tKgq3dZddlmZmZmZmZddlmZddlZ ddlZ ddl m Z dZ ddlZddlmZd d gZd Zd Zd ZdZdZddZy#e $r ddlZdZ Y/wxYw)z1Basic linear factorizations needed by the solver.)bmat csc_matrixeyeissparse)LinearOperatorN cholesky_AAtTF)warn orthogonality projectionscvtjj|}t|r,tj jj|d}n!tjj|d}|dk(s|dk(rytjj|j |}|||zz }|S)aMeasure orthogonality between a vector and the null space of a matrix. Compute a measure of orthogonality between the null space of the (possibly sparse) matrix ``A`` and a given vector ``g``. The formula is a simplified (and cheaper) version of formula (3.13) from [1]_. ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. fro)ordr)nplinalgnormrscipysparsedot)Agnorm_gnorm_Anorm_A_gorths r/home/alanp/www/video.onchill/myenv/lib/python3.12/site-packages/scipy/optimize/_trustregion_constr/projections.pyr r s$YY^^A F{$$))!)7u-{fkyy~~aeeAh'H vf} %D KcR t fd} fd} fd}|||fS)zLReturn linear operators for matrix A using ``NormalEquation`` approach. c8j|}|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_space@s 1558   N Aq!H,I~  quuQx AACCGGAJA FA Aq!H,rc2j|SNrr$rr(s r least_squaresz2normal_equation_projections..least_squaresRsaeeAhrcFjj|Sr-r#rr/s r row_spacez.normal_equation_projections..row_spaceVssswwvay!!rr) rmnr*r)tolr+r0r3r(s ` `` @rnormal_equation_projectionsr79s,!_F$ " }i //rc l tttjgdgg tj j j   fd} fd} fd}|||fS#t$r.tddtj|cYSwxYw)z;Return linear operators for matrix A - ``AugmentedSystem``.NzVSingular Jacobian matrix. Using dense SVD decomposition to perform the factorizations. stacklevelctj|tj g} |}|d }d}t| kDrC| k\r |S|j |z } |}||z }|d }|dz }t| kDrC|Sr )rhstackzerosr r)r$r%lu_solr&r'new_v lu_updaterKr4r)r5r*solves rr+z0augmented_system_projections..null_spacers IIq"((1+& 'q 2AJ Aq!H,I~f %Ee I i Fr A FAAq!H, rcxtj|tjg}|}|zSr-rr=r>)r$r%r?r4r5rCs rr0z3augmented_system_projections..least_squaress; IIq"((1+& 'qa!}rcrtjtj|g}|}|dSr-rE)r$r%r?r5rCs rr3z/augmented_system_projections..row_spaces7 IIrxx{A& 'qbqzr) rrrr#rrr factorized RuntimeErrorr svd_factorization_projectionstoarray) rr4r5r*r)r6r+r0r3rBrCs ````` @@raugmented_system_projectionsrK\s 4#a&!##D 234A = ##..q1D }i //M = + -QYY[-.8-6= = =s)A<<4B32B3cX 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.r9r:cjj|}tjj |d}t j }||<|jj|z }d}t| kDr}| k\r |Sjj|}tjj |d}||<|jj|z }|dz }t| kDr}|S)NFlowerrr!)r#rrrsolve_triangularrr>r ) r$aux1aux2r%r&r'rPQRr4r)r*s rr+z0qr_factorization_projections..null_spacessswwqz||,,QE,B HHQK!  N Aq!H,I~33771:D<<00D0FDAaDACCGGAJA FAAq!H,rcjj|}tjj |d}t j }||<|S)NFrR)r#rrrrTrr>)r$rUrVr&rWrXrYr4s rr0z3qr_factorization_projections..least_squaressJsswwqz||,,QE,B HHQK!rcz|}tjj|dd}j|}|S)NFr#)rStrans)rrrTr)r$rUrVr&rWrXrYs rr3z/qr_factorization_projections..row_spacesCt||,,Q3836-8 EE$Kr) rrqrr#rrinfr rI) rr4r5r*r)r6r+r0r3rWrXrYs `` `` @@@rqr_factorization_projectionsr_sllooaccDzoBGAq! 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$rUrVr%r&r'rUVtr)r*ss rr+z1svd_factorization_projections..null_spacesvvays4x EE$K  N Aq!H,I~66!9DQ3t8Dd AACCGGAJA FAAq!H,rc\j|}dz |z}j|}|SNr!r.r$rUrVr&rcrdres rr0z4svd_factorization_projections..least_squaress/vvays4x EE$Krcjj|}dz |z}jj|}|Srgr2rhs rr3z0svd_factorization_projections..row_spaces7sswwqzs4x DDHHTNr)rrsvd) rr4r5r*r)r6r+r0r3rcrdres ` `` @@@rrIrIsx||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 matrix (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)NormalEquationrlz%Method not allowed for sparse matrix.rmzmOnly accepts 'NormalEquation' option when scikit-sparse is available. Using 'AugmentedSystem' option instead.r9r:QRFactorization)rnSVDFactorizationz#Method not allowed for dense array.ro)rshaperr ValueErrorsksparse_availablewarningsr ImportWarningr7rKr_rIr) rmethodr*r)r6r4r5r+r0r3ZLSYs rr r #ssT 88A;DAq sax qM{ >&F > >DE E % %.@ 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=r9gV瞯<)__doc__ scipy.sparserrrrscipy.sparse.linalgr scipy.linalgrsksparse.cholmodr rr ImportErrorrsnumpyrr __all__r r7rKr_rIr rrrs|7::.-  F 0FP0f;0|30ltssA AA