+ /i4Rt^RIHtHtHtHt^RIHt^RIt ^RIt ^RI H t H t Rt ^RIt^RIHtHtRR .tR tR tR tR tRtRRltR# ]d ^RItRt L5i;i)z1Basic linear factorizations needed by the solver.) block_array csc_array eye_arrayissparse)LinearOperatorN) cholesky_AAtCholmodTypeConversionWarningTF)warncatch_warnings orthogonality projectionsc\PPV4p\V4'd-\P PPVRR7pM!\PPVRR7pV^8XgV^8Xd^#\PPVP V44pWCV,, pV#)a_Measure orthogonality between a vector and the null space of a matrix. Compute a measure of orthogonality between the null space of the (possibly sparse) matrix ``A`` and a given vector ``g``. The formula is a simplified (and cheaper) version of formula (3.13) from [1]_. ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. fro)ord)nplinalgnormrscipysparsedot)Agnorm_gnorm_Anorm_A_gorths&& l/var/www/html/photoedit/myenv/lib/python3.14/site-packages/scipy/optimize/_trustregion_constr/projections.pyr r s$YY^^A F{{$$))!)7u-{fkyy~~aeeAh'H f} %D Kcaaaa \R\R7;_uu_4\S4o RRR4VV VV3RlpVV 3RlpVV 3RlpWgV3# +'giL+;i)zLReturn linear operators for matrix A using ``NormalEquation`` approach. ignore)actioncategoryNc<<S!SPV44pVSPPV4, p^p\SV4S8dOVS8dV#S!SPV44pVSPPV4, pV^, pK_V#)rTr )xvzkrfactor max_refinorth_tols& r null_space/normal_equation_projections..null_spaceDs 1558   N Aq!H,I~  quuQx AACCGGAJA FArc2<S!SPV44#Nrr&rr*s&r least_squares2normal_equation_projections..least_squaresVsaeeAhrcF<SPPS!V44#r0r%rr2s&r row_space.normal_equation_projections..row_spaceZssswwvay!!r)r rr) rmnr,r+tolr-r3r7r*s f&&ff& @rnormal_equation_projectionsr<9sM x2N O Oa P$ " i //; P Os A  A c faaaaaa a \\S4SP.SR..RR7o \PP P S 4o TT TTTTT 3RlpTTT 3RlpTT 3RlpYgT3# \d/\R^R7\SP4SSSST4u#i;i) z;Return linear operators for matrix A - ``AugmentedSystem``.Ncsc)formatzVSingular Jacobian matrix. Using dense SVD decomposition to perform the factorizations. stacklevelc$<\P!V\P!S 4.4pS !V4pVRS p^p\SV4S 8dCVS 8dV#VSP V4, pS !V4pW&, pVRS pV^, pKSV#r0)rhstackzerosr r)r&r'lu_solr(r)new_v lu_updaterKr9r+r:r,solves& rr-0augmented_system_projections..null_spacevs IIq"((1+& 'q 2AJ Aq!H,I~f %Ee I  Fr A FArc<\P!V\P!S4.4pS!V4pVSSS,#r0rrCrD)r&r'rEr9r:rIs& rr33augmented_system_projections..least_squaress; IIq"((1+& 'qa!}rcv<\P!\P!S4V.4pS!V4pVRS#r0rL)r&r'rEr:rIs& rr7/augmented_system_projections..row_spaces7 IIrxx{A& 'qbqzr) rrr%rrr factorized RuntimeErrorr svd_factorization_projectionstoarray) rr9r:r,r+r;r-r3r7rHrIs fffff& @@raugmented_system_projectionsrT`s ilACC(1d)4UCA = ##..q1D i //M = + -QYY[-.8-6= = =s)A776B0/B0c`aaaaa a a \PPSPRRR7wo o o \PP S R,\P 4V8d\R ^R7\SSVSSV4#VV V V VVV3RlpV V V V3RlpV V V 3RlpWgV3#) zMReturn linear operators for matrix A using ``QRFactorization`` approach. Teconomic)pivotingmoder@c<SPPV4p\PP S VRR7p\ P !S 4pW#S&VSPPV4, p^p\SV4S 8dyVS 8dV#SPPV4p\PP S VRR7pW#S&VSPPV4, pV^, pKV#F)lower)r%rrrsolve_triangularrrDr ) r&aux1aux2r'r(r)rPQRr9r+r,s & rr-0qr_factorization_projections..null_spacessswwqz||,,QE,B HHQK!  N Aq!H,I~33771:D<<00D0FDaDACCGGAJA FArc<SPPV4p\PP SVRR7p\ P !S4pW#S&V#rZ)r%rrrr\rrD)r&r]r^r(r_r`rar9s& rr33qr_factorization_projections..least_squaressHsswwqz||,,QE,B HHQK!rc<VS,p\PPSVRRR7pSPV4pV#)Fr%)r[trans)rrr\r)r&r]r^r(r_r`ras& rr7/qr_factorization_projections..row_spacesCt||,,Q3836-8 EE$Kr)NNNzPSingular Jacobian matrix. Using SVD decomposition to perform the factorizations.) rrqrr%rrinfr rR) rr9r:r,r+r;r-r3r7r_r`ras ff&ff& @@@rqr_factorization_projectionsrlsllooaccDzoBGAq! yy~~ah'#- + -Q1-5-6-02 2 2 i //rcaaaa a a \PPSRR7wo o o S RS V83,o S S V8R3,o S S V8,o VV V VVV 3RlpV V V 3RlpV V V 3RlpWgV3#)zNReturn linear operators for matrix A using ``SVDFactorization`` approach. F) full_matricesric<SPV4p^S , V,pSPV4pVSPPV4, p^p\SV4S 8djVS 8dV#SPV4p^S , V,pSPV4pVSPPV4, pV^, pKzV#r$) r&r]r^r'r(r)rUVtr+r,ss & rr-1svd_factorization_projections..null_spacesvvays4x EE$K  N Aq!H,I~66!9DQ3t8Dd AACCGGAJA FArcl<SPV4p^S, V,pSPV4pV#rpr1r&r]r^r(rrrsrts& rr34svd_factorization_projections..least_squaress/vvays4x EE$Krc<SPPV4p^S, V,pSPPV4pV#rpr6rws& rr70svd_factorization_projections..row_spaces7sswwqzs4x DDHHTNr)rrsvd) rr9r:r,r+r;r-r3r7rrrsrts f&&ff& @@@rrRrRsv||7HAq" !QW* A AGQJB !c' A0 i //rc^\P!V4wrVWV,^8Xd \V4p\V4'dLVfRpVR 9d \ R4hVR8Xd,\ 'g \ P!R\^R7RpMVfRpVR 9d \ R4hVR8Xd\WWbW44wrxp MGVR8Xd\WWbW44wrxp M/VR8Xd\WWbW44wrxp MVR8Xd\WWbW44wrxp \Wf3X4p \WV3X4p \We3X 4p WV 3#) a Return three linear operators related with a given matrix A. Parameters ---------- A : sparse array (or ndarray), shape (m, n) Matrix ``A`` used in the projection. method : string, optional Method used for compute the given linear operators. Should be one of: - 'NormalEquation': The operators will be computed using the so-called normal equation approach explained in [1]_. In order to do so the Cholesky factorization of ``(A A.T)`` is computed. Exclusive for sparse matrices. - 'AugmentedSystem': The operators will be computed using the so-called augmented system approach explained in [1]_. Exclusive for sparse matrices. - 'QRFactorization': Compute projections using QR factorization. Exclusive for dense matrices. - 'SVDFactorization': Compute projections using SVD factorization. Exclusive for dense matrices. orth_tol : float, optional Tolerance for iterative refinements. max_refin : int, optional Maximum number of iterative refinements. tol : float, optional Tolerance for singular values. Returns ------- Z : LinearOperator, shape (n, n) Null-space operator. For a given vector ``x``, the null space operator is equivalent to apply a projection matrix ``P = I - A.T inv(A A.T) A`` to the vector. It can be shown that this is equivalent to project ``x`` into the null space of A. LS : LinearOperator, shape (m, n) Least-squares operator. For a given vector ``x``, the least-squares operator is equivalent to apply a pseudoinverse matrix ``pinv(A.T) = inv(A A.T) A`` to the vector. It can be shown that this vector ``pinv(A.T) x`` is the least_square solution to ``A.T y = x``. Y : LinearOperator, shape (n, m) Row-space operator. For a given vector ``x``, the row-space operator is equivalent to apply a projection matrix ``Q = A.T inv(A A.T)`` to the vector. It can be shown that this vector ``y = Q x`` the minimum norm solution of ``A y = x``. Notes ----- Uses iterative refinements described in [1] during the computation of ``Z`` in order to cope with the possibility of large roundoff errors. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. AugmentedSystemNormalEquationz$Method not allowed for sparse array.zmOnly accepts 'NormalEquation' option when scikit-sparse is available. Using 'AugmentedSystem' option instead.r@QRFactorizationSVDFactorizationz#Method not allowed for dense array.)r~r})rr)rshaperr ValueErrorsksparse_availablewarningsr ImportWarningr<rTrlrRr) rmethodr,r+r;r9r:r-r3r7ZLSYs &&&&& rr r 'sMT 88A;DA sax aL{{ >&F > >CD D % %.@.@ MM>(A 7'F >&F @ @BC C !!)!YL - 9 $ $*1iM - 9 $ $*1iM - 9 % %+A!yN - 9 vz*A  .Bvy)A !8Or)Ng-q=gV瞯<)__doc__ scipy.sparserrrrscipy.sparse.linalgr scipy.linalgrsksparse.cholmodrrr ImportErrorrnumpyrr r __all__r r<rTrlrRr rrrs|7DD.K)  F$0NP0f;0|30lt{s A AA