EiD1SSKJr SSKrSSKJrJrJrJrJrJ r J r J r J r J r SSKJr SSKJr SSKJr SSKJrJr S /r\"S 5SS j5rSS jrSS jrSSjrg))warnN) atleast_2darange zeros_likeimagdiag iscomplexobjtriltriuargsort empty_like)ComplexWarning)_apply_over_batch)_asarray_validated)get_lapack_funcs_compute_lworkldl)Ac[[XS95nURSURS:wa [S5eURS:Xa.[ U5[ U5[ R"/[S94$URSn[U5(a[O[nU[LaHU(aASup[ R"[[U555(a[S[ SS 9 OS up[#X4U45up[%XUS 9n U "X\UUS 9upnUS:a [UR'5S U*S35e[)XS 9unn[+U UXS9unn[-UUUUS 9unnUUU4$)a Computes the LDLt or Bunch-Kaufman factorization of a symmetric/ hermitian matrix. This function returns a block diagonal matrix D consisting blocks of size at most 2x2 and also a possibly permuted unit lower triangular matrix ``L`` such that the factorization ``A = L D L^H`` or ``A = L D L^T`` holds. If `lower` is False then (again possibly permuted) upper triangular matrices are returned as outer factors. The permutation array can be used to triangularize the outer factors simply by a row shuffle, i.e., ``lu[perm, :]`` is an upper/lower triangular matrix. This is also equivalent to multiplication with a permutation matrix ``P.dot(lu)``, where ``P`` is a column-permuted identity matrix ``I[:, perm]``. Depending on the value of the boolean `lower`, only upper or lower triangular part of the input array is referenced. Hence, a triangular matrix on entry would give the same result as if the full matrix is supplied. Parameters ---------- A : array_like Square input array lower : bool, optional This switches between the lower and upper triangular outer factors of the factorization. Lower triangular (``lower=True``) is the default. hermitian : bool, optional For complex-valued arrays, this defines whether ``A = A.conj().T`` or ``A = A.T`` is assumed. For real-valued arrays, this switch has no effect. overwrite_a : bool, optional Allow overwriting data in `A` (may enhance performance). The default is False. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- lu : ndarray The (possibly) permuted upper/lower triangular outer factor of the factorization. d : ndarray The block diagonal multiplier of the factorization. perm : ndarray The row-permutation index array that brings lu into triangular form. Raises ------ ValueError If input array is not square. ComplexWarning If a complex-valued array with nonzero imaginary parts on the diagonal is given and hermitian is set to True. See Also -------- cholesky, lu Notes ----- This function uses ``?SYTRF`` routines for symmetric matrices and ``?HETRF`` routines for Hermitian matrices from LAPACK. See [1]_ for the algorithm details. Depending on the `lower` keyword value, only lower or upper triangular part of the input array is referenced. Moreover, this keyword also defines the structure of the outer factors of the factorization. .. versionadded:: 1.1.0 References ---------- .. [1] J.R. Bunch, L. Kaufman, Some stable methods for calculating inertia and solving symmetric linear systems, Math. Comput. Vol.31, 1977. :doi:`10.2307/2005787` Examples -------- Given an upper triangular array ``a`` that represents the full symmetric array with its entries, obtain ``l``, 'd' and the permutation vector `perm`: >>> import numpy as np >>> from scipy.linalg import ldl >>> a = np.array([[2, -1, 3], [0, 2, 0], [0, 0, 1]]) >>> lu, d, perm = ldl(a, lower=0) # Use the upper part >>> lu array([[ 0. , 0. , 1. ], [ 0. , 1. , -0.5], [ 1. , 1. , 1.5]]) >>> d array([[-5. , 0. , 0. ], [ 0. , 1.5, 0. ], [ 0. , 0. , 2. ]]) >>> perm array([2, 1, 0]) >>> lu[perm, :] array([[ 1. , 1. , 1.5], [ 0. , 1. , -0.5], [ 0. , 0. , 1. ]]) >>> lu.dot(d).dot(lu.T) array([[ 2., -1., 3.], [-1., 2., 0.], [ 3., 0., 1.]]) ) check_finiterrz%The input array "a" should be square.dtype)hetrf hetrf_lworkzscipy.linalg.ldl(): The imaginary parts of the diagonalare ignored. Use "hermitian=False" for factorization ofcomplex symmetric arrays.r) stacklevel)sytrf sytrf_lwork)lower)lworkr overwrite_azB exited with the internal error "illegal value in argument number z0". See LAPACK documentation for the error codes.)r hermitian)rrshape ValueErrorsizer nparrayintr complexfloatanyrrrrrrupper_ldl_sanitize_ipiv_ldl_get_d_and_l_ldl_construct_tri_factor)rr r#r"ranr_or_csslsolver solver_lworkr!ldupivinfoswap_arr pivot_arrdluperms U/home/mande/repo/quber/.venv/lib/python3.13/site-packages/scipy/linalg/_decomp_ldl.pyrrsr\ %aCDAwwqzQWWQZ@AAvv{!}jmRXXb-DDD  A$QWUFY& 66$tAw- -.< L'+QGaT:F <% 8EA%(35NCd axAGGI;'/04ug6001 1-S>Hi S)5 NEAr(XyNHB q$;cURn[U5n[U[S9nSnU(aSSSUS4O SSUS- SS4upgpn [ XU 5Hjn U(aSnMX n U S:aXS-:wa X diag_indsblkincs r@r/r/s0  D T#YA  A E6DAc2T#q\Bq I B)AIk !8"%Aguw&6"7AAguw  &)'57*:&;&@&@&B'57"#&)'57*:&;'57"##%BQw " 5LrAcvURSn[U5nU(aUS- SS4OSUS4upgn[XgU5Hln Xn X:wdMU(aU OSn U(aUOU S-n X)U(aSOS:XaX(aSOS- n X(aSOS- n X U /X24X U /X24'XYU /XZU /'Mn U[U54$)a! Helper function to construct explicit outer factors of LDL factorization. If lower is True the permuted factors are multiplied as L(1)*L(2)*...*L(k). Otherwise, the permuted factors are multiplied as L(k)*...*L(2)*L(1). See LAPACK documentation for more details. Parameters ---------- lu : ndarray The triangular array that is extracted from LAPACK routine call with ones on the diagonals. swap_vec : ndarray The array that defines the row swapping indices. If the kth entry is m then rows k,m are swapped. Notice that the mth entry is not necessarily k to avoid undoing the swapping. pivs : ndarray The array that defines the block diagonal structure returned by _ldl_sanitize_ipiv(). lower : bool, optional The boolean to switch between lower and upper triangular structure. Returns ------- lu : ndarray The square outer factor which satisfies the L * D * L.T = A perm : ndarray The permutation vector that brings the lu to the triangular form Notes ----- Note that the original argument "lu" is overwritten. rrrCr)r$rrDr ) r>swap_vecrQr r2r?rJrKrLrMs_indcol_scol_es r@r0r0.sF  A !9D"'!A#r2aAYJBBRR   < CaEACEEy%QQ/u!+e*,.U|U[/H,IBs|U[( )!%El!3D ! wt} rA)TTFT)T)TT)warningsrnumpyr'rrrrrr r r r r numpy.exceptionsrscipy._lib._utilr_decomprlapackrr__all__rr.r/r0rAr@rds^BBB+.'4 '8NNbRj5p6rA