bi$ddlZddlmZddlmZddlmcmZ ddlm Z ddl m Z d dZ dZdZdZdZGdd Ze j&d fd Zy) N) csc_arrayctj|dd\}}}tjtjtj ||kDd}|ddd|fj |jd|f}|S)zCReturn a matrix whose columns are an orthonormal basis for range(V)economicT)modepivotingaxisNr)laqrnp count_nonzerosumabsreshapeshape)VtolQRpranks Q/home/cdr/jupyterlab/.venv/lib/python3.12/site-packages/cvxpy/utilities/linalg.pyorthr soeeAJ6GAq!  BFF266!9s?; 0. rr)rrr iscomplexobjeyeconjT)rnQ1rPQ2s ronb_for_orthogonal_complementr%s  A aB 88A;D t8O8 q FF1IRWWY[[( ( FF1IRTT ! aB Irchtj|r7|tj|jz }|j }nQt |t jr,|t jt j|z }n tdt j|dS)NzUnsupported matrix type.r) sparissparse diags_arraydiagonaltoarray isinstancer ndarraydiag ValueErrorallclose)Aoff_diagonal_elementss r is_diagonalr3&s }}Q !D$4$4QZZ\$B B 5 = = ? Arzz " !BGGBGGAJ$7 7344 ;;,a 00rct|rytrattr#t j j | k\St jt j}|| k\Sfd} || }t j|jr5|t j j"j$z }|| }t j || k\S#tj$rB}d}t|d|}tj||j|jd}~wwxYw)a Return True if we can certify that A is PSD (up to tolerance "tol"). First we check if A is PSD according to the Gershgorin Circle Theorem. If Gershgorin is inconclusive, then we use an iterative method (from ARPACK, as called through SciPy) to estimate extremal eigenvalues of certain shifted versions of A. The shifts are chosen so that the signs of those eigenvalues tell us the signs of the eigenvalues of A. If there are numerical issues then it's possible that this function returns False even when A is PSD. If you know that you're in that situation, then you should replace A by A = cvxpy.atoms.affine.wraps.psd_wrap(A). Parameters ---------- A : Union[np.ndarray, spar.sparray] Symmetric (or Hermitian) NumPy ndarray or SciPy sparse array. tol : float Nonnegative. Something very small, like 1e-10. Tc2ttjdr tjjd}ntjj d}j d}|j dd|}tjd|d|d S) N default_rng{rg?)locscalesizerSAF)ksigmawhichv0return_eigenvectors) hasattrr randomr6 RandomStaternormalsparlaeigsh)r>gr!r@r1s rSA_eigshz#is_psd_within_tol..SA_eigshVsz 299m , %%c*A %%c*A GGAJ XX#SqX 1||A%t057 7ra CVXPY note: This failure was encountered while trying to certify that a matrix is positive semi-definite (see [1] for a definition). In rare cases, this method fails for numerical reasons even when the matrix is positive semi-definite. If you know that you're in that situation, you can replace the matrix A by cvxpy.psd_wrap(A). [1] https://en.wikipedia.org/wiki/Definite_matrix z N)gershgorin_psd_checkr3r,rr alldataminr.rFArpackNoConvergencestr eigenvalues eigenvectorsisnananyfinfodtypeeps) r1rmin_diag_entryrIevemessageerror_with_notetemps ` ris_psd_within_tolr]2s4As#1~ a #66!&&SD.) )VVBGGAJ/N!cT) ) 74Y sd^$ xx|RXXagg&*** te_ 66"* /  % %Y !VHD 2((!--XXYs C;;E=E  Ectj|r|j}tj|| kry|tj |z }tj |}tj|jdj}tj||z | k\St|tjrtj|}tj|| kry|tj|z }tj |}|jd}tj||z | k\St)a> Use the Gershgorin Circle Theorem https://en.wikipedia.org/wiki/Gershgorin_circle_theorem As a sufficient condition for A being PSD with tolerance "tol". The computational complexity of this function is O(nnz(A)). Parameters ---------- A : Union[np.ndarray, spar.sparray] Symmetric (or Hermitian) NumPy ndarray or SciPy sparse array. tol : float Nonnegative. Something very small, like 1e-10. Returns ------- True if A is PSD according to the Gershgorin Circle Theorem. Otherwise, return False. Frr )r'r(r*r rSr)rarrayrravelrKr,r-r.r/)r1rr.A_shiftradiis rrJrJs. }}Qzz| 66$#+ d&&t,,&&/!,-335vvdUlsd*++ Arzz "wwqz 66$#+ bggdm#&&/  #vvdUlsd*++lrc$eZdZdZdZdZdZdZdZy)SparseCholeskyMessagesz;Input matrix is not symmetric to within provided tolerance.z7Input matrix is neither positive nor negative definite.zCholesky decomposition failed.z8The only allowed Expression inputs are Constant objects.z4Non-Expression inputs must be SciPy sparse matrices.zInput matrix must be real.N) __name__ __module__ __qualname__ ASYMMETRIC INDEFINITE EIGEN_FAIL NOT_CONST NOT_SPARSENOT_REALrrrdrds NJJJ1JJIGJ+HrrdFc ddlmcmcm}ddlmcm}t||jr?t||jsttj|j}tj|sttj t#j$|rttj&|s||j(z }|j*j,}|dkDrAt/j0|j*||dzzkDrttj2|j5}t#j6|dkD}t#j6|dk} |s| sttj8| rt;| |d\} } } d| | fStj<|} |j>d}|jA| jB}|jA| jD}|jG| j*}|jA}|jA}|jA}|jG} |jI||||||||t#jP|}t#jP|}t#jP|}t#jP|}tjR|||ff||f} d| |fS#tJ$rO}|jLdtjNk(rt|jLtK|jLd}~wwxYw) a The input matrix A must be real and symmetric. If A is positive definite then Eigen will be used to compute its sparse Cholesky decomposition with AMD-ordering. If A is negative definite, then the analogous operation will be applied to -A. If Cholesky succeeds, then we return a lower-triangular matrix L in CSR-format and a permutation vector p so (L[p, :]) @ (L[p, :]).T == A within numerical precision. We raise a ValueError if Eigen's Cholesky fails or if we certify indefiniteness before calling Eigen. While checking for indefiniteness, we also check that ||A - A'||_Fro / sqrt(n) <= sym_tol, where n is the order of the matrix. rNg?T) assume_posdefg)rr8)*"cvxpy.utilities.cpp.sparsecholesky utilitiescppsparsecholeskycvxpy.expressions.cvxtypes expressionscvxtypesr, expressionconstantr/rdrkvaluer'r(rlr rrmr rLr;r normrhr*rKrisparse_cholesky coo_matrixr IntVectorrowcol DoubleVectorsparse_chol_from_vecs RuntimeErrorargsrjr_ csr_array)r1sym_tolrpspcholrwsymdiffszd maybe_posdef maybe_negdef_LrA_coor!inrowsincolsinvalsoutpivsoutrowsoutcolsoutvalsrYs rr|r|s8711!X((*+!X..013==> > GG == /::;; q/8899 acc' \\   6bgggll+gS.AA3>>? ? JJLvva!e} vva!e}  3>>? ? %qb'FGAq!A:  OOA E  A  eii (F   eii (F   ,F G G G!!#G '$$ vvv Wgw hhwGhhwGhhwGhhwG '7!34QFCA 7? ' 66!9.99 9QVV$ $qvv& & 's%K33 M rsS$$"! & 1Wt(V,, (44EGr