Ei4SrSSKJrJrJrJr SSKJr SSKr SSKr SSK J r J r Sr SSKrSSKJrJr S S /rS rS rS rSrSrSSjrg!\a SSKrSr N3f=f)z1Basic linear factorizations needed by the solver.) block_array csc_array eye_arrayissparse)LinearOperatorN) cholesky_AAtCholmodTypeConversionWarningTF)warncatch_warnings orthogonality projectionscv[RRU5n[U5(a)[R RRUSS9nO[RRUSS9nUS:XdUS:Xag[RRUR U55nXCU-- nU$)a_Measure orthogonality between a vector and the null space of a matrix. Compute a measure of orthogonality between the null space of the (possibly sparse) matrix ``A`` and a given vector ``g``. The formula is a simplified (and cheaper) version of formula (3.13) from [1]_. ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. fro)ordr)nplinalgnormrscipysparsedot)Agnorm_gnorm_Anorm_A_gorths k/home/mande/repo/quber/.venv/lib/python3.13/site-packages/scipy/optimize/_trustregion_constr/projections.pyr r s$YY^^A F{{$$))!)7u-{fkyy~~aeeAh'H f} %D Kc^^^^ [S[S9 [T5m SSS5 UU UU4SjnUU 4SjnUU 4SjnXgU4$!,(df  N)=f)zLReturn linear operators for matrix A using ``NormalEquation`` approach. ignore)actioncategoryNc@>T"TRU55nUTRRU5- nSn[TU5T:aUUT:aU$T"TRU55nUTRRU5- nUS- n[TU5T:aMUU$NrrTr )xvzkrfactor max_refinorth_tols r null_space/normal_equation_projections..null_spaceDs 1558   N Aq!H,I~  quuQx AACCGGAJA FA Aq!H,rc2>T"TRU55$Nrr(rr,s r least_squares2normal_equation_projections..least_squaresVsaeeAhrcF>TRRT"U55$r2r'rr4s r row_space.normal_equation_projections..row_spaceZssswwvay!!r)r r r) rmnr.r-tolr/r5r9r,s ` `` @rnormal_equation_projectionsr>9sJ x2N Oa P$ " i //; P Os A Ac X^^^^^^ ^ [[T5TR/TS//SS9m [RR R T 5m UU UUUUU 4SjnUUU 4SjnUU 4S jnXgU4$![a+ [SSS9 [TR5TTTTU5s$f=f) z;Return linear operators for matrix A - ``AugmentedSystem``.Ncsc)formatzVSingular Jacobian matrix. Using dense SVD decomposition to perform the factorizations. stacklevelc(>[R"U[R"T 5/5nT "U5nUST nSn[TU5T :aEUT :aU$UTR U5- nT "U5nX&- nUST nUS- n[TU5T :aMEU$r$)rhstackzerosr r)r(r)lu_solr*r+new_v lu_updaterKr;r-r<r.solves rr/0augmented_system_projections..null_spacevs IIq"((1+& 'q 2AJ Aq!H,I~f %Ee I  Fr A FAAq!H, rc|>[R"U[R"T5/5nT"U5nUTTT-$r2rrFrG)r(r)rHr;r<rLs rr53augmented_system_projections..least_squaress; IIq"((1+& 'qa!}rcv>[R"[R"T5U/5nT"U5nUST$r2rO)r(r)rHr<rLs rr9/augmented_system_projections..row_spaces7 IIrxx{A& 'qbqzr) rrr'rrr factorized RuntimeErrorr svd_factorization_projectionstoarray) rr;r<r.r-r=r/r5r9rKrLs ````` @@raugmented_system_projectionsrW`s ilACC(1d)4UCA = ##..q1D i //M = + -QYY[-.8-6= = =s)A442B)(B)cR^^^^^ ^ ^ [RRTRSSS9um m m [RR T SSS24[R 5U:a[SSS9 [TTUTTU5$UU U U UUU4S jnU U U U4S jnU U U 4S jnXgU4$) zMReturn linear operators for matrix A using ``QRFactorization`` approach. Teconomic)pivotingmodeNzPSingular Jacobian matrix. Using SVD decomposition to perform the factorizations.rBrCc>TRRU5n[RR T USS9n[ R "T 5nX#T'UTRRU5- nSn[TU5T :a|UT :aU$TRRU5n[RR T USS9nX#T'UTRRU5- nUS- n[TU5T :aM|U$)NFlowerrr%)r'rrrsolve_triangularrrGr ) r(aux1aux2r)r*r+rPQRr;r-r.s rr/0qr_factorization_projections..null_spacessswwqz||,,QE,B HHQK!  N Aq!H,I~33771:D<<00D0FDaDACCGGAJA FAAq!H,rc>TRRU5n[RR TUSS9n[ R "T5nX#T'U$)NFr^)r'rrrr`rrG)r(rarbr*rcrdrer;s rr53qr_factorization_projections..least_squaressHsswwqz||,,QE,B HHQK!rct>UTn[RRTUSSS9nTRU5nU$)NFr')r_trans)rrr`r)r(rarbr*rcrdres rr9/qr_factorization_projections..row_spacesCt||,,Q3836-8 EE$Kr) rrqrr'rrinfr rU) rr;r<r.r-r=r/r5r9rcrdres `` `` @@@rqr_factorization_projectionsrnsllooaccDzoBGAq! yy~~aAh'#- + -Q1-5-6-02 2 2 i //rc^^^^ ^ ^ [RRTSS9um m m T SS2T U:4m T T U:SS24m T T U:m UU U UUU 4SjnU U U 4SjnU U U 4SjnXgU4$)zNReturn linear operators for matrix A using ``SVDFactorization`` approach. F) full_matricesNc>TRU5nST - U-nTRU5nUTRRU5- nSn[TU5T :ahUT :aU$TRU5nST - U-nTRU5nUTRRU5- nUS- n[TU5T :aMhU$)Nr%rr&) r(rarbr)r*r+rUVtr-r.ss rr/1svd_factorization_projections..null_spacesvvays4x EE$K  N Aq!H,I~66!9DQ3t8Dd AACCGGAJA FAAq!H,rc\>TRU5nST- U-nTRU5nU$Nr%r3r(rarbr*rrrsrts rr54svd_factorization_projections..least_squaress/vvays4x EE$Krc>TRRU5nST- U-nTRRU5nU$rwr8rxs rr90svd_factorization_projections..row_spaces7sswwqzs4x DDHHTNr)rrsvd) rr;r<r.r-r=r/r5r9rrrsrts ` `` @@@rrUrUsv||7HAq" !QW* A AGQJB !c' A0 i //rc8[R"U5upVXV-S:Xa [U5n[U5(aDUcSnUS;a [ S5eUS:Xa'[ (d[ R"S[SS9 SnOUcS nUS ;a [ S 5eUS:Xa[XXbX45upxn ODUS:Xa[XXbX45upxn O-US :Xa[XXbX45upxn OUS :Xa[XXbX45upxn [Xf4W5n [XV4W5n [Xe4W 5n XU 4$) 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. rAugmentedSystem)NormalEquationr~z$Method not allowed for sparse array.rzmOnly accepts 'NormalEquation' option when scikit-sparse is available. Using 'AugmentedSystem' option instead.rBrCQRFactorization)rSVDFactorizationz#Method not allowed for dense array.r)rshaperr ValueErrorsksparse_availablewarningsr ImportWarningr>rWrnrUr) rmethodr.r-r=r;r<r/r5r9ZLSYs 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=rBgV瞯<)__doc__ scipy.sparserrrrscipy.sparse.linalgr scipy.linalgrsksparse.cholmodrr r ImportErrorrnumpyrr r __all__r r>rWrnrUr rrrs|7DD.K)  F$0NP0f;0|30lt{s A AA