^Mh0ddlmZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZmZdgZdd Zdd Zdd ZddZdS))warnN) atleast_2darange zeros_likeimagdiag iscomplexobjtriltriuargsort empty_like)ComplexWarning)_asarray_validated)get_lapack_funcs_compute_lworkldlTFctt||}|jd|jdkrtd|jdkr8t |t |t jgtfS|jd}t|rtnt}|turM|rKd\}} t j tt|rtdt d nd \}} t#|| f|f\} } t%| || } | || || \} }}|dkr(t|d | dt)|| \}}t+| |||\}}t-|||| \}}|||fS)aG 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.) 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)Arr!r ranr_or_csslsolver solver_lworkrldupivinfoswap_arr pivot_arrdluperms X/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/linalg/_decomp_ldl.pyrr sZ %alCCCDDAwqzQWQZ@AAAv{{!}}jmmRXb-D-D-DDD  A$Q 2WWUFY&2 6$tAww--  L -.< L L L L'2+QGaT::FL <% 8 8 8EVAU%(3555NCd axxAGGII0004u00011 1-S>>>Hi S)5I N N NEAr(XyNNNHB q$;c|j}t|}t|t}d}|rddd|dfn dd|dz ddf\}}}} } t || | D]|} |rd}|| } | dkr| | dzkr|| dz || <d|| <2| dkr6| || |zkr'| | dzkr|| dz || |z<d|| |z<d}nt d||fS) a This helper function takes the rather strangely encoded permutation array returned by the LAPACK routines ?(HE/SY)TRF and converts it into regularized permutation and diagonal pivot size format. Since FORTRAN uses 1-indexing and LAPACK uses different start points for upper and lower formats there are certain offsets in the indices used below. Let's assume a result where the matrix is 6x6 and there are two 2x2 and two 1x1 blocks reported by the routine. To ease the coding efforts, we still populate a 6-sized array and fill zeros as the following :: pivots = [2, 0, 2, 0, 1, 1] This denotes a diagonal matrix of the form :: [x x ] [x x ] [ x x ] [ x x ] [ x ] [ x] In other words, we write 2 when the 2x2 block is first encountered and automatically write 0 to the next entry and skip the next spin of the loop. Thus, a separate counter or array appends to keep track of block sizes are avoided. If needed, zeros can be filtered out later without losing the block structure. Parameters ---------- a : ndarray The permutation array ipiv returned by LAPACK lower : bool, optional The switch to select whether upper or lower triangle is chosen in the LAPACK call. Returns ------- swap_ : ndarray The array that defines the row/column swap operations. For example, if row two is swapped with row four, the result is [0, 3, 2, 3]. pivots : ndarray The array that defines the block diagonal structure as given above. rFrrrTznWhile parsing the permutation array in "scipy.linalg.ldl", invalid entries found. The array syntax is invalid.)r$rrr'ranger#) r0rr1swap_pivotsskip_2x2xyrsreriindcur_vals r?r,r,sL` A 1IIE S ) ) )FH+0J1aAb"ac2r5JAq"b"RR  DD  H C& Q;;#a%"719-c F3KK q[[W#a%00x3q5  $gXaZ0c!e F3q5MHHCDD D &=r@c"t|}tt|}|jd}d}|rdnd\}} |rt|dnt |d} t |} d| | | f<||dkD]} || z} | dkru|||z|| zf|||z|| zf<|r0|r.|||z|| zf||| z||zf<n|||z|| zf||| z||zf<d| ||z|| zf<| }|| fS)a Helper function to extract the diagonal and triangular matrices for LDL.T factorization. Parameters ---------- ldu : ndarray The compact output returned by the LAPACK routing pivs : ndarray The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For every 2 there is a succeeding 0. lower : bool, optional If set to False, upper triangular part is considered. hermitian : bool, optional If set to False a symmetric complex array is assumed. Returns ------- d : ndarray The block diagonal matrix. lu : ndarray The upper/lower triangular matrix r)rr)rrrBrrg)r rr"r r rconj)r7pivsrr!is_cr<r1blk_irGrHr= diag_indsblkincs r?r-r-sk0   D T#YYA  A E &66DAq 1c2T#q\\Bq I By)DAIck !88"%eAguQw&6"7AeAguQw  <  <&)%'57*:&;&@&@&B&B%'57"##&)%'57*:&;%'57"##%BuQwa  b5Lr@ct|jd}t|}|r|dz ddfnd|df\}}}t|||D]g} || } | | krW|r| nd} |r|n| dz} || |rdndkr| |rdndz } | |rdndz } || | g| | f|| | g| | f<|| | g|| | g<h|t|fS)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. rrrBr)r"rrCr ) r=swap_vecrPrr1r>rIrJrKrLs_indcol_scol_es r?r.r.+sF  A !99D"'6!A#r2aAYJBBRR  4 4  C<< 'CCaE)AACEECy%.QQQ//u+!+e**,.U|U5[/H,IBs|U5[( )!%sEl!3D%  wt}} r@)TTFT)T)TT)warningsrnumpyr%rrrrrr r r r r scipy._lib._utilr_decomprlapackrr__all__rr,r-r.r@r?rbs[BBBBBBBBBBBBBBBBBBBBBBBB++++++''''''44444444 'NNNNbRRRRj5555p666666r@