(ph5SrSSKrSSKJrJrJrJr SSKJrJ r SSK J r /SQr SSjr SS jrSS jrSS jrSS jrSS jrg)z!Cholesky decomposition functions.N)asarray_chkfiniteasarray atleast_2d empty_like) LinAlgError _datacopied)get_lapack_funcs)cholesky cho_factor cho_solvecholesky_bandedcho_solve_bandedcbU(a [U5O [U5n[U5nURS:wa[ SURS35eUR SUR S:wa[ SUR S35eUR S:Xa=[[R"SURS95Rn[XVS9U4$U=(d [XP5n[S U45unU"XQX#S 9upU S:a[S U -5eU S:a[ S U *S 35eX4$)z,Common code for cholesky() and cho_factor().z*Input array needs to be 2D but received a zd-array.rrz8Input array is expected to be square but has the shape: .dtype)potrf)lower overwrite_acleanz9%d-th leading minor of the array is not positive definitez$LAPACK reported an illegal value in z -th argumenton entry to "POTRF".)rrrndim ValueErrorshapesizer npeyerrr r r) arrr check_finitea1dtrcinfos P/var/www/html/venv/lib/python3.13/site-packages/scipy/linalg/_decomp_cholesky.py _choleskyr&s9". 1 71:B BB ww!|EbggYhWXX xx{bhhqk!'')xxj34 4 ww!| bffQbhh/ 0 6 6"'..3R!3K j2% 0FEBJGA ax%'+,- - ax?wG001 1 8Oc"[XUSUS9upAU$)a 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). 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. 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]]) Trrrr r&rrrr r#s r%r r .sn[&24HA Hr'c$[XUSUS9upAXA4$)a 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. 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) 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 : 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. 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]]) >>> 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 Fr)r*r+s r%r r js!N[&24HA 8Or'cUupEU(a[U5n[U5nO[U5n[U5nURS:wd URSURS:wa [ S5eURSURS:wa&[ SURSURS35eUR S:Xa[[ [R"SURS9S 4[R"SURS95Rn[XgS9$U=(d [Xa5n[S XF45unU"XFXRS 9upU S:wa[ S U *-5eU $) aSolve the linear equations A x = b, given the Cholesky factorization of A. Parameters ---------- (c, lower) : tuple, (array, bool) Cholesky factorization of a, as given by cho_factor b : array Right-hand side overwrite_b : bool, optional Whether to overwrite data in b (may improve performance) check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- x : array The solution to the system A x = b See Also -------- cho_factor : Cholesky factorization of a matrix Examples -------- >>> import numpy as np >>> from scipy.linalg import cho_factor, cho_solve >>> A = np.array([[9, 3, 1, 5], [3, 7, 5, 1], [1, 5, 9, 2], [5, 1, 2, 6]]) >>> c, low = cho_factor(A) >>> x = cho_solve((c, low), [1, 1, 1, 1]) >>> np.allclose(A @ x - [1, 1, 1, 1], np.zeros(4)) True rrrz$The factored matrix c is not square.zincompatible dimensions (z and )rT)potrsr overwrite_bz0illegal value in %dth argument of internal potrs)rrrrrrr rrronesrr r ) c_and_lowerbr1r r#rb1r"r/xr$s r%r r sCHJQ q ! a  QZ AJvv{aggajAGGAJ.?@@wwqzRXXa[ 4QWWIU288*ANOO ww!| q14813385 "''3R!3K j1' 2FEA@GA qyK 5!" " Hr'cfU(a [U5nO [U5nURS:XaA[[R "SS/SS//UR S95R n[XS9$[SU45unU"XUS9upgUS:a[SU-5eUS:a[SU*-5eU$)a Cholesky decompose a banded Hermitian positive-definite matrix The matrix a is stored in ab either in lower-diagonal or upper- diagonal ordered form:: ab[u + i - j, j] == a[i,j] (if upper form; i <= j) ab[ i - j, j] == a[i,j] (if lower form; i >= j) Example of ab (shape of a is (6,6), u=2):: upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * Parameters ---------- ab : (u + 1, M) array_like Banded matrix overwrite_ab : bool, optional Discard data in ab (may enhance performance) lower : bool, optional Is the matrix in the lower form. (Default is upper form) check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- c : (u + 1, M) ndarray Cholesky factorization of a, in the same banded format as ab See Also -------- cho_solve_banded : Solve a linear set equations, given the Cholesky factorization of a banded Hermitian. Examples -------- >>> import numpy as np >>> from scipy.linalg import cholesky_banded >>> from numpy import allclose, zeros, diag >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) >>> A = A + A.conj().T + np.diag(Ab[2, :]) >>> c = cholesky_banded(Ab) >>> C = np.diag(c[0, 2:], k=2) + np.diag(c[1, 1:], k=1) + np.diag(c[2, :]) >>> np.allclose(C.conj().T @ C - A, np.zeros((5, 5))) True rrr)pbtrf)r overwrite_abz)%d-th leading minor not positive definitez1illegal value in %d-th argument of internal pbtrf) rrrrrarrayrrr rr)abr9rr r"r8r#r$s r%rrsx r " R[ ww!| RXX1v1v&6bhhG H N N"'' j2% 0FEB,?GA axELMM axL 5!" " Hr'cFUupEU(a[U5n[U5nO[U5n[U5nURSURS:wa [S5eURS:Xal[ [ R"SS/SS//URS95n[US4[ R"SURS95Rn[XS9$[SXA45unU"XAXRS 9upU S:a[S U -5eU S:a[S U *-5eU $) a Solve the linear equations ``A x = b``, given the Cholesky factorization of the banded Hermitian ``A``. Parameters ---------- (cb, lower) : tuple, (ndarray, bool) `cb` is the Cholesky factorization of A, as given by cholesky_banded. `lower` must be the same value that was given to cholesky_banded. b : array_like Right-hand side overwrite_b : bool, optional If True, the function will overwrite the values in `b`. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- x : array The solution to the system A x = b See Also -------- cholesky_banded : Cholesky factorization of a banded matrix Notes ----- .. versionadded:: 0.8.0 Examples -------- >>> import numpy as np >>> from scipy.linalg import cholesky_banded, cho_solve_banded >>> Ab = np.array([[0, 0, 1j, 2, 3j], [0, -1, -2, 3, 4], [9, 8, 7, 6, 9]]) >>> A = np.diag(Ab[0,2:], k=2) + np.diag(Ab[1,1:], k=1) >>> A = A + A.conj().T + np.diag(Ab[2, :]) >>> c = cholesky_banded(Ab) >>> x = cho_solve_banded((c, False), np.ones(5)) >>> np.allclose(A @ x - np.ones(5), np.zeros(5)) True rz&shapes of cb and b are not compatible.rrTr)pbtrsr0z(%dth leading minor not positive definitez0illegal value in %dth argument of internal pbtrs)rrrrrrrr:rrr2rr r) cb_and_lowerr4r1r cbrmr"r>r6r$s r%rrGs\KR r " a  R[ AJ xx|qwwqz!ABB vv{ BHHq!fq!f%5RXXF G q$i!'')B C I I!&& j2' 2FEB@GA axDtKLL axK 5!" " Hr')FFTT)FFT)FT)__doc__numpyrrrrr_miscrr lapackr __all__r&r r r rrr'r%rHsN'DD,$ 8<@9 xIX> BM `G r'