FA QB q2vz R U +B 8 C q2vQV S11u==A Q1 qA A A QRZABJ'A?oomAq!,-1a01 E A !n,"9 x7U $!Q z]T! A & (mc*o%!#&>>1a!12$ )Q  Aga13388:>**!cchhj1n!13388:~!!12, Ay, &&r) is_hermitianr}r)static_argnamesF)rr}r dynamic_shapecrtjt|d}|d}ntjt|d}|j\}}||kr t d||\}}t |||f}n||}}tjd5t|||||\} } } } ddd    fS#1swYxYw)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_errorboolintr ValueErrorrrdefault_matmul_precisionr) rrr}rrrBrCr0r=r<rrrs rqdwhrs>'' L<=,N++ ^>?N $!QU H II DAq a!QA aqA &&y1H$)!Q>3$G!Aq)\H Ay, &&HHs B--B6)r)rrrrr)rrr}z int | Nonerz float | Nonerztuple[int, int] | None)__doc__ __future__rrtr r jax._srcrrrrrr jax._src.laxr r6jax._src.numpyrgrr)r1rIrUrrujitrr[rrrs #!-/ @: A1*8p'hGGF !%,0 3'3' 3'  3' * 3'3'r