ftj AxdZddlZddlmZmZmZmZddlm Z m Z gdZ ddZ d Z d Zd ZGd d e ZdS)z2Nearly exact trust-region optimization subproblem.N)normget_lapack_funcssolve_triangular cho_solve)_minimize_trust_regionBaseQuadraticSubproblem)_minimize_trustregion_exact estimate_smallest_singular_valuesingular_leading_submatrixIterativeSubproblemc |tdt|stdt||f|||td|S)a Minimization of scalar function of one or more variables using a nearly exact trust-region algorithm. Options ------- initial_trust_radius : float Initial trust-region radius. max_trust_radius : float Maximum value of the trust-region radius. No steps that are longer than this value will be proposed. eta : float Trust region related acceptance stringency for proposed steps. gtol : float Gradient norm must be less than ``gtol`` before successful termination. subproblem_maxiter : int, optional Maximum number of iterations to perform per subproblem. Only affects trust-exact. Default is 25. .. versionadded:: 1.17.0 Nz9Jacobian is required for trust region exact minimization.z?Hessian matrix is required for trust region exact minimization.)argsjachess subproblem) ValueErrorcallablerr )funx0rrrtrust_region_optionss _/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/optimize/_trustregion_exact.pyr r sv0 {/00 0 D>>0/00 0 !#r :#D-@ : :$8 : ::ctj|}|j\}}||krtdtj|}tj|}t |D]}d||z |j||fz }d||z |j||fz }||dzd|j|dzd|f|zz}||dzd|j|dzd|f|zz} t|t|dzt|t| dzkr|||<|||dzd<|||<| ||dzd<t||} t| } t|} | | z } | | z }| |fS)aYGiven upper triangular matrix ``U`` estimate the smallest singular value and the correspondent right singular vector in O(n**2) operations. Parameters ---------- U : ndarray Square upper triangular matrix. Returns ------- s_min : float Estimated smallest singular value of the provided matrix. z_min : ndarray Estimated right singular vector. Notes ----- The procedure is based on [1]_ and is done in two steps. First, it finds a vector ``e`` with components selected from {+1, -1} such that the solution ``w`` from the system ``U.T w = e`` is as large as possible. Next it estimate ``U v = w``. The smallest singular value is close to ``norm(w)/norm(v)`` and the right singular vector is close to ``v/norm(v)``. The estimation will be better more ill-conditioned is the matrix. References ---------- .. [1] Cline, A. K., Moler, C. B., Stewart, G. W., Wilkinson, J. H. An estimate for the condition number of a matrix. 1979. SIAM Journal on Numerical Analysis, 16(2), 368-375. z.A square triangular matrix should be provided.rN) np atleast_2dshaperzerosemptyrangeTabsrr)Umnpwkwpwmpppmvv_normw_norms_minz_mins rr r 0sD aA 7DAqAvvIJJJ  A  A1XX  !fAqD !1gQT " qsttWqs1Q3447|B & qsttWqs1Q3447|B & r77T"a[[ CGGd2qkk$9 9 9AaDAacddGGAaDAacddGG AA !WWF !WWF VOE JE %<rctj|}tj|}tjtj|d}tj||z|z }tj||z |z}||fS)a Given a square matrix ``H`` compute upper and lower bounds for its eigenvalues (Gregoshgorin Bounds). Defined ref. [1]. References ---------- .. [1] Conn, A. R., Gould, N. I., & Toint, P. L. Trust region methods. 2000. Siam. pp. 19. r)axis)rdiagr$summinmax)HH_diag H_diag_abs H_row_sumslbubs rgershgorin_boundsr@suWQZZFJq ***J #j0 1 1B #j0 1 1B r6MrcRtj|d|dz |dz fdz||dz |dz fz }t|}tj|}d||dz <|dkr;t |d|dz d|dz f|d|dz |dz f |d|dz <||fS)a  Compute term that makes the leading ``k`` by ``k`` submatrix from ``A`` singular. Parameters ---------- A : ndarray Symmetric matrix that is not positive definite. U : ndarray Upper triangular matrix resulting of an incomplete Cholesky decomposition of matrix ``A``. k : int Positive integer such that the leading k by k submatrix from `A` is the first non-positive definite leading submatrix. Returns ------- delta : float Amount that should be added to the element (k, k) of the leading k by k submatrix of ``A`` to make it singular. v : ndarray A vector such that ``v.T B v = 0``. Where B is the matrix A after ``delta`` is added to its element (k, k). Nr)rr7lenr r)Ar%r*deltar'r/s rr r s6 F1TacT1Q3Y<? # #a!QqSk 1E AA  A AacF Avv"1TacT4AaC4Z=1TacT1Q3Y<-@@$1Q3$ !8OrcfeZdZdZdZdZejej Z d fd Z dZ d Z xZS) r aQuadratic subproblem solved by nearly exact iterative method. Notes ----- This subproblem solver was based on [1]_, [2]_ and [3]_, which implement similar algorithms. The algorithm is basically that of [1]_ but ideas from [2]_ and [3]_ were also used. References ---------- .. [1] A.R. Conn, N.I. Gould, and P.L. Toint, "Trust region methods", Siam, pp. 169-200, 2000. .. [2] J. Nocedal and S. Wright, "Numerical optimization", Springer Science & Business Media. pp. 83-91, 2006. .. [3] J.J. More and D.C. Sorensen, "Computing a trust region step", SIAM Journal on Scientific and Statistical Computing, vol. 4(3), pp. 553-572, 1983. g{Gz?N皙?皙?c Lt||||d|_d|_d|_||_||_||jn||_|jdkrtdtd|j f\|_ t|j |_t|j \|_|_t%|j t&j|_t%|j d|_|j|jz|jz|_dS)NrrzJmaxiter must not be set to a negative number, use np.inf to mean infinite.)potrffro)super__init__previous_tr_radius lambda_lbniterk_easyk_hardMAXITER_DEFAULTmaxiterrrrcholeskyrC dimensionr@hess_gershgorin_lbhess_gershgorin_ubrrinfhess_infhess_froEPS CLOSE_TO_ZERO) selfxrrrhessprRrSrU __class__s rrNzIterativeSubproblem.__init__s Cd+++#%   07t++G ?? ? **tylCC TY&7 &B&B $  #TY// TY.. "^dh6Frc td|j|z t|j |j|jz}tdt|j |j|z t|j|j|jz }||j krt|j |}|dkrd}n3ttj ||z||j ||z zz}|||fS)zGiven a trust radius, return a good initial guess for the damping factor, the lower bound and the upper bound. The values were chosen accordingly to the guidelines on section 7.3.8 (p. 192) from [1]_. r)r9jac_magr8rXr\r[rdiagonalrYrOrPrsqrt UPDATE_COEFF)r_ tr_radius lambda_ubrPlambda_initials r_initial_valuesz#IterativeSubproblem._initial_valuess4< 1C9P8P8< 8< 5G5GGHH C 2 2 4 4555 Y.T5L59]59]2D2DDEE t. . .DNI66I >>NN Y)>!?!?!*T-> )@S-T!TVVNy)33rc||\}}}|j}d}d}d|_|j|jkr|rd}n;|j|t j|zz}||ddd\} } |xjdz c_| dkr|j|j krt| df|j } t| } | |kr |dkrd}nt| | d} t| }| |z dz| |z z|z }||z}| |krNt| \}}|| ||\}}t!||gt" }t j| t j|| }|dz|dzz|||dzzzz }||jkr | ||zz } n'|}t)|||dzz }|j|t j|zz}||ddd\}} | dkr|}d}nt)||}t)t jt j||z||j||z zz}n`t#| |z |z }||jkrnO|}|}n8| dkr|j|j kr|dkrt j|} d}nt| \}}|}||z} |dz|dzz|j|z|dzzkrn|}t)|||dzz }t)t j||z||j||z zz}nt3|| | \}}t|}t)||||dzz z}t)t jt j||z||j||z zz}|j|jk||_||_||_| |fS) zSolve quadratic subproblemTFr)lower overwrite_acleanrr#)transrB)key)rkrWrQrUrreyerVrdr^rrrrr get_boundaries_intersectionsr8r$dotrSr9rfrgrRr r rPlambda_currentrO)r_rhrurPrir' hits_boundaryalready_factorizedr:r%infor(p_normr)r1 delta_lambda lambda_newr2r3tatbstep_lenquadratic_termrelative_errorcrEr/r0s rsolvezIterativeSubproblem.solve1s04/C/CI/N/N, 9 N " j4<''" 4%*""InRVAYY66--49.2(444 JJ!OJJqyyT\D,>>>q%j48)44aY&&>Q+>+>$)M%Q555a!'v 1VI5EFyP +l: I%%#CA#F#FLE5!>>q%?HJJFB #B8555H&(VArva||%<%$> *-GBF9y+@$A$ABB%(99Y;N(OO**&)));%<%E1H+DEE "%GI 122 19Y3F GG""6aDAAqa >E&!)O+KLL "%GBF9y#899:: 19Y3F GG""Oj4<''X#,"+-r)NrHrIN)__name__ __module__ __qualname____doc__rgrTrfinfofloatepsr]rNrkr __classcell__)rbs@rr r s,LO "(5// C0415/G/G/G/G/G/Gb444>Y Y Y Y Y Y Y rr )rNN)rnumpyr scipy.linalgrrrr _trustregionrr __all__r r r@r r rrrrs88%%%%%%%%%%%%KKKKKKKK " " " : : : :FLLL^*'''TL L L L L 1L L L L L r