Џkh4dZddlmZmZmZmZddlmZddlZ ddlZ ddl m Z m Z dZ ddlZddlmZmZd d gZd Zd Zd ZdZdZddZy#e$r ddlZdZ Y1wxYw)z1Basic linear factorizations needed by the solver.) block_array csc_array eye_arrayissparse)LinearOperatorN) cholesky_AAtCholmodTypeConversionWarningTF)warncatch_warnings orthogonality projectionscvtjj|}t|r,tj jj|d}n!tjj|d}|dk(s|dk(rytjj|j |}|||zz }|S)aMeasure orthogonality between a vector and the null space of a matrix. Compute a measure of orthogonality between the null space of the (possibly sparse) matrix ``A`` and a given vector ``g``. The formula is a simplified (and cheaper) version of formula (3.13) from [1]_. ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. fro)ordr)nplinalgnormrscipysparsedot)Agnorm_gnorm_Anorm_A_gorths j/var/www/teggl/fontify/venv/lib/python3.12/site-packages/scipy/optimize/_trustregion_constr/projections.pyr r s$YY^^A F{$$))!)7u-{fkyy~~aeeAh'H vf} %D Kc tdt5t ddd fd} fd} fd}|||fS#1swY"xYw)zLReturn linear operators for matrix A using ``NormalEquation`` approach. ignore)actioncategoryNc8j|}|jj|z }d}t|kDrR|k\r |Sj|}|jj|z }|dz }t|kDrR|SNrrTr )xvzkrfactor max_refinorth_tols r null_spacez/normal_equation_projections..null_spaceDs 1558   N Aq!H,I~  quuQx AACCGGAJA FA Aq!H,rc2j|SNrr(rr,s r least_squaresz2normal_equation_projections..least_squaresVsaeeAhrcFjj|Sr1r'rr3s r row_spacez.normal_equation_projections..row_spaceZssswwvay!!r)r r r) rmnr.r-tolr/r4r7r,s ` `` @rnormal_equation_projectionsr;9sL x2N O!a!$ " }i //;!!s AA c ^ ttjgdggd tjj j   fd} fd} fd }|||fS#t$r.tddtj|cYSwxYw) z;Return linear operators for matrix A - ``AugmentedSystem``.Ncsc)formatzVSingular Jacobian matrix. Using dense SVD decomposition to perform the factorizations. stacklevelctj|tj g} |}|d }d}t| kDrC| k\r |S|j |z } |}||z }|d }|dz }t| kDrC|Sr$)rhstackzerosr r)r(r)lu_solr*r+new_v lu_updaterKr8r-r9r.solves rr/z0augmented_system_projections..null_spacevs IIq"((1+& 'q 2AJ Aq!H,I~f %Ee I i Fr A FAAq!H, rcxtj|tjg}|}|zSr1rrCrD)r(r)rEr8r9rIs rr4z3augmented_system_projections..least_squaress; IIq"((1+& 'qa!}rcrtjtj|g}|}|dSr1rK)r(r)rEr9rIs rr7z/augmented_system_projections..row_spaces7 IIrxx{A& 'qbqzr) rrr'rrr factorized RuntimeErrorr svd_factorization_projectionstoarray) rr8r9r.r-r:r/r4r7rHrIs ````` @@raugmented_system_projectionsrQ`s ilACC(1d)4UCA = ##..q1D }i //M = + -QYY[-.8-6= = =s)A554B,+B,cX tjjjdd\ tjj dddftj |krtddt||S fd } fd } fd }|||fS) zMReturn linear operators for matrix A using ``QRFactorization`` approach. Teconomic)pivotingmodeNzPSingular Jacobian matrix. Using SVD decomposition to perform the factorizations.r?r@cjj|}tjj |d}t j }||<|jj|z }d}t| kDr}| k\r |Sjj|}tjj |d}||<|jj|z }|dz }t| kDr}|S)NFlowerrr%)r'rrrsolve_triangularrrDr ) r(aux1aux2r)r*r+rPQRr8r-r.s rr/z0qr_factorization_projections..null_spacessswwqz||,,QE,B HHQK!  N Aq!H,I~33771:D<<00D0FDAaDACCGGAJA FAAq!H,rcjj|}tjj |d}t j }||<|S)NFrX)r'rrrrZrrD)r(r[r\r*r]r^r_r8s rr4z3qr_factorization_projections..least_squaressJsswwqz||,,QE,B HHQK!rcz|}tjj|dd}j|}|S)NFr')rYtrans)rrrZr)r(r[r\r*r]r^r_s rr7z/qr_factorization_projections..row_spacesCt||,,Q3836-8 EE$Kr) rrqrr'rrinfr rO) rr8r9r.r-r:r/r4r7r]r^r_s `` `` @@@rqr_factorization_projectionsresllooaccDzoBGAq! yy~~aAh'#- + -Q1-5-6-02 2 2 }i //rc tjjd\ dd |kDf |kDddf |kD  fd} fd} fd}|||fS)zNReturn linear operators for matrix A using ``SVDFactorization`` approach. F) full_matricesNcj|}d z |z}j|}|jj|z }d}t| kDre| k\r |Sj|}d z |z}j|}|jj|z }|dz }t| kDre|S)Nr%rr&) r(r[r\r)r*r+rUVtr-r.ss rr/z1svd_factorization_projections..null_spacesvvays4x EE$K  N Aq!H,I~66!9DQ3t8Dd AACCGGAJA FAAq!H,rc\j|}dz |z}j|}|SNr%r2r(r[r\r*rirjrks rr4z4svd_factorization_projections..least_squaress/vvays4x EE$Krcjj|}dz |z}jj|}|Srmr6rns rr7z0svd_factorization_projections..row_spaces7sswwqzs4x DDHHTNr)rrsvd) rr8r9r.r-r:r/r4r7rirjrks ` `` @@@rrOrOsx||7HAq" !QW* A AGQJB !c' A0 }i //rc:tj|\}}||zdk(r t|}t|r=|d}|dvr t d|dk(r8t s2t jdtdd}n|d }|d vr t d |dk(rt||||||\}}} nM|dk(rt||||||\}}} n3|d k(rt||||||\}}} n|d k(rt||||||\}}} t||f} t||f} t||f } | | | fS) 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)NormalEquationrrz$Method not allowed for sparse array.rszmOnly accepts 'NormalEquation' option when scikit-sparse is available. Using 'AugmentedSystem' option instead.r?r@QRFactorization)rtSVDFactorizationz#Method not allowed for dense array.ru)rshaperr ValueErrorsksparse_availablewarningsr ImportWarningr;rQrerOr) rmethodr.r-r:r8r9r/r4r7ZLSYs rr r 'ssT 88A;DAq sax aL{ >&F > >CD D % %.@ MM>(A 7'F >&F @ @BC C !!)!Q8YL - M9 $ $*1aHiM - M9 $ $*1aHiM - M9 % %+Aq!Xy#N - M9 1vz*A A .B1vy)A b!8Or)Ng-q=r?gV瞯<)__doc__ scipy.sparserrrrscipy.sparse.linalgr scipy.linalgrsksparse.cholmodrr rx ImportErrorrynumpyrr r __all__r r;rQrerOr rrrs|7DD.K)  F$0NP0f;0|30lt{s A AA