a CCCf0@sddlmZddlZddlmZmZmZmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZmZdgZdd dZdd d ZdddZdddZdS))warnN) atleast_2darange zeros_likeimagdiag iscomplexobjtriltriuargsort empty_like)ComplexWarning)_asarray_validated)get_lapack_funcs_compute_lworkldlTFcCsHtt||d}|jd|jdkr,td|jdkrRt|t|tjgtdfS|jd}t |rht nt }|t ur|rd\}} t t t|rtdtdd nd \}} t|| f|f\} } t| ||d } | || ||d \} }}|dkrt|d | dt||d \}}t| |||d\}}t||||d \}}|||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.Zdtype)ZhetrfZ hetrf_lworkzscipy.linalg.ldl(): The imaginary parts of the diagonalare ignored. Use "hermitian=False" for factorization ofcomplex symmetric arrays.) stacklevel)ZsytrfZ 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 nparrayintrcomplexfloatanyrrrr rrupper_ldl_sanitize_ipiv_ldl_get_d_and_l_ldl_construct_tri_factor)ArrrranZr_or_csslZsolverZ solver_lworkrldupivinfoZswap_arrZ pivot_arrdlupermr3T/var/www/html/django/DPS/env/lib/python3.9/site-packages/scipy/linalg/_decomp_ldl.pyr s6m       c Cs|j}t|}t|td}d}|r0ddd|dfndd|dddf\}}}} } t|| | D]} |rfd}qX|| } | dkr| | dkr|| d|| <d|| <qX| dkr| || |kr| | dkr|| d|| |<d|| |<d}qXtdqX||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.)rrrr ranger) r)rr*Zswap_ZpivotsZskip_2x2xyrsreriindZcur_valr3r3r4r%s*0 .    r%cCst|}tt|}|jd}d}|r*dnd\}} |r@t|dnt|d} t|} d| | | f<||dkD]} || } | dkr||||| f||||| f<|r|r||||| f||| ||f<n$||||| f||| ||f<d| |||| f<| }qj|| fS)a Helper function to extract the diagonal and triangular matrices for LDL.T factorization. Parameters ---------- ldu : ndarray The compact output returned by the LAPACK routing pivs : ndarray The sanitized array of {0, 1, 2} denoting the sizes of the pivots. For every 2 there is a succeeding 0. lower : bool, optional If set to False, upper triangular part is considered. hermitian : bool, optional If set to False a symmetric complex array is assumed. Returns ------- d : ndarray The block diagonal matrix. lu : ndarray The upper/lower triangular matrix r)rr)rrr5rrg)rrrr r rZconj)r-pivsrrZis_cr0r*Zblk_ir7r8r1Z diag_indsZblkincr3r3r4r&s$    $*$r&c Cs|jd}t|}|r$|dddfnd|df\}}}t|||D]} || } | | kr@|r\| nd} |rh|n| d} || |r~dndkr| |rdnd7} | |rdnd7} || | g| | f|| | g| | f<|| | g|| | g<q@|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. rrr5r)rrr6r ) r1Zswap_vecr=rr*r2r9r:r;r<Zs_indZcol_sZcol_er3r3r4r'+s# " $r')TTFT)T)TT)T)warningsrnumpyrrrrrrrr r r r Zscipy._lib._utilr _decomprZlapackrr__all__rr%r&r'r3r3r3r4s 0    U 8