o  hq3@sdZddlmZmZmZmZddlmZddlZ ddlZ z ddl m Z dZ Wne y5ddlZdZ YnwddlZddlmZd d gZd d Zd d ZddZddZddZddd ZdS)z1Basic linear factorizations needed by the solver.)bmat csc_matrixeyeissparse)LinearOperatorN cholesky_AAtTF)warn orthogonality projectionscCsntj|}t|rtjjj|dd}ntjj|dd}|dks$|dkr&dStj||}|||}|S)aMeasure orthogonality between a vector and the null space of a matrix. Compute a measure of orthogonality between the null space of the (possibly sparse) matrix ``A`` and a given vector ``g``. The formula is a simplified (and cheaper) version of formula (3.13) from [1]_. ``orth = norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``. References ---------- .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal. "On the solution of equality constrained quadratic programming problems arising in optimization." SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395. fro)ordr)nplinalgnormrscipysparsedot)Agnorm_gnorm_Anorm_A_gorthrb/var/www/vscode/kcb/lib/python3.10/site-packages/scipy/optimize/_trustregion_constr/projections.pyr s  c s@tfdd}fdd}fdd}|||fS)zLReturn linear operators for matrix A using ``NormalEquation`` approach. csv|}|j|}d}t|kr9|kr |S|}|j|}|d7}t|ks|SNrrTr )xvzkrfactor max_refinorth_tolrr null_space@sz/normal_equation_projections..null_spacecs|SNrr rr%rr least_squaresRsz2normal_equation_projections..least_squarescsj|Sr)rrr+r,rr row_spaceVsz.normal_equation_projections..row_spacer rmnr'r&tolr(r-r/rr$rnormal_equation_projections9s  r4c stttjgdggz tjjWnty2t dddt |YSwfdd}fdd}fd d }|||fS) z;Return linear operators for matrix A - ``AugmentedSystem``.NzVSingular Jacobian matrix. Using dense SVD decomposition to perform the factorizations. stacklevelcst|tg}|}|d}d}t|krD|kr$ |S||}|}||7}|d}|d7}t|ks|Sr)rhstackzerosr r)r r!lu_solr"r#new_v lu_updaterKr1r&r2r'solverrr(rs  z0augmented_system_projections..null_spacecs,t|tg}|}|Sr)rr8r9r r!r:)r1r2r?rrr-sz3augmented_system_projections..least_squarescs(tt|g}|}|dSr)r@rA)r2r?rrr/s z/augmented_system_projections..row_space) rrrrrrr factorized RuntimeErrorr svd_factorization_projectionstoarrayr0rr=raugmented_system_projections\s   " rFc stjjjddd\tjdddftj|kr,tdddt||Sfd d }fd d }fd d}|||fS)zMReturn linear operators for matrix A using ``QRFactorization`` approach. Teconomic)pivotingmodeNzPSingular Jacobian matrix. Using SVD decomposition to perform the factorizations.r5r6csj|}tjj|dd}t}||<|j|}d}t|krV|kr0 |Sj|}tjj|dd}||<|j|}|d7}t|ks)|S)NFlowerrr)rrrrsolve_triangularrr9r r aux1aux2r!r"r#rPQRr1r&r'rrr(s"    z0qr_factorization_projections..null_spacecs4j|}tjj|dd}t}||<|S)NFrK)rrrrrMrr9r rOrPr")rRrSrTr1rrr-s  z3qr_factorization_projections..least_squarescs*|}tjj|ddd}|}|S)NFr)rLtrans)rrrMrrU)rRrSrTrrr/s  z/qr_factorization_projections..row_space) rrqrrrrinfr rDr0rrQrqr_factorization_projectionss  rYc stjjdd\dd|kf|kddf|kfdd}fdd}fdd }|||fS) zNReturn linear operators for matrix A using ``SVDFactorization`` approach. F) full_matricesNcs|}d|}|}|j|}d}t|krK|kr( |S|}d|}|}|j|}|d7}t|ks!|S)NrrrrNrUVtr&r'srrr(s      z1svd_factorization_projections..null_spacecs$|}d|}|}|SNrr*rUr\r]r^rrr-s   z4svd_factorization_projections..least_squarescs(j|}d|}j|}|Sr_r.rUr`rrr/s   z0svd_factorization_projections..row_space)rrsvdr0rr[rrDs  rD-q=r5V瞯t|\}}||dkrt|}t|r4|durd}|dvr#td|dkr3ts3tjdtdd d}n|dur:d }|d vrBtd |dkrSt ||||||\}}} n2|dkrdt ||||||\}}} n!|d krut ||||||\}}} n|d krt ||||||\}}} 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. rNAugmentedSystem)NormalEquationrdz%Method not allowed for sparse matrix.rezmOnly accepts 'NormalEquation' option when scikit-sparse is available. Using 'AugmentedSystem' option instead.r5r6QRFactorization)rfSVDFactorizationz#Method not allowed for dense array.rg)rshaperr ValueErrorsksparse_availablewarningsr ImportWarningr4rFrYrDr) rmethodr'r&r3r1r2r(r-r/ZLSYrrrr #sDJ      )Nrbr5rc)__doc__ scipy.sparserrrrscipy.sparse.linalgr scipy.linalgrsksparse.cholmodrrj ImportErrorrknumpyrr __all__r r4rFrYrDr rrrrs.    ##S>6