ó î&]\c@`sóddlmZmZmZddlmZddlZddlmZm Z m Z m Z m Z m Z mZmZmZmZmZddlmZddlmZmZdgZeeeed „Zed „Zeed „Zed „ZdS( i(tdivisiontprint_functiontabsolute_import(twarnN( t atleast_2dtComplexWarningtaranget zeros_liketimagtdiagt iscomplexobjttrilttriutargsortt empty_likei(t_asarray_validated(tget_lapack_funcst_compute_lworktldlcC`sçtt|d|ƒƒ}|jd|jdkrAtdƒ‚n|jdkrxt|ƒt|ƒtjgdtƒfS|jd}t |ƒr—t nt }|t krï|rïd\}} tj t t|ƒƒƒrûtdtd d ƒqûn d\}} t|| f|fƒ\} } t| |d |ƒ} | |d| d |d|ƒ\} }}|dkrƒtdj|jƒ| ƒƒ‚nt|d |ƒ\}}t| |d |d|ƒ\}}t|||d |ƒ\}}|||fS(sG 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. 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.]]) 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 See also -------- cholesky, lu 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 t check_finiteiis%The input array "a" should be square.tdtypethetrft hetrf_lworks‡scipy.linalg.ldl(): The imaginary parts of the diagonalare ignored. Use "hermitian=False" for factorization ofcomplex symmetric arrays.t stacklevelitsytrft sytrf_lworktlowertlworkt overwrite_asv{} exited with the internal error "illegal value in argument number {}". See LAPACK documentation for the error codes.t hermitian(RR(RR(RRtshapet ValueErrortsizeRtnptarraytintR tcomplextfloattanyRR RRRRtformattuppert_ldl_sanitize_ipivt_ldl_get_d_and_lt_ldl_construct_tri_factor(tARRRRtatntr_or_ctstsltsolvert solver_lworkRtldutpivtinfotswap_arrt pivot_arrtdtlutperm((s7lib/python2.7/site-packages/scipy/linalg/_decomp_ldl.pyRs0m(     !c C`s_|j}t|ƒ}t|dtƒ}t}|rHddd|dfndd|dddf\}}}} } xât|| | ƒD]Î} |r›t}qƒn|| } | dkrã| | dkrÖ|| d|| R<RR)R*R+(((s7lib/python2.7/site-packages/scipy/linalg/_decomp_ldl.pyts L ‘ U8