doi ArSrSSKrSSKJrJrJrJr SSKJ r J r /SQr S Sjr Sr S rS r"S S \ 5rg)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 |Uc [S5e[U5(d [S5e[X4X#U[S.UD6$)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 z9Jacobian 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/land-ocr/venv/lib/python3.13/site-packages/scipy/optimize/_trustregion_exact.pyr r sY0 {/0 0 D>>/0 0 !# :D-@ :$8 ::c[R"U5nURupX:wa [S5e[R"U5n[R "U5n[ U5HnSX5- URXU4- nSX5- URXU4- nX5S-SURUS-S2U4U--nX5S-SURUS-S2U4U--n [U5[US5-[U5[U S5-:a XdU'XUS-S&MXtU'XUS-S&M [X5n [U 5n [U5n X- n X- nX4$)aGiven 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 0seD aA 77DAvIJJ  A  A1XfAD !gQT " stWqss1Q347|B & stWqss1Q347|B & r7T"a[ CGd2qk$9 9aDacdGaDacdG A !WF !WF OE JE <rc[R"U5n[R"U5n[R"[R"U5SS9n[R"X-U- 5n[R "X- U-5nXE4$)z 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?siWWQZFJq *J #j0 1B #j0 1B 6Mrc([R"USUS- 2US- 4S-5XS- US- 4- n[U5n[R"U5nSXRS- 'US:wa/[ USUS- 2SUS- 24USUS- 2US- 4*5USUS- &X54$)a Compute term that makes the leading ``k`` by ``k`` submatrix from ``A`` singular. Parameters ---------- A : ndarray Symmetric matrix that is not positive definite. U : ndarray Upper triangular matrix resulting of an incomplete Cholesky decomposition of matrix ``A``. k : int Positive integer such that the leading k by k submatrix from `A` is the first non-positive definite leading submatrix. Returns ------- delta : float Amount that should be added to the element (k, k) of the leading k by k submatrix of ``A`` to make it singular. v : ndarray A vector such that ``v.T B v = 0``. Where B is the matrix A after ``delta`` is added to its element (k, k). Nr)rr6lenrr)Ar$r)deltar&r.s rr r s6 FF1TacT1Q3Y<? #a!QqSk 1E AA  A AcF Av"1TacT4AaC4Z=1TacT1Q3Y<-@$1Q3 8Orc^\rSrSrSrSrSr\R"\ 5Rr S U4Sjjr Sr SrSrU=r$) 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?c \>[T U]XX45 SUlSUlSUlX`lXplUc UROUUlURS:a [S5e[SUR45uUl [UR5Ul[UR5uUlUl[%UR[&R(5Ul[%URS5UlURUR.-UR*-Ulg)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_DEFAULTmaxiterrrrcholeskyrB dimensionr?hess_gershgorin_lbhess_gershgorin_ubrrinfhess_infhess_froEPS CLOSE_TO_ZERO) selfxrrrhessprPrQrS __class__s rrLIterativeSubproblem.__init__s +#%   07t++G <? ? **tyylC TYY&7 &B $  #TYY/ TYY. "^^dhh6Frc ([SURU- [UR*URUR 5-5n[S[UR R55*URU- [URURUR 5- 5nXR:a[URU5nUS:XaSnO3[[R"X2-5X0RX#- --5nXCU4$)zGiven a trust radius, return a good initial guess for the damping factor, the lower bound and the upper bound. The values were chosen accordingly to the guidelines on section 7.3.8 (p. 192) from [1]_. r)r8jac_magr7rVrZrYrdiagonalrWrMrNrsqrt UPDATE_COEFF)r] tr_radius lambda_ubrNlambda_initials r_initial_values#IterativeSubproblem._initial_valuess4<< 1C9P9P8P8< 8< 5GGH C 2 2 455 Y.T5L5L59]]59]]2DDE .. .DNNI6I >N )>!?!*->-> @S-T!TVN)33rcURU5up#nURnSnSnSUlURUR:GaU(aSnO:URU[ R "U5--nURUSSSS9upU=RS- slW S:XGaURUR:Ga[W S4UR*5n [U 5n X::a US:XaSnGO[XSS9n [U 5nX- S-X- -U- nX/-nX:Ga?[U 5unnURU UU5unn[!UU/["S 9n[ R$"U [ R$"WU 55nUS-US--UX!S---- nUUR&::a U UU-- n GO)Un[)X2US-- 5nURU[ R "U5--nURUSSSS9unn U S:XaUnSnGO[)UU5n[)[ R*"[ R""X4-55X0R,XC- --5nGO][#X- 5U- nUUR.::aGOXUnUnGO5U S:XaURUR::aUS:Xa[ R0"U5n SnGO[W 5unnUnUU-n US-US--UR&U-US--::aOUn[)X2US-- 5n[)[ R*"X4-5X0R,XC- --5nOv[3WW U 5unn[U5n[)X2UUS-- -5n[)[ R*"[ R""X4-55X0R,XC- --5nURUR:aGMX0lX lXlW U4$) zSolve quadratic subproblemTFr)lower overwrite_acleanrr")transrA)key)rjrUrOrSrreyerTrcr\rrrrr get_boundaries_intersectionsr7r#dotrQr8rerfrPrr rNlambda_currentrM)r]rgrurNrhr& hits_boundaryalready_factorizedr9r$infor'p_normr(r0 delta_lambda lambda_newr1r2tatbstep_lenquadratic_termrelative_errorcrDr.r/s rsolveIterativeSubproblem.solve1s04/C/CI/N,9 NN " jj4<<'"%*"IInRVVAY66--49.2(4 JJ!OJqyT\\D,>,>>q%j488)4a&>Q+>$)M%Q5a!' 1V5EFyP +: %#CA#FLE5!>>q%?HJFB #B85H&(VVArvva|% *-GGBFF9+@$AB%(9(99;N(OO*&));%Y Y rr )NN)rnumpyr scipy.linalgrrrr _trustregionrr __all__r r r?r r rrrrsF8%%K " :FL^*'TL 1L r