ó î&]\c@`sædZddlmZmZmZddlmZmZmZddl m Z m Z ddl m Z ddd d d gZeeeed „Zeeed „Zeeed„Zeed„Zeeed„Zeed„ZdS(s!Cholesky decomposition functions.i(tdivisiontprint_functiontabsolute_import(tasarray_chkfinitetasarrayt atleast_2di(t LinAlgErrort _datacopied(tget_lapack_funcstcholeskyt cho_factort cho_solvetcholesky_bandedtcho_solve_bandedc C`sD|rt|ƒn t|ƒ}t|ƒ}|jdkrTtdj|jƒƒ‚n|jd|jdkr‰tdj|jƒƒ‚n|jdkr¨|jƒ|fS|pºt ||ƒ}t d |fƒ\}||d|d|d |ƒ\}}|dkrt d |ƒ‚n|dkr:td j| ƒƒ‚n||fS( s,Common code for cholesky() and cho_factor().is?Input array needs to be 2 dimensional but received a {}d-array.iis;Input array is expected to be square but has the shape: {}.tpotrftlowert overwrite_atcleans9%d-th leading minor of the array is not positive definitesFLAPACK reported an illegal value in {}-th argumenton entry to "POTRF".(R( RRRtndimt ValueErrortformattshapetsizetcopyRRR( taRRRt check_finiteta1Rtctinfo((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyt _choleskys(   $    c C`s.t|d|d|dtd|ƒ\}}|S(sí 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.linalg import cholesky >>> a = np.array([[1,-2j],[2j,5]]) >>> L = cholesky(a, lower=True) >>> L array([[ 1.+0.j, 0.+0.j], [ 0.+2.j, 1.+0.j]]) >>> L @ L.T.conj() array([[ 1.+0.j, 0.-2.j], [ 0.+2.j, 5.+0.j]]) RRRR(RtTrue(RRRRR((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyR /s+c C`s4t|d|d|dtd|ƒ\}}||fS(sA 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. Examples -------- >>> from scipy.linalg import cho_factor >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]]) >>> c, low = cho_factor(A) >>> c array([[3. , 1. , 0.33333333, 1.66666667], [3. , 2.44948974, 1.90515869, -0.27216553], [1. , 5. , 2.29330749, 0.8559528 ], [5. , 1. , 2. , 1.55418563]]) >>> np.allclose(np.triu(c).T @ np. triu(c) - A, np.zeros((4, 4))) True RRRR(RtFalse(RRRRR((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyR _s;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( s2Solve 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 Examples -------- >>> from scipy.linalg import cho_factor, cho_solve >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]]) >>> c, low = cho_factor(A) >>> x = cho_solve((c, low), [1, 1, 1, 1]) >>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4)) True iiis$The factored matrix c is not square.sincompatible dimensions.tpotrsRt overwrite_bs1illegal value in %d-th argument of internal potrs(R (RRRRRRR( t c_and_lowertbR!RRRtb1R txR((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyR Ÿs"#    )! cC`s—|rt|ƒ}n t|ƒ}td|fƒ\}||d|d|ƒ\}}|dkrstd|ƒ‚n|dkr“td| ƒ‚n|S(sU 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 See also -------- cho_solve_banded : Solve a linear set equations, given the Cholesky factorization of a banded hermitian. Examples -------- >>> from scipy.linalg import cholesky_banded >>> from numpy import allclose, zeros, diag >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) >>> A = A + A.conj().T + np.diag(Ab[2, :]) >>> c = cholesky_banded(Ab) >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :]) >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5))) True tpbtrfRt overwrite_abis)%d-th leading minor not positive definites1illegal value in %d-th argument of internal pbtrf(R&(RRRRR(tabR'RRR&RR((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyR Øs:   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( sv Solve the linear equations ``A x = b``, given the Cholesky factorization of the banded hermitian ``A``. Parameters ---------- (cb, lower) : tuple, (ndarray, 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_like 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 Examples -------- >>> from scipy.linalg import cholesky_banded, cho_solve_banded >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) >>> A = A + A.conj().T + np.diag(Ab[2, :]) >>> c = cholesky_banded(Ab) >>> x = cho_solve_banded((c, False), np.ones(5)) >>> np.allclose(A @ x - np.ones(5), np.zeros(5)) True iÿÿÿÿis&shapes of cb and b are not compatible.tpbtrsRR!s)%d-th leading minor not positive definites1illegal value in %d-th argument of internal pbtrs(R)(RRRRRR( t cb_and_lowerR#R!RtcbRR)R%R((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyR !s -    !  N(t__doc__t __future__RRRtnumpyRRRtmiscRRtlapackRt__all__RRRR R R R R (((s;lib/python2.7/site-packages/scipy/linalg/decomp_cholesky.pyts    0@9I