+ nDjqARt^RIt^RIHtHtHt^RIHtHtH t ^RI H t H t ^RI HtHtHt^RIHt.ROtRtRRltRR ltRR ltRR lt]!RR4R 4t]!R4RR l4tRRlt]!RR4R4tR#)z!Cholesky decomposition functions.N)asarray_chkfiniteasarray empty_like)_asarray_validated_apply_over_batch_deprecate_dtypes) LinAlgError _datacopied)get_lapack_funcs_normalize_lapack_dtype_ensure_aligned_and_native)_batched_linalgc RT RVUu.uF q"R,NK up RVUu.uF q"R,NK up R2p\V4huupiuupi)z Internal z return info = lapack_infoz for slices num.)r) routine_nameerr_lstemsgs&& W/data/cameron/venvs/s3viz/lib/python3.14/site-packages/scipy/linalg/_decomp_cholesky.py_check_format_errors_warningsrsf L>G1TGqM2B2BG1T0UV)01Axx12! 5 c 2U1s A A c\WR7p\P!V4pVPR ,VPR ,8wd\ RVP: 24h\ RV4\ WR4wrR\WR4wrRT;'dFVP^8H;'d/VPR,;'gVPR,pVPR ,^8Xd;VPRR p\P!VR ,VPR7pV#\P!WQW#4wrxV'd \RV4V#) z,Common code for cholesky() and cho_factor().) check_finitez8Expected a square matrix or batch thereof, got a1.shape=zlinalg.cholesky F_CONTIGUOUS C_CONTIGUOUSNdtypepotrf)r!)rnp atleast_2dshape ValueErrorrr r ndimflagszerosrr _choleskyr) alower overwrite_acleanra1 batch_shapecrs &&&&& rr)r)s A 9B r B xx|rxx|#T288+VWW',-b>OB0AOBNNBGGqLNN0LLBHH^4L xx|qhhsm HH[6) :!**2kIJA%gw7 Hc$\WVRVR7pV#)av 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. During decomposition, only the selected half of the matrix is referenced. Default is upper-triangular. overwrite_a : bool, optional Whether to overwrite data in `a` (may improve performance). Default is False. See :ref:`tutorial_linalg_overwrite` for details. check_finite : bool, optional Whether to check that the entire 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. See Also -------- cho_factor : Compute the Cholesky decomposition without explicitly setting the other triangle to 0. Notes ----- During the finiteness check (if selected), the entire matrix `a` is checked. During decomposition, `a` is assumed to be symmetric or Hermitian (as applicable), and only the half selected by option `lower` is referenced. Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still succeed if the symmetric/Hermitian matrix represented by the selected half is positive definite, yet it may fail if an element in the other half is non-finite. Examples -------- >>> import numpy as np >>> 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]]) Tr+r,r-r)r))r*r+r,rr0s&&&& rcholeskyr4:s| !k) +A Hr1cr\WVRVR7pVPRRp\P!WR7pWF3#)ay 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:: If `overwrite_a` is disabled, this function matches `cholesky` and zeros the other triangle. If it is enabled, the returned matrix may contain random data in the entries not used by the Cholesky decomposition, saving out on the explicit putting to 0 done by `cholesky`. Parameters ---------- a : (..., M, M) array_like Matrix to be decomposed lower : bool, optional Whether to compute the upper or lower triangular Cholesky factorization. During decomposition, only the selected half of the matrix is referenced. (Default: upper-triangular) overwrite_a : bool, optional Whether to overwrite data in ``a`` (may improve performance). Default is False. See :ref:`tutorial_linalg_overwrite` for details. check_finite : bool, optional Whether to check that the entire 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 : ndarray, bool Flag indicating whether the factor is in the lower or upper triangle Raises ------ LinAlgError Raised if decomposition fails. See Also -------- cholesky : Compute the Cholesky decomposition and explicitly put the other triangle to 0. cho_solve : Solve a linear set equations using the Cholesky factorization of a matrix. Notes ----- During the finiteness check (if selected), the entire matrix `a` is checked. During decomposition, `a` is assumed to be symmetric or Hermitian (as applicable), and only the half selected by option `lower` is referenced. Consequently, if `a` is asymmetric/non-Hermitian, `cholesky` may still succeed if the symmetric/Hermitian matrix represented by the selected half is positive definite, yet it may fail if an element in the other half is non-finite. Examples -------- >>> import numpy as np >>> 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]], ... dtype=float, order="F") >>> A_ref = np.copy(A) >>> c, low = cho_factor(A) >>> c array([[3. , 1. , 0.33333333, 1.66666667], [0. , 2.44948974, 1.90515869, -0.27216553], [0. , 0. , 2.29330749, 0.8559528 ], [0. , 0. , 0. , 1.55418563]]) >>> np.allclose(c.T @ c - A, np.zeros((4, 4))) True >>> c, low = cho_factor(A, overwrite_a=True) >>> 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_ref, np.zeros((4, 4))) True Fr3N)repsr )r)r$r"tile)r*r+r,rr0r/ ret_lowers&&&& r cho_factorr9}s?n !k!- /A''#2,K0I <r1c&VwrE\WAWRVR7#)aSolve the linear equations A x = b, given the Cholesky factorization of A. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- c_and_lower : tuple, (array, bool) Cholesky factorization of an array, as given by cho_factor. The first element of the tuple is the matrix containing the Cholesky factor, and the second element is a boolean flag indicating whether the factor is in the lower or upper triangle. b : array Right-hand side overwrite_b : bool, optional Whether to overwrite data in b (may improve performance) See :ref:`tutorial_linalg_overwrite` for details. 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 -------- >>> import numpy as np >>> 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  overwrite_br) _cho_solve) c_and_lowerbr<rr0r+s&&&& r cho_solver@sZHA aE VVr1cV'd\V4p\V4pM\V4p\V4pVP^8wg*VP^,VP^,8wd \ R4hVP^,VP^,8wd'\ RVP RVP R24hVP ^8Xde\ \P!^VPR7R3\P!^VPR74Pp\WVR7#T;'g \WQ4p\R W34wpV!WW#R7wrV ^8wd\ RV ) R 24hV#) z$The factored matrix c is not square.zincompatible dimensions (z and )rTr+r<illegal value in zth argument of internal potrs)potrs)rrr&r$r%sizer@r"eyeronesrr r ) r0r?r+r<rb1dtrFxinfos &&&&& rr=r=s: q ! a  QZ AJvv{aggajAGGAJ.?@@wwqzRXXa[ 4QWWIU288*ANOO ww!| q14813385 "''33R!3K j1' 2FEA@GA qy,dUG3PQRR Hr1cV'd \V4pM \V4pVP^8XdH\\P !^^.^^..VP R74P p\WR7#\RV34wpV!WVR7wrgV^8d\V R24hV^8d\RV R24hV#)a 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) See :ref:`tutorial_linalg_overwrite` for details. 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 -------- >>> import numpy as np >>> 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 r)r+ overwrite_abz'-th leading minor not positive definiterEz-th argument of internal pbtrf)pbtrf) rrrGcholesky_bandedr"arrayrrr rr%)abrOr+rrKrPr0rMs&&&& rrQrQ,s| r " R[ ww!| RXX1v1v&6bhhG H N N"'' j2% 0FEB,?GA axTF"IJKK ax,TF2PQRR Hr1c&VwrE\WAWRVR7#)a Solve the linear equations ``A x = b``, given the Cholesky factorization of the banded Hermitian ``A``. The documentation is written assuming array arguments are of specified "core" shapes. However, array argument(s) of this function may have additional "batch" dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see :ref:`linalg_batch` for details. Parameters ---------- cb_and_lower : tuple, (ndarray, bool) The first element of the tuple is the matrix containing the Cholesky factor, as provided by `cholesky_banded`. The second element is a boolean flag indicating whether the factor is in the lower triangle, as provided *to* `cholesky_banded`. b : array_like Right-hand side overwrite_b : bool, optional If True, the function will overwrite the values in `b`. See :ref:`tutorial_linalg_overwrite` for details. 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 -------- >>> import numpy as np >>> 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 r;)_cho_solve_banded) cb_and_lowerr?r<rcbr+s&&&& rcho_solve_bandedrX}s lKR RE*6 88r1crV'd\V4p\V4pM\V4p\V4pVPR,VP^,8wd \R4hVP^8Xdv\ \ P!^^.^^..VPR74p\VR3\ P!^VPR74Pp\WR7#\R W34wpV!WW#R7wrV ^8d\V R24hV ^8d\RV ) R24hV#) z&shapes of cb and b are not compatible.rTrDz&th leading minor not positive definiterEzth argument of internal pbtrsr)pbtrs)rrr$r%rGrQr"rRrrXrIrr r) rWr?r+r<rmrKr[rLrMs &&&&& rrUrUs r " a  R[ AJ xx|qwwqz!ABB vv{ BHHq!fq!f%5RXXF G q$i!'')B C I I!&& j2' 2FEB@GA axTF"HIJJ ax,dUG3PQRR Hr1)r4r9r@rQrX)FFTT)FFT)FT)r0rB)r?z1|2)rSrB)rWrB)__doc__numpyr"rrrscipy._lib._utilrrr_miscrr lapackr r r r __all__rr)r4r9r@r=rQrXrUr1rres'88VU+  @@ F^B.Wb8\* + 89M M `88v9l+ , r1