ó ˽÷Xc@`sàdZddlmZmZmZddlmZmZddlm Z m Z ddl m Z ddd d d gZ eeeed „Zeeed „Zeeed„Zeed„Zeeed„Zeed„ZdS(s!Cholesky decomposition functions.i(tdivisiontprint_functiontabsolute_import(tasarray_chkfinitetasarrayi(t LinAlgErrort _datacopied(tget_lapack_funcstcholeskyt cho_factort cho_solvetcholesky_bandedtcho_solve_bandedc C`sö|rt|ƒ}n t|ƒ}t|jƒdksP|jd|jdkr_tdƒ‚n|pqt||ƒ}td |fƒ\}||d|d|d|ƒ\}}|dkrÌtd |ƒ‚n|dkrìtd | ƒ‚n||fS( s,Common code for cholesky() and cho_factor().iiisexpected square matrixtpotrftlowert overwrite_atcleans)%d-th leading minor not positive definites1illegal value in %d-th argument of internal potrf(spotrf(RRtlentshapet ValueErrorRRR( taRRRt check_finiteta1R tctinfo((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyt _choleskys /$  c C`s.t|d|d|dtd|ƒ\}}|S(sø Compute the Cholesky decomposition of a matrix. Returns the Cholesky decomposition, :math:`A = L L^*` or :math:`A = U^* U` of a Hermitian positive-definite matrix A. Parameters ---------- a : (M, M) array_like Matrix to be decomposed lower : bool, optional Whether to compute the upper or lower triangular Cholesky factorization. Default is upper-triangular. overwrite_a : bool, optional Whether to overwrite data in `a` (may improve performance). check_finite : bool, optional Whether to check that the input matrix contains 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 ------- c : (M, M) ndarray Upper- or lower-triangular Cholesky factor of `a`. Raises ------ LinAlgError : if decomposition fails. Examples -------- >>> from scipy import array, linalg, dot >>> a = array([[1,-2j],[2j,5]]) >>> L = linalg.cholesky(a, lower=True) >>> L array([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]]) >>> dot(L, L.T.conj()) array([[ 1.+0.j, 0.-2.j], [ 0.+2.j, 5.+0.j]]) RRRR(RtTrue(RRRRR((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyR%s+c C`s4t|d|d|dtd|ƒ\}}||fS(s= Compute the Cholesky decomposition of a matrix, to use in cho_solve Returns a matrix containing the Cholesky decomposition, ``A = L L*`` or ``A = U* U`` of a Hermitian positive-definite matrix `a`. The return value can be directly used as the first parameter to cho_solve. .. warning:: The returned matrix also contains random data in the entries not used by the Cholesky decomposition. If you need to zero these entries, use the function `cholesky` instead. Parameters ---------- a : (M, M) array_like Matrix to be decomposed lower : bool, optional Whether to compute the upper or lower triangular Cholesky factorization (Default: upper-triangular) overwrite_a : bool, optional Whether to overwrite data in a (may improve performance) check_finite : bool, optional Whether to check that the input matrix contains 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 ------- c : (M, M) ndarray Matrix whose upper or lower triangle contains the Cholesky factor of `a`. Other parts of the matrix contain random data. lower : bool Flag indicating whether the factor is in the lower or upper triangle Raises ------ LinAlgError Raised if decomposition fails. See also -------- cho_solve : Solve a linear set equations using the Cholesky factorization of a matrix. RRRR(RtFalse(RRRRR((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyR Us.c C`s|\}}|r-t|ƒ}t|ƒ}nt|ƒ}t|ƒ}|jdksn|jd|jdkr}tdƒ‚n|jd|jdkr¦tdƒ‚n|p¸t||ƒ}td ||fƒ\}|||d|d|ƒ\}} | dkrtd | ƒ‚n|S( sSolve the linear equations A x = b, given the Cholesky factorization of A. Parameters ---------- (c, lower) : tuple, (array, bool) Cholesky factorization of a, as given by cho_factor b : array Right-hand side overwrite_b : bool, optional Whether to overwrite data in b (may improve performance) 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 ------- x : array The solution to the system A x = b See also -------- cho_factor : Cholesky factorization of a matrix iiis$The factored matrix c is not square.sincompatible dimensions.tpotrsRt overwrite_bs1illegal value in %d-th argument of internal potrs(spotrs(RRtndimRRRR( t c_and_lowertbRRRRtb1RtxR((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyR ˆs"    )! cC`s—|rt|ƒ}n t|ƒ}td|fƒ\}||d|d|ƒ\}}|dkrstd|ƒ‚n|dkr“td| ƒ‚n|S(sÍ Cholesky decompose a banded Hermitian positive-definite matrix The matrix a is stored in ab either in lower diagonal or upper diagonal ordered form:: ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j) Example of ab (shape of a is (6,6), u=2):: upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * Parameters ---------- ab : (u + 1, M) array_like Banded matrix overwrite_ab : bool, optional Discard data in ab (may enhance performance) lower : bool, optional Is the matrix in the lower form. (Default is upper form) check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- c : (u + 1, M) ndarray Cholesky factorization of a, in the same banded format as ab tpbtrfRt overwrite_abis)%d-th leading minor not positive definites1illegal value in %d-th argument of internal pbtrf(spbtrf(RRRRR(tabR$RRR#RR((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyR ¸s)   c C`sê|\}}|r-t|ƒ}t|ƒ}nt|ƒ}t|ƒ}|jd|jdkrntdƒ‚ntd ||fƒ\}|||d|d|ƒ\}}|dkrÆtd|ƒ‚n|dkrætd| ƒ‚n|S( sžSolve the linear equations A x = b, given the Cholesky factorization of A. Parameters ---------- (cb, lower) : tuple, (array, bool) `cb` is the Cholesky factorization of A, as given by cholesky_banded. `lower` must be the same value that was given to cholesky_banded. b : array Right-hand side overwrite_b : bool, optional If True, the function will overwrite the values in `b`. 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 ------- x : array The solution to the system A x = b See also -------- cholesky_banded : Cholesky factorization of a banded matrix Notes ----- .. versionadded:: 0.8.0 iÿÿÿÿis&shapes of cb and b are not compatible.tpbtrsRRs)%d-th leading minor not positive definites1illegal value in %d-th argument of internal pbtrs(spbtrs(RRRRRR( t cb_and_lowerR RRtcbRR&R"R((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyR ðs    !  N(t__doc__t __future__RRRtnumpyRRtmiscRRtlapackRt__all__RRRRR R R R (((s;/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp_cholesky.pyts    0308